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Full text of "THE MAN WHO COUNTED - ENGLISH - MALBA TAHAN"
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THE MAN WHO COUNTED 

A Collection of Mathematical Adventures 

MALE A TAHAN 

i 

A MEETING OF THE MINDS 



Of the amusing circumstances of my encounter with a strange traveler on the road 
from Samarra to Baghdad. 

In the name of Allah, the All Merciful! My name is <PRESIDIO_ANONYMIZED_PERSON>. Once I was 
returning, at my camel’s slow pace, along the road to Baghdad after an excursion to the 
famous city of Samarra, the ha nk s of the Tigris, when I saw a modestly dressed traveler 
who was seated on a rock, apparently resting from the fatigue of the journey. 

I was about to offer the perfunctory salaam of travelers when, to my great surprise, he 
rose and said ceremoniously, “One million, four hundred and twenty-three thousand, 
seven hundred and forty-five.” He quickly sat down and lapsed into silence, his head 
resting in his hands, as if he were absorbed in meditation. I stopped at some distance and 
stood watching him, as if he were a historic monument to the legendary past. 

A few moments later, the man again rose to his feet and. in a clear, deliberate voice, 
called out another, equally fabulous number, “Two million, thirty and twenty-one 
thousand, eight hundred and sixty-six.” 

Several times more the strange traveler rose and uttered a number in the millions, 
before sinking down again on the rough stone by the roadside. Unable to restrain my 


curiosity, I approached the stranger and, after greeting him in the name of Allah, asked 
him the meaning of these fantastic sums. 

“Stranger,” replied the Man Who Counted, “I do not disapprove of this curiosity that 
disturbs the peace of my thoughts and calculations. And now that you have spoken to me 
with such courtesy and graciousness, I am going to accede to your wishes. But first I 
must tell you the story of my life.” 

And he told me the following, which, for your entertainment, I transcribe exactly as I 
heard it. 


2 

SOMEONE TO COUNT ON 



In which Beremiz Samir, the Man Who Counted, tells the story of his life. How I 
learned of the brilliance of his calculations, and how we became traveling companions. 

My name is <PRESIDIO_ANONYMIZED_PERSON>. I was born in the little village of Khoi, in Persia, in the 
shadow of the huge pyramid of Mount Ararat. While still very young, I began work as a 
shepherd in the service of a rich gentleman from Khamat. 

“Every day, at first lights I took the vast flock of sheep to graze and was required to 
bring them back to their fold before nightfall. For fear of losing a stray lamb and being 
severely punished as a consequence, I counted them several times a day. 

“I became so good at counting that I could sometimes count the whole flock correctly 
at a glance. I went on to count, for practice, flight of birds in the sky. Little by little I 
began to develop a great skill in this art. After a few months — tha nk s to new and 
continuing practice counting ants and other insects — I performed the remarkable feat of 


counting all the bees in a swarm. This prodigious calculation, however, was as nothing 
compared with the many others I later achieved. My generous master owned, in two or 
three far-off oases, huge date plantations, and, informed of my mathematical agility, he 
charged me with overseeing the sale of his fruit, which I counted in clusters, one by one. I 
worked thus under the date palms, for almost ten yeans. Pleased with the profits I secured 
for him, my good master rewarded me with four months of rest, and I am now on my way 
to Baghdad to visit some of my family and to see the beautiful mosques and sumptuous 
palaces of the famous city. And, so as not to waste time, I have practiced throughout my 
journey counting the trees in this region, the flowers that perfume it, and the birds that fly 
among its clouds.” 

And, pointing to an old fig tree quite close, he went on, ‘That tree, for example, has 
two hundred and eighty-four branches. Given that each branch has, on the average, three 
hundred and forty-seven leaves; it is easy to conclude that that tree has a total of ninety- 
eight thousand, five hundred and forty-eight leaves. Well, my friend?” 

“Wonderful!” I cried in astonishment. “It is incredible that a man can count, at a 
glance, all the branches in a tree, all the flowers in a garden. That skill can bring immense 
riches to anyone.” 

“Do you think so?” exclaimed Beremiz, “It has never occurred to me that counting 
millions of leaves and swarms of bees could make money. Who could possibly be 
interested in how many branches there are in a tree, how many birds in a flight that 
crosses the sky?” 

“Your wondrous skill,” I explained, “could be used in twenty thousand different ways. 
In a great capital like Constantinople, or even in Baghdad, you would be of invaluable 
help to the government. You could count populations, annies, and flocks. It would be 
easy for you to sum up the resources of the country, the value of its harvest, its taxes, its 
commodities, and all the wealth of the state. Through my connections — for I am from 
Baghdad — I assure you that it will not be difficult to find some distinguished post in the 
service of Caliph al-Mutasim, our ford and master. Perhaps you might become treasurer, 
or fulfill the function of secretary to the Muslim household.” 

“If that is truly so, then my mind is made up,” replied the counting man. “I am going 
to Baghdad.” 

And without more ado, he mounted behind me on my camel — the only one we had — 
and we set out on the long road to the splendid city. From that point on, united by that 
casual meeting on a country road, we became friends and inseparable companions. 

Beremiz was a man of happy and talkative disposition. Still young (he was not yet 
twenty-six), he was blessed with a most lively intelligence and a remarkable aptitude for 
the-science of numbers. From the most trivial of happenings, he would make unlikely 
analogies that demonstrated his mathematical acuity. He also knew how to tell stories and 
anecdotes that illustrated his conversation, already odd and attractive in itself. 

At times, he would not speak for several hours, wrapped in an impenetrable silence, 
pondering prodigious calculations. On those occasions, I took pains not to disturb him. I 
left him in peace, to make, in his exceptional mind, fascinating discoveries in the arcane 
mysteries of mathematics, the science that the Arab race so developed and extended. 



3 

BEASTS OF BURDEN 



Of the singular episode of the thirty-five camels that were to be divided between three 
Arab brothers. How Beremiz Samir, the Man Who Counted, made an apparently 
impossible division that left the quarreling brothers completely satisfied. The unexpected 
profit that the transaction brought us. 

We had been traveling for a few hours without stopping when there occurred an 
episode worth retelling, wherein my companion Beremiz put to use his talents as an 
esteemed cultivator of algebra. 

Close to an old half abandoned inn, we saw three men arguing heatedly beside herd of 
camel. Amid the shouts and insults the men gestured wildly in fierce debate and we could 
hear their angry cries: 

“It cannot be!” 

“That is robbery!” 

“But I do not agree!” 

The intelligent Beremiz asked them why they were quarreling. 

“We are brothers,” the oldest explained, “And we received thirty-five camels as our 
inheritance. According to the express wishes of my father half of them belong to me, one- 
third to my brother Hamed, and one-ninth to Harim, the youngest. Nevertheless we do 
not know how to make the division, and whatever one of us suggests the other two 
disputes. Of the solutions tried so far, none have been acceptable. If half of 35 is 17.5 if 
neither one-third nor one-ninth of this amount is a precise-number, then how can we 
make the division?” 


“Very simple,” said the Mar, Who Counted. “I promise to make the division fairly, but 
let me add to the inheritance of 35 camels this splendid beast that brought us here at such 
an opportune moment.” 

At this point I intervened. 

“But I cannot permit such madness. How are we going to continue on our journey if 
we are left without a camel?” 

“Do not worry, my Baghdad friend,” Beremiz, said in a whisper. “I know exactly what 
I am doing. Give me your camel, and you will see what results.” 

And such was the tone of confidence in his voice that, without the slightest hesitation, 
I gave over my beautiful Jamal, which was then added to the number that had to be 
divided between the three brothers. 

“My friends,” he said, “I am going to make a fair and accurate division of the camels 
as you can see, now number 36.” 

Turning to the eldest of the brothers, he spoke thus: “You would have half of 35 — that 
is 17.5. Now you will receive half of 36 — that is 18. You have nothing to complain about 
because you gain by this division.” 

Turning to the second heir, he continued, “And you, Hamed, you would have received 
one-third of 35 — that is, 11 and some. Now you will receive one-third of 36 that is 12. 
You cannot protest as you too gain by this division. 

Finally he spoke to the youngest, “And you young Harim Namir, according to your 
father’s last wishes you were to receive one-ninth of 35 or three camels and part of 
another. Nevertheless, I will give you one-ninth of 36, or 4. You have benefited 
substantially and should be grateful to me for it.” 

And he concluded with the greatest confidence, “By this advantageous division, which 
has benefited everyone, 18 camels belong to the oldest, 12 to the next, and 4 to the 
youngest, which comes out to — 18 + 12 + 4 = 34 camels. Of the 36 camels, therefore, 
there are 2 extra. One, as you know, belongs to my friend from Baghdad. The other 
rightly belongs to me for having resolved the complicated problem of the inheritance to 
everyone’s satisfaction.” 

“Stranger, you are a most intelligent man,” exclaimed the oldest of the three brothers, 
“and we accept your solution with the confidence that it was achieved with justice and 
equity.” 

The clever Beremiz the Man Who Counted, took possession of one of the finest 
animals in the herd and, handing me the reins of my own animal, said, “Now, dear friend, 
you can continue the journey on your camel, comfortable and content. I have one of my 
own to carry me.” 

And we traveled on towards Baghdad. 


4 

FOOD FOR THOUGHT 



Of our coming across a rich sheik, wounded and hungry. The offer he made to us for our 
eight loaves of bread, and how the division of the eight coins we received in payment was 
resolved in a surprising manner. Beremiz’s three type of division: simple division, exact 
division and perfect division. Praise for the Man Who Counted - from an illustrious 
Vizier. 

Three days later, we were approaching the ruins of a small village called Sippar when 
we found sprawled on the ground a poor traveler, his clothes in rags and he apparently 
badly hurt. His condition was pitiful. We went to the aid of the unfortunate man, and he 
later told us the story of his misfortune. 

His name was Salem Nasair and he was one of the richest merchants in Baghdad. On 
the way back from Basra a few days before bound for el-Hillah, his large caravan had 
been attacked and looted by a band of Persian desert nomads, and almost everyone had 
perished at their hands. He, the head, managed to escape miraculously hiding in the sand 
among the bodies of his slaves. 

When he had finished his tale of woe, he asked us in a trembling voice, “Do you by 
some chance have anything to eat? I am dying of hunger.” 

“I have three loaves of bread.” I answered. 

“I have five,” said the Man Who Counted. 

“Very well,” answered the sheik. “I beg you to share those loaves with me. Let me 
make an equitable arrangement. I promise to pay for the bread with eight pieces of gold, 
when I get to Baghdad.” 

So we did. 

The following day, in late afternoon, we entered the famous city of Baghdad, Pearl of 
the East. Crossing the bustling square, we were held up by a resplendent entourage, at 
whose head, on an elegant sorrel horse, rode the powerful Ibrahim Maluf, one of the 
viziers. Seeing Sheik Salem Nasair in our company, he brought his dazzling retinue to a 
halt and called out to him, “What happened to you, my friend? How is it that you arrive 
in Baghdad in rags, in the company of these two strangers?” 

The poor sheik told him in detail everything that had happened on his journey, 
praising us effusively. 


“Pay these two strangers at once,” ordered the vizier. Taking from his purse eight gold 
coins, he gave them to Salem Nasair, saving. “I will take you to the palace at once, since 
the defender of the faithful unquestionably want to be infonned of this new affront by 
Bandit and Bedouins, attacking our friends and sacking one of our caravans in the 
caliph’s territories.” 

Then Salem Nazair said to us, “I take leave of you my friends. I wish however to 
thank you once more for your help and, as promised, to repay your generosity.” Turning 
to the Man Who Counted, he said, “Here are rive gold pieces for your. 

To my great surprise, the Man Who Counted made a respectful objection. “Forgive 
me, O Sheik! Such a division, although apparently simple, is not mathematically correct. 
Since I gave five loaves, I should receive seven coins. My friend, who supplied three 
loaves, should receive only one.” 

“In the name of Muhammad!” exclaimed the vizier, showing a lively interest. “How 
can this stranger justify such an absurd division?” 

The Man Who Counted approached the minister and spoke as follows: 

“Let me show you, O Vizier, that my proposal is mathematically correct. During the 
journey, when we were hungry, I took out a loaf and divided it into three pieces. We each 
ate one piece. My five loaves, then, yielded up fifteen pieces, correct? My friend’s three 
loaves added nine pieces, making a total of twenty-four pieces. Of my fifteen pieces, I 
consumed eight, so that actually I contributed seven. Of my friend’s nine pieces, he also 
consumed eight, so contributing one. My seven nieces and my friend’s one made up the 
eight that went to Sheik Salem Nasair. So it is right that I receive seven coins and my 
friend only one.” 

The grand vizier, after praising the Man Who Counted extravagantly, ordered that he 
be given seven coins and I one. “This division, seven for me and one for my friend, is, as 
I just proved, mathematically perfect; but is not perfect in the eyes of the Almighty.” 

And gathering the coins again, he divided them equally, giving me four of them and 
keeping four. 

“This is an extraordinary man,” declared the vizier. “He did not accept the proposed 
division of the eight coins into five and three. He then proved that he had a right to seven 
and his companion to one. But then he divides the coins into two equal parts and gives 
one of them to his friend.” 

He added with enthusiasm, “By the Almighty! This young man, besides being wise 
and quick in the ways of arithmetic, is a fine and generous friend. He shall be my 
Secretary this very day.” 

“Great Vizier,” said the Man Who Counted. “I notice that you have just expressed, in 
thirty words and 125 letters, the highest praise I have ever heard. May Allah eternally 
bless and protect you!” 

My friend Beremiz’s skill went as far as the words and letters being used. We all of us 
marveled at that display of enviable genius. 



5 

IN SO MANY WORDS 



Of the prodigious calculations Beremiz Samir performed, as we made our way to the 
Golden Goose Inn, in order to determine the exact number of words, spoken in the course 
of our journey and the average number of words per minute. How the Man Who Counted 
resolves a problem. 

After leaving Sheik Nasair and Vizier Maluf, we went along to a small inn called the 
Golden Goose, in the neighborhood of the Mosque of Suleiman. There we sold our 
camels to a camel driver of long acquaintance who lived nearby. 

On the way, I said to Beremiz, “Now you see my friend, that I was right when I said 
that a master of calculation with your talents could easily find a good job in Baghdad. As 
soon as you arrived, they asked you to accept the post of secretary to a vizier. Now you 
do not have to return to the sad, stony village of Khoi.” 

“Although I may prosper and become rich here,” the counting man answered, “I want 
to return to Persia someday in order to see my native land again. Ungrateful is he who 
would forget his country and childhood friends when he finds happiness in an oasis of 
prosperity and good fortune.” 

And, taking my arm, he added, “We have traveled together for eight days exactly. 
During that time, clarifying points and mulling over things that interest me, I have spoken 
exactly 414,720 words. Since in eight days there are 1 1,520 minutes, one can deduce that 
during the journey I uttered an average of 36 words per minute — that is, 2,160 per hour. 
These numbers show that I spoke little, that I was discreet and did not waste your time 
with pointless discourses. A man of few words, excessively silent, becomes a 
disagreeable creature; but those who talk endlessly bore and annoy their companions. 
Therefore, we should avoid fruitless chatter, without becoming terse and, therefore, 
ungracious. To that end, I will tell you a very curious case.” 


After a brief pause, the counting man recounted the following: “In Teheran, in Persia, 
there was an old merchant who had three sons. One day the merchant called his children 
together and said, ‘He who can go a day without speaking one pointless word, I will 
reward with a prize of twenty-three gold coins.’ 

“At nightfall three sons presented themselves before their ancient parent. Said the first, 
“Today, my father, I avoided all pointless words. I hope, therefore, that I have earned the 
promised reward. The prize as you will recall, amounted to twenty-three pieces. 

“The second approached the old man, kissed his hands and said no more than, “Good 
evening father.” 

The youngest son said not a word. He approached his father and extended his open 
hand to receive the prize. The merchant, after observing the behavior of his three sons 
said. “The oldest, on coming before me, lost my attention with various pointless words 
the youngest showed himself too terse. The prize then, to my second son, who was 
circumspect but not verbose, simple, without pretension.” 

When he finished, Beremiz asked me, “Do you not think that the old man judged his 
three sons fairly?” 

I did not answer. I thought it better not to discuss the twenty-three coins with that 
amazing man who was always reducing everything to numbers, calculating averages, and 
solving problems. 

A few moments later we arrived at the Golden Gods? The innkeeper was named Salim 
and had once worked for my father. When he saw me, he grinned and cried. “Allah be 
with you, little one. I await your wishes now and forever.” 

I told him that I needed a room for myself and for my friend Beremiz Samir, the 
calculator, secretary to the vizier Maluf. 

“This man is a mathematician?” asked old Salim. “Then he has come at just the right 
moment to help me out of a difficult spot. I have just had a dispute with a jeweler. We 
argued for a long time and yet we still have a problem that we can’t resolve. 

Having been informed that a great mathematician had arrived at the inn several 
curious people drew around. The jeweler was summoned and he declared extremely 
interested in a solution to the problem. 

“What is the cause of the argument?” Beremiz asked. 

“This man,” old Salim said pointing to the jeweler “came from Syria to sell precious 
stones in Baghdad. He promised he would pay 20 dinars for his lodgings if he sold all of 
his jewels for 100 dinars and 35 dinars if he sold them for 200. After several days of 
wandering about, he ended up selling all of them for 140 dinars. How much does he owe 
me according to our agreement?” 

“24-1/2 dinars! It stands to reason!” answered the Syrian. “If after selling the jewels 
for 200 dinars, I owed you 35, then, logically, if I had sold them for 20 dinars — ten times 
less — I would owe only 3-1/2 dinars. But, as you well know, I sold them for 140 dinars, 
and 20 goes into 140 7 times, if my calculation is correct. Therefore, if I had sold the 
jewels for 20 and would have owed you 3-1/2, having then sold them for 140, 1 now owe 
you an amount equal to 7 times 3-1/2 dinars, or 24-1/2 dinars. 



PROPORTION SUGGESTED BY THE JEWELER 

200 : 35 :: 140 : x 
x = (35x140)/ 200 =24-1/2 


“You are wrong,” old Salim replied in irritation. “According to my count, it if is 28. 
Listen! If for a sale price of 100 I was to receive 20, then for 140 I should get 28. Its 
obvious! I will show you.” 

Old Salim reasoned this way: “If for 100 dinars, I was to receive 20, then for 10 - 
which is l/10 th of 100 - I would receive 1/1 0 th of 20, that is, 2. 

Therefore, for 10 I would receive 2. How many times does 10 go into 140? Fourteen. 
Therefore, for a sale of 140 I should receive 14 times 2, which is 28, as I already said 


Old Salim, after all these calculations, exclaimed energetically, “I should receive 28! 
That is the correct amount!” 

“Calm down, my friends,” broke in the counting man. “We should settle matters 
quietly, with some decorum. Haste leads to anger and mistakes. The solutions you 
suggest are wrong, as I will show.” 

“According to the agreement that the two of you made, you,” he said, speaking to the 
Syrian, “were to pay 20 dinars for your lodgings if you had sold the jewels for 100 dinars, 
but 35 it you had sold them for 200 dinars. Therefore, we have the following: 


“You notice that a difference of 100 dinars in the sale price corresponds to a difference of 
15 in the price of lodgings. Is that clear?” 

“As clear as camel’s milk,” agreed the two men. 

“Then.” the mathematician went on, “if an increase of 100 in the sale price in lodgings 
if the price increased by only 40?’ If the difference were 20, which is one-fifth of 100, the 
increase in lodgings would be 3, since 3 is one-fifth of 15. For a difference of 40, which 
is double 20, the increase in lodgings would be 6. Therefore, after selling the jewels for 
140, the bill would be 26 dinars. 


PROPORTION SUGGESTED BY OLD SALIM 

100 : 20 :: 140 : x 
x = (20x140)/ 100 =28 


Sale Price 
200 
- 100 
100 


Price of Lodgings 


35 

-20 

15 


PROPORTION SUGGESTED BY BEREMIZ 

100: 15:: 40: x 
x = (15x40)/ 100 = 6 

“My friends, numbers, in their utter simplicity, dazzle even the wisest of men.” Even 
divisions that seem to us perfect are, at times, beset with error. From the uncertainty of 
calculations comes the undeniable prestige of the mathematician. According to the terms 
of the agreement, the man will have to pay 26 dinars and not 24-1/2, as he at first thought 
“Even in. the final solution to the problem, there was a small difference that should not be 
taken lightly, the size of which I cannot express in numbers alone." 

“The gentleman is right,” agreed the jeweler. “I see that my calculation was wrong.” 

And, without hesitating he took 26 dinars from his bag and gave them to old Salim. At 
the same time offering the clever Beremiz a beautiful gold ring with dark stones, along 
with expressions of affection. All those gathered in the inn admired the wisdom of the 
counting man. 


6 

TRIAL BY NUMBERS 



What happened during our visit to the Vizier Maluf. Our meeting with the poet who 
did not believe in the wonders of calculation. The Man Who Counted demonstrates an 
original way of counting the camels in an extensive caravan. The age of the betrothed 
and a camel’s ear. Beremiz discovers “quadratic friendship’’ and speaks of King 
Solomon. 

After the second prayers, we left the Golden Goose and hurried to the house of Vizier 
Ibrahim Maluf, the king’s minister. Entering his property, I was truly amazed. 


We passed through the heavy iron gate and down a narrow corridor, led by a huge 
black slave with gold armbands, into the splendid inner garden of the palace. This garden 
laid out exquisite taste, was shaded by two rows of orange trees. 

Various doors opened on to it, some of which must have led to the harem. At the sight 
of the two infidel slaves who were picking flowers fled among the flower beds and 
disappeared behind the columns. From the elegant garden, through a narrow door in a 
high wall, we crossed to a patio in its center, splendidly tiled. In the middle were three 
water jets three liquid curves sparkling in the sun. 

Still following the slave with gold armbands we crossed the patio and went into the 
palace itself. Passing through a variety of richly furnished rooms, hung with tapestries 
fringed in silver thread. We at last reached the rooms of the king’s minister, who was 
reclining on large cushions and conversing with two friends. 

One of them I recognized as Sheik Salem Nasair, our companion in the desert 
adventure. The other was a small, round-faced man, with a kindly expression, and a trace 
of gray in his beard. Finely dressed, he wore a rectangular medallion one-half of which 
was golden yellow and the other half dark, like bronze. 

Vizier Maluf received us with obvious goodwill and, turning to the man with the 
medal, said with a smile, “My dear poet, here is our great calculator. The young man with 
him is a citizen of Baghdad whom he met by chance while wandering in the ways of 
Allah.” 

We both gave a deep salaam to the noble sheik. We learned later that his companion 
was the famous poet Abdul Hajmid, close friend of Caliph al-Mutasim. The curious 
medal he had received from the hands of the caliph, as a prize for writing a poem of 
30,200 verses without once using the letters kaf, lam, or ayn. 

“I have some trouble believing, friend Maluf, in the prodigious feats of this Persian 
calculator.” laughed the poet, “when numbers are combined, certain trickery, and 
algebraic subtlety, comes into play. Once there appeared before King el-Harit, the son of 
Modad, a wise man who declared he could read destinies in the sand. “Do you make 
exact calculations?” the king asked him and before the magus could recover from his 
surprise he added. If you do not know how to calculate exactly, your visions are worth 
nothing; if you arrive at them through calculation alone, I disbelieve them. In India I 
learned a proverb that says, ‘Distrust the calculation seven times and test them a hundred 
times.” 

“To quiet this distrust,” suggested the vizier, “let us put our guest to a decisive test.” 

So saying, he rose from his cushions and, taking Beremiz lightly by the arm, led him 
to one of the balconies of the palace. He opened the casement onto yet another patio, 
which at that moment was full of camels, fine animals, almost all of them of a good 
breed. I noticed two or three white ones, from Mongolia, and a few clear-skinned carehs. 

“Here is a fine collection of camels I bought yesterday,” said the vizier. “I want to 
send them as a gift to the father of my betrothed. I know exactly how many there are. Can 
you tell me?” 



And the vizier, to make the test more interesting, whispered the total to his friend. I 
was taken aback. There were a great many camels, constantly milling about. If my friend 
were to make a mistake, our visit would be a woeful disaster. 

But after running his eyes over the restless herd Beremiz said. “According to my 
calculations, O Vizier, there are 257 camels in this patio.” 

“Exactly! Quite correct!” the vizier confirmed. “Two hundred and fifty-seven camels 
by Allah.” 

“How did you manage to count them so swiftly and so exactly?” asked the poet, 
beside himself with curiosity. 

“Simple,” explained Beremiz. “Counting the camels one by one would be to me an 
uninteresting chore. So I went about it in the following way: I counted first the hooves 
and then the ears. They came to a total of 1,541. To this I added 1 and then divided the 
result by six. The exact quotient is 257.” 

“Great heavens!” exclaimed the vizier, in delight. “How original! Who could have 
imagined that he would count the ears and hooves just for interest’s sake?” 

“I should say, Vizier,” Beremiz added, “that calculations are sometimes made difficult 
by carelessness or inability on the calculator’s part. Once, in Khoi, when I was watching 
my master’s flock, a cloud of butterflies passed. Another shepherd asked whether I could 
count them. ‘Eight hundred and fifty-six,’ I answered. ‘Eight hundred and fifty-six?’ he 
exclaimed, as if he found the total exaggerated. Only then did I realize that I had counted 
not the butterflies but their wings. I divided by two and got the correct total.” 

At this the vizier laughed aloud, a sound to my ears like sweet music. 

“There’s one point in all this that is beyond me,” said the poet, very seriously. “I 
understand that dividing by 6 - 4 legs and 2 ears — would yield the total number of 
camels. But I do not understand why he added 1 before dividing his total of 1,541.” 

“Quite simple,” replied Beremiz. “In counting the ears, I noticed that one of the 
camels had a slight defect, it lacked an ear so, to round out the total, I had to add.” 

Then, turning to the vizier, he asked. “Would it be indiscreet or imprudent to enquire 
O Vizier, how old your betrothed might be?” 

“No at all,” replied the minister smiling. “Astir is 16.” And he added, with a trace of 
suspicion, “But I do not see any relation between her age and the camel. I am about to 
present to my future father-in-law.” 

“I wanted to make a small suggestion,” replied Beremiz. “If you were to remove from 
the herd the defective animal, the total would be 256 which is the square of 16 (16 x 16). 
The gift offered to the father of the delightful Astir would then have a mathematical 
perfection, the number of camels equal to the square of the age of the beloved. The 
number 256 is an exact power of 2 — number held to be symbolic by the ancients — 
while 257 is a prime number. These relations between squared numbers are good 
auguries for lovers. There is an interesting legend about squared numbers. Would you 
“like to hear it?” 

“With pleasure,” replied the vizier. “Good stories well told are a pleasure to listen to, 
and I am always eager to hear them.” 



Flattered, the Man Who Counted inclined his head graciously and began, “It is told of 
King Solomon that, as proof of his courtesy and his wisdom, he gave to his betrothed, the 
beautiful Belkis, queen of Sheba, a box containing 529 pearls. Why 529? Because 529 is 
the square of 23 — that is 23 times 23 gives 529 — and 23 was the queen’s age. In young 
Astirs case, 256 has the advantage over 529.” 

Everyone looked at the Man Who Counted in some astonishment. Serenely, he went 
on, “The digits of the number - 256 add up to 13. The square of 13 is 169. The digits of 
169 add up to 16. As a result, the numbers 13 and 16 have a curious relation, which we 
could call a quadratic friendship. If numbers could speak, we might overhear the 
following dialogue: Sixteen says to Thirteen, ‘I wish to offer you an homage to our 
friendship. My square is 256, and the sum of its digits is 16.’ I think that I have amply 
justified the preference we should grant to the number 256, which is much more singular 
than 257.” 

“Your idea is a rare one,” replied the vizier. “I shall do it, although I could be accused 
of plagiarizing the great Solomon.” And, turning to the poet, he said, “I see that this 
man’s intelligence is no less than his skill at finding analogies and inventing stories. I did 
well when decided to make him my secretary.” 

“I regret to have to infonn you, illustrious lord,” replied Beremiz, “that I can accept 
your noble offer only if there is also a place for my friend Hanak Tade Maia, who is now 
without work or funds.” 

I was astonished and delighted by the graciousness of the Man Who Counted, who 
sought for me in this way the worthy protection of the powerful vizier. 

“Your request is a fair one,” replied the vizier. “Your friend may undertake the work 
of scribe, with its corresponding salary.” 

I accepted promptly and then expressed my gratitude to the vizier and to the good 


7 

GOING TO MARKET 



Of our visit to the marketplace. Beremiz and the blue turban. The case of “the four 
fours. ” The problem of the fifty dinars. Beremiz solves the problem and receive a very 
beautiful gift. 

Several days later, having finished our day’s work in the vizier’s palace, we went for a 
stroll in the marketplace and gardens of Baghdad. There was an unusually intense activity 
that afternoon because, only a few hours earlier, the rich caravans had arrived from 
Damascus. “The arrival of the caravans was always a great event. It offered the only way 
to find out what-other countries were producing and to mingle with the foreign 
tradesman. The city was unusually active, bustling with life. It was impossible to enter 
into the bazaar, for example there were sacks and boxes of new merchandise in all the 
patios and storerooms. Foreigners from Damascus, wearing huge and multicolored 
turbans and sporting weapons at their belts strolled casually about the market, gazing 
indifferently at the stalls. There was a pronounced odor or incense, kit and spice. The 
vegetable merchants were quarreling, nearly coming to blows and hurling insults at one 
another. 

A young guitarist from Mosul, sitting on some sacks, was singing a sad, droning tune. 

What matters the life of a man 
If people, for better or worse, 

Live as simply as they can. 

Here ends my verse. 

The shopkeepers, in the doors of their shops, were hawking their wares, making 
exaggerated claims with the fertile imagination of Arabs. 

“This cloth! Look at it. Worthy of an emir!” 

“Friends! Here is a delicious perfume that recalls the love of your wife.” 

“Look, O master, at these slippers and this beautiful caftan that the genies 
recommend to the angels!” 

Beremiz was interested in an elegant, bright blue turban that a hunchbacked Syrian 
was selling for four dinars. This merchant’s shop was quite unusual as well, for 
everything in it — turbans, boxes, dagger, bracelets, and so on - cost only four dinars. 
There was a sign that said in bright letters: 


The Four Fours 

Seeing that Beremiz was interested in buying the turban. I said, “Such an extravagance 
seems mad to me. We have only a little money, and we have not yet paid for our 
lodgings.” 

“It is not the turban that interests me,” replied Beremiz. “Did you notice that this shop 
is called The Four Fours. This is a coincidence of unusual importance.” 

“A coincidence? Why?” 

“The name of this business recalls one of the wonders of calculus: using four fours, we 
can get any number whatsoever.” 


And before I could ask him about this mystery, Beremiz explained by writing in the 
fine sand that was scattered on the floor. 

“Do you want to get zero? Nothing simpler. Just write: 

44-44 

Here you have four fours in an equation that equals zero. “What about the number 1? 
Here is the easiest way: 

44/44 

This fraction represents the quotient of 44 divided by 44, or 1 . 

“Would you like to see the number 2? We can easily use the four fours and write: 

4/4 + 4/4 

The sum of the two fractions is exactly 2. Three is even easier. Just write the 
expression 

(4 + 4 + 4)/ 4 

Notice that the dividend adds up to 12, which when divided by 4 yields 3. So 3, too, 
can be made from lour fours.” 

“And how are you going to get the number 4?” I asked him. 

“Nothing could be easier,” Beremiz explained. “We can get 4 in several different 
ways. Here I have one expression equal to 4: 

4 + (4 -4)/ 4 

You can see that the right-hand part equals nothing, and that the total is 4. The 
expression is equivalent to 4 + 0, that is, 4.” 

I saw that the Syrian merchant was listening carefully to Beremiz’s explanation, as if 
the combinations of four fours fascinated him. 

Beremiz continued, “If I want to get the number 5 — no problem. We write: 

[(4x4) + 4] / 4 

This fraction shows 20/4 and the quotient is 5. Here we have 5 written out with four 
fours. 

“Now we go on to six, and here is a most elegant fonnula: 

[(4 + 4)/ 4] +4 

“A slight alteration in that fonnula gives us 7: 

44/4 - 4 

“To get 8 by means of four fours is easy: 

4 + 4 + 4-4 

“And the number 9 is also interesting: 

4 + 4 + 4/4 

“And now I will show you a beautiful expression, 

(44 -4)/ 4 



equaling 10, made with four fours.” 


At that point, the hunchbacked shopkeeper, who had been following Beremiz’s 
explanations in respectful silence, interrupted. “From what I have just heard, the 
gentleman must be a distinguished mathematicians If he can solve a certain riddle that I 
came across in a mathematical problem two years ago, 1 will give him the blue turban 
that he wanted to buy.” 

And the shopkeeper told the following “Once I lent 100 dinars, 50 to a Sheikh from 
Medina and another 50 to a merchant from Cairo. 


“The sheik paid the debt 
5 that is 

in tour installments, 

in the following amounts: 20, 15, 10 and 

Paid 20 

and still owed 

30 

Paid 15 

and still owed 

15 

Paid 10 

and still owed 

5 

Paid 5 

and still owed 

0 

Total 50 

Total 

50 

“Notice, my friend, that the total of the payments and the total of his debt balance 
were both 50.” 

“The merchant from Cairo also paid the debt of 50 dinars in four installments, in the 
following amounts: 

Paid 20 

and still owed 

30 

Paid 18 

and still owed 

12 

Paid 3 

and still owed 

9 

Paid 9 

and still owed 

0 


Total 50 


Total 51 


“Note that the first total is 50 — as in the previous case — while the other total is 51. 
Apparently this should not have occurred. I do not know how to explain the difference of 
1 in the second manner of repayment; I know that I was not cheated, as I was paid all of 
the debt, but how to explain the difference between the total of 5 1 in the second case and 
50 in the first?” 

“My friend,” began Beremiz, “I can explain this in a few words. The debt balance has 
nothing to do with the total debt. Let us agree that the debt of 50 was paid in three 
installments - the first 10, the second 5, and the third. 35. The bill with the balance would 
be 

Paid 10 and still owed 40 

Paid 5 and still owed 35 



Paid 35 


and still owed 


0 


Total 50 Total 75 


In this example, the first total is still 50, while the total of the debt balance is, as you 
can see, 75; it could be 80, 99, 100, 260, 800, or any number at all. Only by chance would 
the total be exactly 50, as in the case of the sheik, or 5 1, as in the case of the merchant.” 

The shopkeeper was satisfied now that he understood Beremiz’s explanation, and he 
kept his promise by giving the Man Who Counted the blue turban worth 4 dinars. 

8 

SEVENTH HEAVEN 



Beremiz holds forth on the forms of geometry. Our happy encounter with Sheik Salem 
Nasair and his friends the sheep rearers. Beremiz solves the problem of the twenty-one 
wine casks. The explanation of the disappearing dinar. 

Beremiz was very satisfied at receiving the tine present from the Syrian merchant. “It 
is very well, made,” he said, turning the turban over and looking at it from all sides, “it 
has however one defect, which could easily he avoided: its fonn is not strictly 
geometric.” 

I looked at him unable to conceal my surprise. That original man had a way of 
transfonning the most ordinary things, to the point of considering even turbans in the 
light of geometric form. 

“It should not be surprising to you my friend,” said the wise Persian, “that I should 
wish for turbans of geometric form. Geometry is everywhere. Consider the ordinary and 
perfect forms of many bodies. Flowers, leaves, and innumerable animals reveal admirable 
symmetry that lightens the spirit of Geometry. I repeat exists everywhere: in the sun’s 


disk, in leaves, in the rainbow, in butterflies, in diamonds, in starfish, in the tiniest grain 
of sand. There is an infinite variety of geometric forms throughout nature. A crow flying 
slowly through the air traces wondrous figures with its sooty body. The blood circulating 
in the veins of a camel also obeys strict geometric principles; its humps, unique among 
mammals, show a singular elliptical form; the stone thrown at an intruding jackal 
describes a perfect curve in the air, known as a parabola: the bee makes its cells in the 
form of hexagonal prisms and uses that geometric form to build its house with the 
greatest possible economy of material. 

“Geometry exists everywhere. It is necessary, however, to have eyes to see it, 
intelligence to understand it, and spirit to wonder at it. The rude Bedouin sees geometric 
forms but does not understand them; the Sunni understands them but does not admire 
them; the artist, finally, perceives the perfection of figures, understands beauty, and 
admires order and harmony. God was the Great Geometer. He geometrized heaven and 
hearth. In Persia there exists one plant, much sought as food by camels and sheep, whose 
seed . . .” 

And so, holding forth enthusiastically on the multiple beauties embraced by geometry, 
Beremiz walked along the long and dusty road from the palace of the merchant to the 
bridge of victory. I accompanied him in silence, enchanted by his curious enlightenment. 

After crossing Muazin Square, which is also known as the shelter of the camel drivers, 
we deserted the beautiful Inn of the Seven Sorrows, much frequented in the hot weather, 
by Bedouins and travelers from Damascus and Mosul. Its most elegant feature was its 
inner patio, with good shade in summer, its four walls covered with plants of all colors 
from the mountains of Libya it had an air of peace and repose. 

On an old wooden sign, beside which the Bedouins tied their camels, we read: 

Inn of the Seven Sorrows. 

“Seven sorrows,” murmured Beremiz. “Strange! Do you by any chance know the 
owner of this inn?” 

“I know him well,” I replied. “He is a former rope merchant from Tripoli whose father 
served under Sultan Quervan. They call him the Tripolitan. He is very well thought of, 
for his simple, open nature, a fine and kindly man. They say that he went to the Sudan, in 
a caravan with soldiers of fortune, and brought back from African territory five black 
slaves who serve him with incredible fidelity. On his return, he gave up his rope business 
and opened his hostelry, with the help of the five slaves.” 

“With or without slaves,” replied Beremiz, “this man, this Tripolitan, must be 
extremely original. He made the number 7 part of the inn’s name, and 7 has always been, 
for Mohammedans, Christians, Jews, idolaters, or pagans, a sacred number — the sum of 
3, which is divine, and 4, which symbolizes the material world. That relation makes for 
strange connections among entities that total seven: 

Seven are the gates of hell 
Seven are the days of the week 
Seven wise men of Greece 
Seven the seas that cover the earth 



Seven the planets, and seven 
The wonders of the world 


He was continuing his strange and eloquent observations on the sacred number when 
we saw at the door of the inn our good friend Sheik Salem Nasair waving to us to 
approach. 

“How happy I am to have found you at this moment, O counting man,” said the sheik, 
smiling, as we drew near. “Your arrival is providential not just for me but also for these 
three friends who are here in the inn.” He added, sympathetically but with keen interest, 
“Come in, come in! This is a most difficult matter.” 

He led us down a shadowy, damp corridor to the welcoming “Brightness of the inner 
patio, which had five or six round tables. At one of them sat three travelers. 

When the sheik and the counting man approached, they raised their heads and made a 
salaam. One of them, who seemed very young, was tall, slender and clear-eyed and wore 
a bright happy turban with a white band, in which sparkled a quite beautiful emerald. The 
two others were stocky and broad shouldered, with the dark skin of African Bedouins. 
Their clothes and appearance set them apart. They were deep in a discussion that, judging 
by their gestures, was perplexing them, as happens when a solution is difficult to find. 

The sheik addressed the three of them: “Here is the esteemed master calculator.” And. 
to Beremiz he added, “Here are my three friends. They are sheep rearers from Damascus. 
They are facing one of the strangest problems I have come across. It is this as payment 
for a small flock of sheep they received here in Baghdad, a quantity of excellent wine, in 
21 identical casks: 

7 full 
7 half-full 
7 empty 

They want to divide so that each receives the same number of casks and the same 
quantity of wine. Dividing up the casks is easy — each would receive 7. The difficulty, as 
I understand it is in dividing the wine without opening them, leaving them just as they 
are. Now, calculator, is it possible to find a satisfactory answer to this problem?” 

Beremiz, after pondering for two or three minutes, replied, “The division of the 21 
casks, O Sheik, can be done without much complication. I am going to suggest the 
simplest possible solution. The first will receive 

3 full casks 
1 half-full 
3 empty 

for a total of seven casks. The second will receive 

2 full casks 

3 half-full 
2 empty 



for a total of 7 casks. The third will also receive 7 casks in the same arrangement. 
According to my division, each party will acquire 7 casks and equal quantity of wine. Let 
us say that a full cask of wine holds two portions, and a half-full cask one. According to 
the division, the first partner will receive 

2 + 2 + 2 + 1 

A total of 7 units and each of the others will receive 
2 + 2 + 1 + 1+1 

also adding up to 7. This proves that my suggested division is exact and just. Although 
the problem appears complicated, its numerical resolution presents no difficulty.” 

His solution was received with much enthusiasm not just by the sheik but also by the 
three men of Damascus. 



This figure shows, in the simplest possible form, the solution to the problem of 
dividing the twenty-one casks of wine. 

“By Allah!” exclaimed the young man with the emerald. “This calculator is amazing! 
In a moment he cleared up a problem that seemed most difficult to us.” Turning to the 
owner of the inn, he asked, in a friendly voice, “Man of Tripoli, what money have we 
spent at this table?” 

“Your total bill, with your food, is 30 dinars,” was the reply. Sheik Nasair wished to 
pay the bill, but the men of Damascus refused, which led to a small discussion and an 
exchange of compliments, with everyone speaking at once. At last it was agreed that 
Sheik Nasair, a guest, should pay nothing and that each of the others should pay 10 
dinars; so 30 dinars were handed to a Sudanese slave for his master. A few moments 
later, the slave returned and said, “My master says he made an error. The bill is 25 dinars, 
and he has asked me to return 5 to you.” 

“That man of Tripoli is most honorable,” remarked Sheik Nasair. And taking the five 
coins, he handed one to each of the three men, so that two remained. After exchanging a 
“lance with the men from Damascus, the sheik handed them as a reward to the Sudanese 
slave who had served them food. 


At that moment, the young man with the emerald rose and, looking gravely at his 
friends, said. “This business of paying over the 30 dinars has left us with a serious 
problem.” 

“Problem? I see no problem.” replied the sheik, astonished. 

“Oh yes.” said the man from Damascus. “A serious and seeming ridiculous problem. 
A dinar has disappeared. Think now. Each one of us paid 9 dinars. Three times nine is 27. 
Adding to these 27 the 2 that the sheik gave to the slave, we have 29 dinars. Of the 30 we 
handed over to the man from Tripoli, only 29 are accounted for. Where, then, is the other 
dinar? Where has it disappeared to?” 

Sheik Nasair thought for a moment. “You are quite right, my friend. It seems clear. If 
each one of you paid 9 dinars, and the slave received 2 that adds up to 29. One of the 
original 30 is missing. How is that?” 

At that moment, Beremiz, silent until then, intervened and addressed the sheik. “You 
are mistaken, O Sheik. The accounting should not be done in that way. Of the 30 dinars 
paid to the man from Tripoli for food, 25 went to the Tripolitan, 3 were returned, and 2 
were a tip to the Sudanese slave. Nothing disappeared, and the account presents no 
problem. Of the 27 dinars paid over, the Tripolitan received 25, and the slave 2.” 

The men from Damascus, hearing Beremiz’s explanation, exploded in loud laughter. 
“By the virtues of the Prophet,” exclaimed the oldest of them, “this counting man has 
solved the mystery of the, disappearing dinar and saved the reputation of this hostelry. 
Thanks be to Allah!” 


9 

IN THE STARS 



In which we receive a visit from Sheik Iezid, the poet. The strange results of an 
astrologer’s predictions. Women and mathematics. Beremiz is invited to teach 
mathematics to a young girl. Her odd situation. Beremiz speaks of his friend and master 
No-Elim the Wise. 




On the last day of the month Muharram, at sunset, the renowned Iezid Abdul Hamid 
came looking for us at the inn. 

“Some new problem to solve, O Sheik?’ Beremiz asked with a smile. 

“You guessed it my friends!” replied our visitor. “I am faced with a serious difficulty. 
I have a daughter, Telassim, with n quick intelligence and an obvious leaning towards her 
studies. When Telassim was bom, I consulted a famous astrologer who knew how to read 
the future by observing clouds and stars. He told me my daughter would he happy for the 
first eighteen years. From that age on, she would be threatened by a series of tragic 
misfortunes. He, however, had a way of keening her bad luck from deeply affecting her 
destiny. Telassim, he said, ought to leam the properties of numbers and their many 
working possibilities. But to master numbers and calculation, it is essential to know the 
science of al-Khwarizmi, that is, mathematics. So I decided to provide a happy future for 
Telassim by making her study the mysteries of calculus and geometry.” 

The kindly sheik paused and then went on, “I looked for various scholars at court but 
could not find one able to teach geometry to a seventeen year-old girl. One of them, an 
extremely gifted man, tried to dissuade me: ‘Would you try to teach a giraffe to sing?’ he 
asked. ‘Its vocal chords cannot produce the least sound. It would be a dreadful waste of 
time, a useless labor. A giraffe will never sing. And the female brain,’ said the holy man, 
‘is incapable of grasping the first principles of geometry. This unique science is founded 
in reason, in the use of equations, and in the application of clear principles with the help 
of logic and proportion. How can a girl w ho has been shut up in her father’s harem leam 
algebraic formulas and geometric theorems? Never! Easier for a whale to make a 
pilgrimage to Mecca than for a woman to leam mathematics. Why take on the 
impossible? If bad luck must come our way, let it be by the will of Allah.’” 

The sheik, looking serious, got up from his cushion and paced from side to side, 
before going on even more gloomily. “Discouragement the great cormptor settled on my 
spirit when I heard these words. Nevertheless visiting my good friend the merchant 
Salem Nasair one day, I heard effusive reference to the new calculator from Persia who 
had arrived in Baghdad. I heard the story of the loaves in all its details and it impressed 
me deeply. I set out to find the Counting Man and went specially to meet him at the 
house of the Vizier Maluf. I was astonished at how originally he solved the question of 
the 257 camels, reduced in the end to 256. Do you remember?” 

And Sheik Iezid, raising his head and looking solemnly at the Man Who Counted, 
added, “Are you, my Arab brother, able to teach the ingenuities of calculating to — my 
daughter Telassim? I will pay you any price you ask; and you can continue in your post 
as secretary to Vizier Maluf.” 

“Most generous Sheik,” replied Beremiz at once. “I see no reason to refuse your noble 
invitation. In a few months I can teach your daughter the workings of algebra and the 
secrets of geometry. The philosophers are doubly in error in their estimation of the 
intellectual capabilities of women. The female intelligence, well directed, can perfectly 
encompass the beauties and secrets of science. It will be simple to disprove the unfair 
ideas of the holy men. History yields up various examples of women who distinguished 
themselves in mathematics. In Alexandria, for example, lived Hypatia who taught 
hundreds of people the science of calculating, wrote a commentary on the work of 
Diophantus, analyzed the difficult texts of Apollonius, and corrected the astronomical 



tables currently in use. Be neither afraid nor uncertain, O Sheik. Your daughter will 
easily grasp the knowledge of Pythagoras, Allah be praised! It remains only to arrange 
the day and hour of her first lesson.” 

And the noble Iezid replied, “As soon as possible. Telassim is already 17. And I am 
anxious to free her from the woeful predictions of the astrologers.” He added, “I must 
warn you however, of a detail that is not unimportant. My daughter within the Harem and 
has never set eyes on a man not of our family. She can attend her lessons in mathematics 
only from behind a thick curtain, with her face veiled, and with two family slaves in 
attendance. Given this condition, do you accept?” 

“With relish,” replied Beremiz. “Clearly the reserve and modesty of a young girl have 
value above algebraic formulas. The philosopher Plato had the following sign hung on 
the door of his school: 

No One May Enter without Knowledge of Geometry 

One day a depraved young man turned up, eager to attend Plato’s Academy. The 
master, however, refused to admit him, declaring, ‘Geometry is pure and simple. Your 
shamelessness is offensive to such a pristine science. In this way, the famous pupil of 
Socrates demonstrated that mathematics cannot accompany depravity and is insulted in 
immorality. It will be a pleasure to instruct your daughter, whom I do not know, and 
whose face I will never have the good fortune to admire. If Allah is willing, I can begin 
tomorrow.” 

“Perfect,” replied the sheik. “One of my servants will come for you a little alter 
second prayers. Farewell.” 

When Sheik Iezid left the inn, I suggested to the Man Who Counted that the task 
might be beyond him. “One thing puzzles me, Beremiz. How can you teach mathematics 
to a young girl when you never really studied from books or attended the classes of the 
sages? How did you learn the power of calculation that you apply so brilliantly and 
opportunely? I know well that you uncovered its mysteries among sheep, fig trees, and 
flights of birds when you were a shepherd ...” 

“You are mistaken my friend.” Beremiz replied serenely. “While I was watching my 
master’s flocks, in Persia I knew an old dervish called No-Elim. I once saved his life 
during a violent sandstorm. From then on, he was my closest friend. He was a wise man, 
and he taught me useful and wonderful things. Having learned from such a man, I can 
teach geometry to the very last book of the unforgettable Euclid of Alexandria.” 


10 

BIRD IN THE HAND 



Our arrival at Iezid’s palace. The ill-tempered Tara Tir questions Beremiz ’s calculations. 
Caged birds and perfect numbers. The Man Who Counted praises the Sheik’s kindness. 
We hear a tender and enchanting song. 

It was a little past four when we left the inn and made our way to the home of Iezid 
Abdul Hamid. With the help of a pleasant and conscientious slave, we quickly 
maneuvered through the winding streets or the Muassan neighborhood and arrived at a 
magnificent palace set in the centre of an elegant park. 

Beremiz marveled at the distinguished appearance of Iezid’s home, in the center of the 
nark stood a great silvered tower on which rays of sunlight made rainbows of color. A 
large patio led into the main house through an iron gate etched in the most beautiful 
artistic detail. A second interior patio, in the center of which was a carefully ordered 
garden, divided the house into two wings. One of these contained the family’s bedrooms, 
the other was given over to public rooms and a salon where the Sheikh often met with 
philosophers, poets, and viziers. 

In spite of its decoration, the sheik’s palace appeared sad and gloomy. Those who 
looked on the barred windows could not imagine the displays of art that lined the interior 
rooms. The two wings were joined by a long gallery supported by ten slender columns of 
white marble, punctuated with horseshoe arches and decorated with a plinth of tile in 
relief and a mosaic floor. Two splendid staircases, also of white marble, led to a garden 
with a large, peaceful pond surrounded by flowers of all colors and scents. A vast cage of 
birds, decorated with mosaics, seemed to be the centerpiece of the garden. There were 
birds of strange voice, rare breed, and brilliant plumage. Some, of unusual beauty, 
belonged to species unknown to me. 

Our host greeted us very cordially as he joined” us from the garden. There was within 
a dark, thin, broad-chested young man, whose manner revealed him to be lacking in 



courtesy. At his waist he sported a rich dagger with a hilt of ivory. He had a piercing, 
belligerent gaze, and his brusque, agitated way of speaking was most disagreeable. 

“So, this is your calculator?” he snorted, disdain in his words. “What a trusting .’ 

you are, Iezid. You are going to allow some passing beggar to approach and speak to the 
beautiful Telassim? That’s the last thing you need! By Allah, how “naive you are!” And 
he gave an ugly laugh. 

His insolence infuriated me, and I wanted to answer this boor’s rudeness with 
fisticuffs. Beremiz however remained cool. Or perhaps the Man Who Counted saw in this 
occasion, in these insults, only another problem to be solved. 

The poet, disturbed by the man’s inconsiderate behavior, said, “Mathematician, please 
forgive the rash judgment of my cousin Tara Tir. He does not know, and therefore cannot 
properly appreciate, your mathematical skills. He is, more than anything else, worried 
about Telassim’s future.” 

“Of course, I don’t know the mathematical skills of this stranger! I couldn’t care less 
how many camels pass through Baghdad searching for shade and food,” the young man 
exclaimed. And then, speaking rapidly, tripping over his words, he continued, “My 
cousin, in just a few minutes I can show that you are completely mistaken about the 
talents of this adventurer. If you will allow me, I will put an end to this man’s science, 
using no more than a couple of trivialities I picked up from a master of Mosul.” 

“Yes, indeed. Go ahead,” agreed Iezid. “You can ask the calculator questions right 
now, or raise any problems that you like.” 

“Problems? What for? Would you compare the science of the sages with that of a 
jackal in a cage?” he retorted crudely. “I assure you that it won’t be necessary to come up 
with some problem to umnask this ignorant Sufi. 1 won’t have to trouble my memory to 
achieve my result, much quicker than you think.” 

And. fixing Beremiz with a cold, unyielding gaze, he pointed to the large birdcage and 
asked, ‘Tell me. ‘fowl calculator,’ how many birds are in that cage?” 

Beremiz Samir crossed his anns and studied the bright flock with intense 
concentration. It would be madness, I thought, to try and count the birds flying restlessly 
around the cage, dashing from one perch to another. 

There was an explained silence. After a few seconds the Man Who Counted turned to 
the gracious Iezid and said, “I implore you, O Sheik, to free three of these birds 
immediately, then it will be simpler and more pleasing to announce the total number.” 

His request seemed not to be dictated by logic. For whoever could count a certain 
number of birds could just as easily count three more. Most intrigued by such an 
unexpected request, Iezid ordered the bird keeper to do as Beremiz had suggested. Freed 
from captivity, three pretty hummingbirds arrowed into the sky. 

“Now there are in this cage,” said Beremiz in a deliberate manner, “496 birds.” 

“Wonderful,” exclaimed Iezid enthusiastically. “The exact number! And Tara Tir 
knows it! I told him there were exactly 500 in my collection. I sent one nightingale to 
Mosul, and now we have freed 3, leaving 496.” 

“A lucky guess,” grumbled Tara Tir, with a gesture of disgust. 



Moved by curiosity, Iezid asked Beremiz, “Can you tell me why you prefer 496 when 
it would have been just as simple to add 496 and 3 to get 499?” 

“I will explain, O Sheik,” Beremiz replied proudly. “Mathematicians always try to 
give preference to remarkable numbers and avoid dull, ordinary sums. But between 499 
and 496 there can be no doubt. The number 496 is a perfect number and therefore 
deserves our preference.” 

“And what do you mean by a perfect number?” the poet asked. “What makes a 
number perfect?” 

“A perfect number” explained Beremiz, “is one that is equal to the sum of its divisors, 
excluding the number itself. So, for example the number 28 has five divisors: 

1, 2, 4, 7, and 14 

The sum of the divisors 

1+2 + 4 + 7 + 14 

is exactly 28. Therefore 20 belongs in the category of perfect numbers. 

“The number 6 is also perfect. The divisors of 6 are 

1, 2, and 3 

whose sum is 6. 

“Along with 6 and 28, there is 496, which is also, as I said, a perfect number.” 

The ill-tempered Tara Tir, unable to bear Beremiz’s further explanations, excused 
himself to Sheik Iezid and left muttering angrily, his rout by the skill of the 
mathematician not inconsiderable. As he passed in front of me, he shot me a glance of 
supreme scorn. 

“I implore you, O calculator,” the noble Iezid apologized again, “not to be offended by 
the words of my cousin Tara Tir. He is a hotheaded man, and since he took over the salt 
mines in al-Derid he has become irritable and violent. He has already suffered five 
murder attempts and several assaults from slaves.” 

It was evident that the intelligent Beremiz did not want to trouble the sheik, and he 
responded with kindness and generosity: “If we want to live in peace with our neighbors, 
we must restrain our anger and cultivate our goodwill! When I feel insulted I try to follow 
the wisdom of Solomon: ‘“A fool’s wrath is presently known, but a prudent man covereth 
shame” (Proverbs 12:16). I can never forget — the teachings of my good-hearted father. 
Whenever he saw me overwrought and ready for revenge, he would say, “He who 
humbles himself before his fellow man is exalted in the eyes or God.” 

And after a short pause, he added, “I am, nevertheless, very grateful to the insolent 
Tara Tir, and I bear him no malice. His unruly nature gave me the opportunity to practice 
new acts of charity.” 

“New acts of charity?” the sheik replied with surprise. “What do you mean?” 

“Every time we tree a caged bird,” .explained Beremiz, “we accomplish three acts of 
charity. The first to the little bird, returning him to the freedom from which he has been 
snatched; the second to our own conscience, and the third to God ...” 

“You mean that if I were to free all of the birds in the cage ...” 



“I assure you, O Sheik, that you would be committing 1,488 acts of supreme charity!” 
exclaimed Beremiz, as if he already knew by heart the product of 496 and 3. 

The generous Iezid was so impressed by the words of Beremiz that he decided to free 
all of the birds in the enormous cage. When they heard his orders, the servants and slaves 
were stunned. The collection, gathered with much patience and effort, was worth a 
fortune. It contained partridges, hummingbirds, brightly plumed pheasants, black gulls, 
ducks from Madagascar, Caucasian owls, and various swallows from China and India, 
exceedingly rare. 

“Free the birds,” the Sheik shouted again, waving his hand with its splendid ring. 

The wide doors opened, and the captive birds fled their prison in a flock, spreading 
over the treetops of the garden 

Then Beremiz said, “Every bird, with its wings outspread, is a book, its pages open to 
heaven. An ugly crime, to rob or destroy this library of God.” 

And then we heard the first notes of a song. The voice was so soft and tender that it 
mingled with the trill of the little swallows and the soft cooing of the doves, the melody 
was enchanting and sad filled with melancholy and longing, like the lament of a lonely 
nightingale. It rose then to a crescendo of complicated roulades; bright trills, and faltering 
cries of love that contrasted with the serenity of the afternoon and hung in the air like 
leaves lifted in the wind. Then it returned to the sad and mournful air of the beginning 
and seemed to hang over the garden like a whisper. 

Though I speak with the tongues of men 
And of angles, 

And have no charity 
lam become a sounding brass 
Or a tinkling symbol 
I am nothing 
lam nothing 


And though I have the gift of prophesy 
And understand all mysteries and all knowledge 
So that I could remove mountains 
And have not charity 
I am nothing 
lam nothing 


And though I bestow all my goods 
To feed the poor 
And though I give my body 
To be burned 



And have not charity 
lam nothing 
lam nothing 


The charm of that voice seemed to envelop the place in a wave of indefinable joy. 
Even the air seemed to lighten. 

“It is Telassim singing,” said the sheik, seeing that we were listening closely, 
captivated by that strange sound. 

The birds were flying away, filling the air with their joyful songs of freedom. There 
were only 496, but they seemed like ten thousand. 

Beremiz was wrapped in silence. The notes of the music penetrated his spirit, joining 
the happiness he felt at the liberation of the birds. Then he raised his eyes, searching for 
the place from where that voice came. 

“And who wrote those beautiful verses?” he asked. 

The sheik answered, “I don’t know. A Christian slave taught them to Telassim, and 
now she cannot forget them. They must be from a Nazarene poet. That is what my wife 
told me, the mother of Telassim.” 


11 

FOR GOOD MEASURE 



How Beremiz started his lessons in mathematics. A phrase from Plato. The unity of 
God. What is measuring? The parts of mathematics. Arithmetic and numbers. Algebra 
and relations. Geometry and forms. Mechanics and astronomy. A dream of the King 
Assad Abu Carib. The “invisible student” prays to Allah. 


The room where Beremiz was to give his lessons was very spacious. It was divided by 
a thick, heavy curtain of red velvet that hung from the ceiling to the floor. The ceiling 
was brightly painted the columns were golden. Over the carpets were scattered large silk 
cushions the edges of which were inscribed with verses from the Koran. The walls were 
decorated with fanciful blue designs intertwined with beautiful lyrics by Antar, the desert 
poet. In the center of the room, between two columns in gold letters against a blue 
background, were these words from a paean by Antar: 

When Allah loves one of his followers 
He leads him to inspiration 

A gentle perfume of incense and roses pervaded the room as dusk approached. The 
polished marble windows were open and looked out toward the garden and rows of 
luxuriant apple trees stretching down to the gray, rough waters of a river. A black slave 
stood by the door, her face exposed, her fingernails colored with henna. 

“Is your daughter here?” Beremiz asked the sheik. 

“Of course,” replied Iezid. “I told her to sit on the other side of the room, behind the 
curtain, w here she will be able to see and hear. Nevertheless, she will be invisible to 
everyone on this side.” 

Truly, everything had been arranged so that one could not make out even the 
silhouette of the young girl who was to be Beremiz’s pupil. Perhaps she watched us 
through some tiny hole in the velvet, imperceptible to us. 

“I think that now we can begin,” the sheik said, continuing affectionately, “try to pay 
attention Telassim, my daughter.” 

“Yes, Father,” answered a well-bred female voice on the other side of the room. 

Then Beremiz got ready to begin his lesson: he crossed his legs and sat down .on one 
of the cushions in the middle of the room. I discreetly found a place in the comer and 
made myself as comfortable as I could. The sheik sat by my side. 

All science proceeds from prayer. Therefore, Beremiz began his class with a prayer: 
“In the name of Allah, the All-Merciful. Praised be the omnipotent Creator of the 
universe! The mercy of God is our supreme gift! We adore you, O Lord and implore your 
grace! Lead us in the path of righteousness, in the path of those blessed by your hand!” 

When he had finished the prayer, Beremiz said, “When we look to the sky on a still, 
clear night we feel that we are incapable of understanding the wondrous works of God. 
Before our astonished eyes, the stars form a luminous caravan that files across an infinite 
desert, Circling the immense nebulas and planets, following eternal laws, from the depths 
of space, and suggesting to us a very precise notion: the idea of ‘number.’ 

“There lived in Greece, when that country was still pagan, a philosopher named 
Pythagoras — How wise is Allah! When a disciple asked him which were the dominant 
forces in the affairs of man, the wise man answered, ‘Numbers rule everything.” 

“Truly. The simplest thought cannot be achieved without including, in many aspects, 
the fundamental concept of number. The Bedouin in the desert, his head bowed in prayer, 
murmurs the name of God, and his spirit is filled with a number: ‘Oneness, Unity.’ Yes, 
God, according to the truth written in the pages of the Holy Book and repeated by the 



Prophet, is one, eternal, and immutable! Therefore, the number appears, in the framework 
of our intelligence, as a symbol of the Creator. 

“From numbers, which are the basis of all reason and understanding, comes another 
notion of indisputable importance: the notion of ‘measurement.’ 

“To measure is to compare. Nevertheless, only those numbers that contain an element 
which to base comparison are susceptible to measurement. Would it be possible to 
measure the vastness of space? Not at all. Space is infinite and therefore has no basis for 
comparison, is it possible to measure eternity? Never. In human terms, time is always 
infinite, and in the calculation of eternity nothing so ephemeral can serve as a unity of 
measurement. 

In many cases nevertheless, it would be possible for us to substitute one dimension 
that does not fit with our system of measurements for one that can be calculated with 
more certainty. This exchange, in order .to simplify the measuring process, constitutes the 
principal object of the science that we call mathematics. 

“In order to reach his goal, the mathematician has to study numbers, their properties 
and permutations. This aspect is called arithmetic. Once numbers are understood, it is 
possible to use them to evaluate dimensions that vary “or that are not known but that can 
be symbolized with formulas and equations. In that way, we get algebra. The 
measurements we make of reality are represented by material things or symbols; in either 
case, these things or symbols are endowed with three attributes: form, size, and position. 
It is important, therefore, to study such attributes. That is the objective of geometry. 

“Mathematics also deals with the laws that govern movement and forces, laws that 
appear in the admirable science called mechanics. Mathematics puts all of its wonderful 
resources at the disposal of a science that raises the spirit and exalts men. That science is 
astronomy. 

“Some think that, within the framework of mathematics, arithmetic, algebra, and 
geometry are entirely distinct disciplines that is a grave error. They all work together, the 
one helping the other in some cases even interchangeable. 

“Mathematics, which teaches man to be simple and humble, is the basis of “all the arts 
and sciences.” 

“Let me repeat to you an incident that occurred to a famous Yemenite-monarch: Assad 
Abu Carib, king of Yemen, while resting one day in the spacious balcony of his palace, 
dreamed that he had met seven young maids walking along a path at a certain point 
overcome with fatigue and thirst, they stopped under the burning desert sun. Just then 
there rose up a beautiful princess who offered the pilgrims a pitcher of water. The kind 
princess quenched their thirst, and they continued on their journey reinvigorated. 

“When he awoke, Assad Abu-Carib was so impressed by this inexplicable dream that 
he decided to call for a famous astrologer, named Sanib, to consult him on the meaning of 
this vision that he — a just and powerful king — had seen in the world of image and 
fantasy. The astrologer Sanib said, ‘Sir, the seven young maids walking along the path 
were the divine arts and the human sciences: Painting, Music, Sculpture, Architecture, 
Rhetoric, Dialectics, and Philosophy. The kind princess who came to their aid was the 
great and prodigious science of Mathematics.’ ‘Without the help of mathematics,’ the 
wise man continued, ‘the arts could not advance and all the sciences would perish.’ 



Moved by these words, the king decided to organize centers for the study of mathematics 
in all the cities, villages, and oases of his country. By the sovereign’s order, able and 
eloquent wise men went forth into the bazaars, inns, and caravansaries to give lessons in 
arithmetic to the traders and nomads. Within a few months the country had become more 
prosperous. In parallel with the advance of science, the real wealth of the country was 
growing; the schools were fdled with students; Commerce expanded rapidly; works of art 
increased; monuments were raised, rare and foreign treasures abounded in the cities. The 
country of Yemen opened itself to progress and wealth but then misfortune! — Maktub! 
his prodigious flowering of work and wealth came to an end. King Assad Abu Carib 
closed his eyes-to the world and was raised by the infidel Asrail to the heaven of Allah. 
The death of the monarch opened two graves one received the body of the glorious king 
and the other was buried the artistic and scientific culture of his people. A vain, indolent 
prince, a man of little intellectual merit, ascended to the throne. He spent more time on 
idle pursuits than on the administrative problems of his country. In just a few months, all 
of the public services had become chaotic, schools were closed, and artists and wise men 
were forced to flee from the threats of malefactors and thieves. The public treasury was 
criminally squandered on idle festivals and lavish banquets. The country was brought to 
ruin by mismanagement and was finally attacked by eager enemies, who quickly 
conquered the Yemenites. 

“The story of Assad Abu Carib shows that the progress of a people is linked to the 
development of their mathematical abilities. In all the universe, mathematics is number 
and measure. Oneness, the symbol of the Creator, is the beginning of everything, which 
would not exist but for the unvarying proportions and relations of numbers. All of life’s 
great enigmas can be reduced to simple combinations of either variable or constant, 
known or unknown, elements that we can solve. 

“So that we can understand this science, we must begin with numbers. We will see 
how to examine them, with the help of Allah, the All-Merciful! 

“Uassalan!” 

With these words the Man Who Counted ended his first class. And then, as an able 
surprise, we heard the voice of the invisible student, hidden behind the curtain, speaking 
the following prayer: 

“O omnipotent God, Creator of heaven and earth, forgive the poverty, the meanness, 
the naivete of our hearts. Listen not to our voices but to our inarticulate cries; attend not 
to our desires but to the clamor of our needs. How many times do we ask for something 
that could never be ours! 

“God is great! 

“O God! We thank you for this world, our great home, its size and wealth, the 
multifarious life of the world of which we are a part. We praise you for the splendor of 
the blue skies and for the evening breeze and for the clouds and for the stars in the 
heavens. We praise you, Lord, for the immense oceans, for the water that runs in the 
streams, for the eternal hills, for the luxuriant trees, and for the carpet of grass that 
soothes our feet. 


“God is merciful! 



“We thank you, Lord, for the many delights that make us feel in our spirit the beauty 
of life and love . . . 

“O God, the All-Merciful! Forgive the poverty, the meanness, the naivete of our 
hearts.” 


12 

CIRCULAR REASONING 



Beremiz reveals his fascination with skip rope. The curve of the marazan and the 
spiders. Pythagoras and the circle. Our meeting with Harim Namir. The problem of the 
sixty melons. How the magistrate lost a bet. The voice of the blind muezzin calls believes 
to prayers at sunset. 

It was close to the time of prayer when we left the poet Iezid’s sumptuous palace. 
Passing the shrine of Ramih, we heard birds twittering in the branches of an old fig tree. 
“Look! These are certainly some of the birds set free today.” I said to Beremiz. “How 
pleasant to hear their joy at recovering their freedom being turned into song!” At that 
moment, however Beremiz was not interested in the singing of the birds. He was 
absorbed in watching some children playing in a nearby street. Two of them held the ends 
of a rope that was about four or five cubits long. The others were trying to jump over it 
while the two turned it lower or higher, depending on the skill of the jumper. 

“Look at the rope, my Baghdad friend!” said the calculator taking my arm. 

“Look at the perfect curve. Do you not think it is worth watching?” 

“What are you talking about? The rope?” I exclaimed. “I don’t see anything 
extraordinary in this game of children taking advantage of the last light of day for their 
enjoyment.” 

“Well then, my friend, your eyes are blind to the beauty and wonder of nature. When 
the children raise the rope, holding it by its ends and then letting it fall freely from its 
own weight, the rope forms a curve on its own, as a result of natural forces. On other 



occasions I have observed this same curve, which the sage N6-Elim called marazan, in 
cloth and in the humps of some dromedaries. Is this curve analogous to the parabola? In 
the future, if Allah wishes, geometers will find ways to trace this curve, point, by point, 
and will rigorously study its properties.” 

“Nevertheless,” he continued, “there are many other, more important curves. First the 
circle. Pythagoras, the Greek philosopher and mathematician, considered the circle the 
most perfect curve, thereby associating the circle with perfection. And the circle, being 
the most perfect curve of all, is the easiest to draw.” 

At that point Beremiz interrupted the lecture he had just started on curves, indicated a 
young man a short distance away, and shouted, “Harim Namir!” 

The young man turned around abruptly and came toward us, smiling. Then I saw that 
he was one of the three brothers we had met in the desert arguing over the inheritance of 
thirty-five camels — a difficult division, complicated with thirds and ninths, that Beremiz 
solved by resorting to a curious trick that I have already explained. 

“Destiny has led us to the great calculator. My brother Hamed is trying to “clear up a 
matter or sixty melons that no one can resolve.” And Harim led us to a house where we 
found his brother Hamed Namir with several merchants. 

Hamed was very pleased to see Beremiz and, turning to the merchants, said, “This 
man who has just arrived is a great mathematician. Thanks to his valuable help, we found 
a solution to a problem that seemed insoluble: how to divide thirty-five camels among 
three people. I am certain that he will be able to explain within minutes the discrepancy 
we have encountered in the sale of sixty melons.” 

Beremiz was carefully apprised of the case. One of the merchants explained, “The two 
brothers Harim and Hamed brought me two batches of melons to sell in the market. 
Harim brought me 30 melons that were to be sold at a price of 3 for 1 dinar; Hamed also 
brought me 30 melons, but they were to be sold at the higher price of 2 melons for’ 1 
dinar. Logically, once the melons were sold, Harim would receive 10 dinars and his 
brother IS. The total sale-price would then be 25 dinars. 

“Yet, when I reached the marketplace, doubts began to trouble my spirit. If I started by 
selling the more expensive melons, I would lose my buyers. If I started by selling the 
cheaper melons, I thought, I would later have trouble in selling the other thirty. The only 
solution was to sell the two batches at the same time. 

‘Having made this decision, I put all the melons together and started to sell them in 
lots of 5 for 2 dinars. The reasoning seemed to me clear: if I sold 3 for 1 dinar and then 2 
for 1, it would he easier to sell 5 for 2 dinars. 

“After selling 60 melons in 12 lots of 5 each, I made 24 dinars. But how could I pay 
both brothers if one was to get 10 and the other 15 dinars? There was a difference of 1 
dinar. I don’t know how to explain this difference since, as I said, the business was 
handled with the utmost care. Isn’t it the same thing to sell 3 for 1 and then 2 for 1 as to 
sell 5 for 2?” 

“The matter would have not been at all important.” broke in Hamed Namir, “if it had 
not been for the absurd meddling of the magistrate who supervises the marketplace. 
Having heard of this matter, the magistrate did not know how to account for the 



difference, and he bet 5 dinars that it was caused by the theft of one melon during the 
sale.” 

“The magistrate is wrong,” said Beremiz, “and he will have to pay the 5 dinars. This is 
how to explain the difference: 

“Harim’s batch consisted of 10 lots of 3 melons each. Each lot should have been sold 
for 1 dinar. The total would have been 10 dinars. 

“Hamed’s batch consisted of 15 lots of 2 melons each, which, at a price of .1 dinar per 
lot, would have brought a total of 15 dinars. 


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This figure clears up the problem of the 60 melons. “A” represents the 30 melons 
which were sold at 3 for a dinar and “B” the 30 melons at 2 for a dinar; the whole shows 
them divided into twelve batches of 5 melons each, sold for 2 dinars. 


“Notice that the number of lots in the two batches is not the same. In order to sell the 
melons in lots of 5, only 10 lots of 5 could have been sold for 2 dinars; these 10 lots 
having been sold, there would have remained 10 melons belonging only to Hamed’s 
batch and these, being more expensive, would have had to be sold at a rate of 2 for 1 
dinar. 

“The difference of one dinar was, then, a result of the sale of the last 10 melons. 
Therefore, there was no theft. The loss of 1 dinar was a result of the difference in price 
between the two batches.” 

At that moment we had to break off our meeting. The voice of the muezzin, echoing 
through the air, was calling the faithful to afternoon prayers. 

“Hai al el-salah! Hai al el-salah!” 

Without losing a moment, each of us began our ritual preparations, in .accordance 
with the Holy Book. The sun had just reached the horizon. It was the hour of maghrib, 
the prayer at dusk. From the tower of the Mosque of Omar, the blind muezzin called the 
faithful to prayer in a deep, deliberate voice: 

“Allah is great and Mohammad, the Prophet, is the true messenger of God. Moslems, 
come to pray! Come to pray! Remember that even thing is dust, except Allah!” 

The merchants, following Beremiz, unrolled their bright rugs, removed their sandals, 
turned toward the Holy City, and cried out: 

“Allah, wise and merciful! Praised be the omnipotent Creator of heaven and earth: 
Lead us in the path of righteousness, in the path of those who are by you protected and 
blessed!” 


13 

FREINDSHIP KNOWS NO BOUNDS 



Which tells of our visit to the caliph ’s palace, and of the audience he was kind enough 
to grant us. Of poets and friendship — friendship between men and between numbers. 
The Man Who Counted is extolled by the caliph of Baghdad. 

Four days later, in the morning, we were informed that we would be received in 
solemn audience by Caliph Abul Abas Ahmed al-Mutasim Billah, emir of all believers, 
vicar of Allah. That message, so gratifying to any Muslim, had been eagerly awaited not 
only by me but by Beremiz as well. 

It is possible that the sovereign, hearing from Sheik Iezid or some of the feats 
performed by the renowned mathematician, might have shown interest in getting to know 
the Man Who Counted. Nothing else can explain our presence in the court among the 
most distinguished figures in Baghdad society. 

Entering the rich palace of the emir, I was dazzled. The great arcades, curving 
harmoniously and supported by tall and slender pairs of columns, were decorated around 
their bases with the fine mosaic. I could see these mosaics were made up of tiny red and 
white tiles, set in stucco. The ceilings of the principal rooms were decorated in blue and 
gold; the walls of all the rooms were covered with embossed tiles and the walks with 
mosaic. The lattices, the carpets, the divans — in fact, all the furnishings of the palace — 
showed off the supreme magnificence of a prince of Hindu legend. 

Outside, in the gardens, the same magnificence was visible, enhanced by the hand of 
nature, perfumed by a thousand different aromas, covered in a carpet of greenness, bathed 
by the river, refreshed by innumerable white marble fountains, beside which thousands of 
slaves labored. 

When we arrived, we were led to a divan by one of the aides of Vizier Ibrahim Maluf. 
We saw the all-powerful monarch seated on a rich throne of ivory and velvet. The 
overpowering beauty of the great room disturbed me somewhat. All the walls were 
decorated with fine inscriptions, put there by the cunning hand of an inspired 
calligrapher. I noted that those inscriptions were verses from our most brilliant poets. 


Even-where stood urns filled with flowers — on the cushions, flowers without petals, 
flowers woven in the carpets or exquisitely worked into the gold platters. 

Many sumptuous columns caught my eye, their caps and shafts elegantly worked by 
the chisels of Moorish artists, unmatched masters of the many geometric variations in the 
design of flower and leaf, of tulips, of lilies, of a multitude of diverse plants, wonderfully 
harmonious, indescribably beautiful. 

In attendance were seven viziers, two judges, several sages, and various well-known 
dignitaries. 

It fell to the noble Maluf to present us. The vizier, elbows at waist, his slender hands 
extended, palms open, spoke as follows: “To comply with your command, O King of the 
Temple, I made sure that today, in this illustrious audience, my present secretary, 
Beremiz Samir, and his friend Hanak Tade Maia, scribe and functionary of the palace, 
should present themselves.” 

“Then be welcome, Muslim brothers!” replied the sultan in a warm and friendly voice. 
“I admire men of wisdom. Anyone versed in mathematics, under the skies of this land, 
can always count on my support and, indeed, my firm protection.” 

“May Allah guide you, O Master!” exclaimed Beremiz, bowing. 

1 remained still, my head bowed, my anns crossed, but since I had not been addressed, 
I did not presume to answer. 

The man who held in his hands the destiny of all Arabs seemed both open and 
generous. His features were fine, tanned by the desert sun, prematurely furrowed. When 
he smiled, -which he did often, he showed regular white teeth. He wore at his waist, in a 
silk girdle, an elegant dagger, its sheath studded with precious stones. His turban was 
green, with thin white stripes. Green, as everyone knows, marks the descendants of the 
Holy prophet Muhammad, to whom all glory and peace! 

“I have many important things to discuss in this audience,” declared the caliph. “I 
have no wish, however, to embark on a discussion of serious matters of politics without 
first having some clear proof that the Persian mathematician commended by my friend 
the poet Iezid is truly a worthy and accomplished calculator.” 

So appealed to by the illustrious monarch, Beremiz felt compelled to justify Sheik 
Iezid’ s confidence in him. Addressing the sultan, he spoke, “O ruler of the faithful, I am 
no more than a simple shepherd who has been graced by your regard.” 

After a short pause, he went on, “Even so, my generous friends consider it justified to 
name me among the men of numbers, and I feel flattered by such high praise. I think, 
however, that men in general are good calculators. So is the soldier on campaign who 
estimates distance in a single glance. So is the poet who counts syllables and verifies the 
cadence of his verses. So is the musician who applies in his compositions the laws of 
perfect hannony. So is the painter who draws with the unvarying proportions of 
perspective in mind. So is the humble rug maker who arranges one by one the threads of 
his labor. All of these, O King, are fine and accomplished calculators.” 

After turning his noble gaze on those who surrounded the throne, Beremiz continued, 
“I see to my infinite pleasure that you, Master, have wise and learned men all around you. 
I see in the shadow of your mighty throne distinguished men who pursue their studies and 



extend the frontiers of science. The company of wise men, O King, is for me the greatest 
of treasures. The worth of a man lies in what he knows. Knowledge is power. Wise men 
teach by example, and there is nothing that captures the human spirit more convincingly 
than example. Man, however, should not pursue knowledge except to do good. 

“The Greek philosopher Socrates lent all the weight of his authority to make this 
point: The only useful knowledge is that which betters us.’ Seneca, another celebrated 
thinker, asked in disbelief. ‘What does it matter to know what a straight line is if one has 
no notion of rectitude.’” Allow me then, O generous and just King, to pay homage to the 
wise and learned ones in this room.” 

The Man Who Counted paused briefly and went on, solemn and eloquent, “In my 
daily labors, taking notice of .all that Allah brought from non-being into being. I learned 
to value numbers and to use them in accordance with practical and reliable laws. I feel, 
however, some difficulty in providing the proof you have just asked for. Still, counting 
on your celebrated generosity, I must nevertheless say-that in this splendid divan 1 see 
nothing but admirable and eloquent demonstrations of mathematics everywhere. The 
walls of this elegant room are decorated with various poems, each one of precisely 504 
words. Some of these words are written in black characters, the rest in red. The 
calligrapher, who traced out the letters of these poems, disposing the 504 words, has 
shown that he has a talent and imagination equal to that of the poets who wrote these 
immortal verses.” 

“Indeed, O King,” Beremiz went on. “And the reason is simple. I find, in these 
incomparable verses that grace this magnificent room, great praise of friendship. There, 
close to the column, I can read the first line of the famous poem of al-Muhalhil: 

If my friends depart me, most wretched will I be, since all my treasures depart me. 

And there, farther on, I read the line of Tarafa: 

The enchantments of life lie solely in the great friendships we form. 

Indeed, all these are sublime, profound, and eloquent. The greater beauty, however, 
lies in the ingenuities of the calligrapher, in demonstrating that the friendship these verses 
extol exists not just among those endowed with life and feeling. Friendship occurs also 
among numbers. 

“You will ask, no doubt, how to recognize among numbers those bound together in 
mathematical friendship. How can geometry distinguish in a numerical series those that 
are so bound? 

“I will explain as briefly as I can the idea of friendship among numbers. Take, for 
example, the numbers 220 and 284; 220 is exactly divisible by the following numbers: 

1, 2, 4, 5, 10,11, 20, 22, 44, 55, and 110 

In turn, 284 is exactly divisible by these numbers: 

1, 2, 4, 71, and 142 

“Among these two numbers, there are remarkable coincidences. If we add up the 
numbers above that go into 220, we get 284; and if we add those that get into 284, we get 
exactly 220. 



“From this result, mathematicians have concluded that the numbers 220 and 284 are 
‘friends,’ that each of them seems to serve, delight, guard, and honor the other. 

And he finished thus: “Now, O generous and just King, the 504 words that make up 
the poems in praise of friendship were traced in the following way: 220 of them in black 
characters and 284 in red characters, numbers that, as I explained, are friends. 

“Now, notice another, no less interesting “connection. The 504 words are formed in 32 
lines, as you see. Well, the difference between 284 and 220 is 64, a number that, apart 
from being a square and a cube, is exactly twice the number of lines. 

“A skeptic would call it mere coincidence, but a believer who follows the teachings of 
the Holy Prophet Muhammad — to whom all prayer and peace! — kn ows that so-called 
coincidences are impossible, unless Allah has already inscribed them in the Book of 
Destiny. I declare, then, that the calligrapher, dividing the number of 504 in two — 220 
and 284 wrote a poem on friendship that must move all men of spirit.” 

Hearing the words of the Man Who Counted, the caliph was overjoyed. It seemed 
incredible to him that he had counted at a glance the 504 words of thirty poems and 
verified that 220 were black and 284 were red. 

“Your words, O man of numbers, have proved to me beyond doubt that you are in 
truth a mathematician of the highest worth. I have been delighted by that absorbing 
connection you call numerical friendship, and I am now interested in finding which 
calligrapher it was who inscribed the verses that adorn the walls of this room. It is quite 
easy to find out whether the disposition of the 504 words into two friendly parts was 
deliberate or a trick of destiny, the work of Allah, the Exalted One.” 

And, summoning one of his secretaries, Sultan al-Mutasim asked him, “Do you 
remember, Nuredin Zarur, which calligrapher worked in this palace?” 

“I kn ow him well, Master. He lives close to the Mosque of Ottoman,” replied the 
sheik. 

“Bring him here as soon as possible,” the caliph ordered. “I want to question him at 
once.” 

“I hear and obey!” 

And the secretary left like an arrow to carry out the order of the sovereign. 


14 

AN ETERNAL TRUTH 



Of the things that occurred in the throne room. The musicians and twin dancers. How 
Beremiz could distinguish Iclimia from Tabula. An envious vizier criticizes Beremiz. The 
Man Who Counted praises theorists and dreamers. The king proclaims theory the victor- 
over the demand of the immediate present. 

After Sheik Nuredin Zarur, the emissary of the king, went in search of the calligrapher 
who had written out the poems that decorated the room, there entered five Egyptian 
musicians who performed with great feeling the most tender Arab songs and melodies. 
While the musicians sang and played their harps, zithers, and flutes, two gracious 
dancers, by their appearance Spanish slaves, danced on a wide, circular stage for the 
pleasure of the assembled company. 

Slave girls who are to become dancers are chosen carefully and are particularly 
valued, for they provide beauty and entertainment, personal satisfaction, and flattery for 
the guests. The dances vary according to the origin of the dancers, and their variety 
indicated the wealth and power of our host. The physical likeness of the two dancers was 
considered a high virtue. In order to find such a pair, a careful and exacting selection was 
required. 

The similarity of the two slave girls astonished everyone. Both had the same slender 
waist, the same dark complexion, eyes outlined identically in black kohl; both wore the 
same necklaces, bracelets, and collars, and, to compound the confusion, there was not the 
slightest difference in their gowns. 

At a certain moment, the caliph, who appeared to be in fine humor, addressed himself 
to Beremiz. 

“What do you think of my pretty slaves? You will already have noticed that they are 
identical. One is named Iclimia and the other Tabessa. They are twins and are worth a 
fortune. I have never found anyone who could distinguish the two when they appear on 
stage. Look closely. Iclimia is on the right now. Tabessa is on the left, next to the column, 
giving us her best smile. By the color of her skin and the delicate perfume she exudes, 
she seems to be a leaf of aloe.” 


“I confess, O Sheik of Islam,” Beremiz replied, “that these dancers are truly 
marvelous. Praised be Allah, the One God, who created beauty from which comes such 
seductive feminine creatures. Of beautiful women, the poet says: 


It is for your luxury- that poets weave a fabric of gold thread and for your beauty- 
painters create fresh immortality. 

To adorn you, to clothe you, to make you more exquisite, the sea gives its pearls, the 
earth its gold, the garden its flowers. 

Over your youth, the desire of man’s heart spreads its glory.” 


“It seems to me, however, easy enough,” offered the Man Who Counted, “to 
distinguish Iclimia from her sister Tabessa. One need only note their dresses.” 

“How is that possible?” replied the Sultan. “There is not the slightest difference 
between their dresses. Both, by my orders, wear identical veils, blouses, and dance 
skirts.” 

“Please forgive me, O generous King,” Beremiz replied politely, “but the seamstresses 
did not respect your orders, with the proper caution. Iclimia’s skirt has 312 fringes while 
Tabessa’s has 309. This difference in the total number of fringes is enough to dispel any 
confusion between the twin sisters.” 

On hearing this, the sultan clapped several times, stopped the dancing, and ordered the 
hakim to count the fringes on the dancers’ costumes one by one. 

Beremiz’s calculations were confirmed. The lovely Iclimia had 312 fringes on her 
costume and her sister Tabessa only 309. 

“By Allah!” exclaimed the caliph. “Even though he’s a poet, Sheik Iezid did not 
exaggerate. This Beremiz is truly a talented counting man. J le counted all the fringes on 
both- costumes while the dancers twirled dizzily across the stage. By Allah! It’s 
incredible!” 

But when envy takes hold of a man, it opens his soul to every despicable, crude 
inclination. 

There was in the court of al-Mutasim a vizier named Nahum ibn-Nahum, an envious, 
bad man. As he watched Beremiz’s prestige rise before the caliph like a wave of sand 
propelled by a desert wind, the sheik was driven to despair and decided to catch out my 
friend and make a fool of him. So he approached the king and slowly spoke thus: 

“I have just noticed, O Emir of the Creators, that the Persian calculator, our guest of 
this afternoon, is a gifted counter of elements and serial figures. He counted the 500 or so 
words written on the walls, indicated the friendship between numbers, spoke of their 
difference — 64, which is both a cube and a square — and ended by counting one by one 
the fringes on the gowns of the beautiful dancers. 

“It would be terrible if our mathematicians spent their time on such childish things 
without any practical use whatsoever. Really, what good is it to know that in the verses 
that we adore there are 220 plus 284 words? The concern of those who admire a poet 
does not lie in counting the letters of the verses or in counting the number of black or red 



words in his poems. Neither does it matter whether we know that in the gown of this 
beautiful and graceful dancer there are 312, 309, or even 1,000 fringes. This is alt 
ridiculous and of very limited interest for those men of feeling who cultivate beauty and 
art. 

“The clever man, supported by science, must dedicate himself to the solution of life’s 
great problems. Wise men, inspired by Allah, the Exalted One, did not erect the dazzling 
edifice of mathematics so that this noble science would be “used in the manner of this 
Persian counting man. It seems to me a crime to reduce the science of Euclid, 
Archimedes, or the extraordinary Omar Khayyam, whom Allah secures in his glory, to 
the pitiful task of merely enumerating things and beings. We are curious to see whether 
this Persian calculator can apply the talents he is said to possess to the solution of genuine 
problems, that is, problems that bear on the demands of everyday life.” 

“I believe you arc slightly mistaken Vizier.” Beremiz replied quickly, “and I would be 
honored if you would allow me to clarify this insignificant error, and therefore I beg the 
generous caliph, our soul and master, to allow me pennission to continue speaking.” 

“It seems to me there is a certain wisdom in Vizier Nahum ibn-Nahum’s criticism,” 
replied the Caliph. “I believe that a clarification of the matter is absolutely required. So 
speak. What you say will shape the opinions of those who listed to you here.” 

There was a long silence in the room. Then the Man Who Counted spoke: “Learned 
men, O King of the Arabs, know that mathematics arose from the awakening of the 
human soul. But it was not bom with utilitarian purposes. The first impulse of this 
science was the desire to solve the mystery of the universe. Its development came, 
therefore, from the effort to penetrate and understand the infinite. And even now, after 
centuries of trying to part the heavy veil, it is the search for the infinite that moves us 
forward. The material progress of man depends on abstract investigations and on present 
day scientists and the material progress of humanity in the future will depend on these 
men of science who work toward purely scientific ends, without considering the practical 
application of their theories.” 

Beremiz paused briefly and then continued, with a smile, “When the mathematician 
makes his calculations or looks for new relations among numbers, he does not look for 
truth with a practical purpose. To cultivate science only for its practical purpose is to 
despoil the soul of science. The theory that we study today, and that appears to us 
impractical, might have implications in the future that are unimaginable to us. Who can 
imagine the repercussions of an enigma through the centuries? Who can solve the 
unknowns of the future with the equations of the present? Only Allah knows the truth. 
And it is possible that the theoretical investigations of today may provide, within one or 
two thousand years, precious practical uses. 

“It is important to bear in mind that mathematics, besides solving problems, 
calculating areas, and measuring volumes, also possesses much more elevated purposes. 
Because it is so valuable in the development of intelligence and reason, mathematics is 
one of the surest ways for a man to feel the power of thought and the magic of the spirit. 

“Mathematics is, in conclusion, one of the eternal truths and, as such, raises the spirit 
to the same level on which we contemplate the great spectacles of nature and on which 
we feel the presence of God, eternal and omnipotent. As I have said, O illustrious Vizier- 



Nahum ibn-Nahum, you have made a slight error. I count the verses of a poem, calculate 
the height of a star, measure the size of a country or the force of a torrent, and thus I 
apply the formulas of algebra and the principles of geometry, without concerning myself 
with the profit I might earn from my calculations and studies. Without dreams or 
imagination, science is impoverished. It is lifeless.” 

The nobles and wise men around the throne were deeply affected by Beremiz’s 
eloquent words. The king approached the Man Who Counted, raised his right hand, and, 
with great authority, exclaimed, “The belief of the scientist-dreamer has triumphed and 
will always triumph over the vulgar opportunism of the ambitious scientist without 
philosophical belief- O word of Allah!” 

On hearing these words, so justly and righteously spoken by the king, the hateful 
Nahum ibn-Nahum bowed, saluted the king, and. without saying a word, left the salon 
with his head bowed. Correct was the poet who wrote: 

Let the imagination soar 
Without illusion, what would life be? 


15 

SQUARED AWAY 



Nuredin, the messenger, returns to the caliph ’s palace. What he learned from a holy 
man. How the poor calligrapher lived. The number square and the chessboard. Beremiz 
discoursed on magic squares. The question of the wise man. The caliph asks Beremiz to 
tell the story of chess. 

Nuredin had no luck in his quest. The calligrapher whom the king wished to question 
on the matter of friendly numbers was nowhere to be found in Baghdad. Telling of the 
steps he had taken to carry out the caliph’s orders, Nuredin spoke as follows: 


“I left the palace with three guards, heading for the Mosque of Ottoman — may Allah 
be praised once more! An old holy man who takes care of the mosque told me that the 
man I was looking for had lived for several months in a house nearby. A few days earlier, 
however, he had left for Basra with a caravan of rug merchants. He also told me that the 
calligrapher, whose name he did not know, lived alone and ventured out very rarely from 
his small and modest dwelling. I thought it wise to visit his old house in case 1 might find 
some clue as to where he had gone. “The house had been abandoned since its previous 
inhabitant departed. Everything indicated the sorriest poverty. A broken bed in a corner 
was its only furnishing. On a rough wooden table, however, there was a chessboard with 
some pieces and, on the wall, a square filled with numbers. I found it strange that a man 
so impoverished, living such a meager life, would be a chess player and would have on 
his wall as decoration mathematical symbols. I decided to bring the chessboard and the 
number square with me so that our worthy sages might study these clues left behind by 
the old calligrapher.” 

The sultan, his interest awakened, ordered Beremiz to take a close look at the 
chessboard and the square, which seemed more appropriate to a follower of al- 
Khwarizmi than to a poor calligrapher. 

After studying both objects carefully, the Man Who Counted spoke as follows: “This 
interesting square of numbers the calligrapher left behind is what we call a magic square. 
Let us take a square and divide it up into four or nine or sixteen equal boxes. Put a 
number in each of these boxes. When the sum of the numbers in any line or column or 
diagonal always adds up to the same result, we have a magic square. The resulting sum is 
called the ‘constant’ of the square, and the number of boxes in any line is the absolute 
value of the square. The numbers in the separate boxes must all be different. It is 
impossible to construct a magic square with only four boxes. 

“No one knows the origin of magic squares. In times past, making them up was a 
favorite occupation for the curious. Just as the ancients attributed magical qualities to 
certain numbers, it was very natural to read magical virtues into these squares. They were 
known to Chinese mathematicians forty-five centuries before Muhammad. In India, many 
people used a magic square as a charm. A wise man of Yemen claimed that magic 
squares could prevent certain diseases. According to certain tribes, a magic square made 
of silver and worn around the neck was a safeguard against the plague. Ancient Persian 
magicians, who also practiced medicine, claimed to cure a sickness .by applying a magic 
square, according to the venerable principle Primum non nocere, or, the first principle is 
not to hurt. 


6 1 

1 8 

75 

mm 4M| 

> 3 

2 $ 

*4 


A magic square of nine numbers. 


“In the field of mathematics, however, magic squares have a curious property. When, 
for example, a magic square can be broken down into other magic squares, it is called 
hypermagical. Certain hypermagical squares are known as diabolicals.” 


4 

5 

16 

9 

14 

11 

2 

7 

1 

8 

13 

12 

IS 

io 

3 

6 


This magic square is one that mathematicians call a diabolical. The constant, 34, is 
obtained not just by adding the numbers in any column, line, or diagonal but also by 
adding four numbers from the square in many different ways. The corner numbers add up 
to 34. In fact, there are 86 different ways of reaching the same total. 

Beremiz’s observations on magic squares were followed closely by the caliph and his 
nobles. A wise old man with bright eyes and a squashed nose but with a warm and merry 
manner, after first praising “the remarkable Beremiz, the Persian,” declared that he 
wished to seek his opinion. . 

His question was as follows: “Is it possible for a man versed in geometry to rind the 
exact relation between the circumference and the diameter of a circle?” 

The Man Who Counted replied thus: “It is not possible to calculate a circumference 
exactly, even when we know the diameter. There should be an exact number, but its 
precise value is unknown to geometers. Ancient astrologers believed that the 
circumference of a circle was three times its diameter, but that was not so. Archimedes 
the Greek found that if the measure of a circumference was 22 cubits, its diameter must 
measure approximately 7 cubits. So, the number sought should be 22 divided by 7. Hindu 
mathematicians do not agree, and the great al-Khwarizmi declared that Archimedes’ law 
was very far from being true in real life.” 

And Beremiz, addressing the sage with the squashed nose, concluded, “That very 
number seems wrapped in mystery, containing qualities that Allah alone can reveal.” 

Then the Man Who Counted picked up the chessboard and, addressing the king, 
spoke, “This old board, divided into sixty-four black and white squares, is used, as you 
know, for an interesting game invented many centuries ago by a Hindu called Lahur 
Sessa to entertain an Indian king. The discovery of chess is bound up with a legend that 
involves numbers, calculations, and notable instruction.” 

“I should like to hear of it,” the caliph declared. “It must be interesting.” 

“I hear and obey,” Beremiz replied. 

And he told the story recounted in the following chapter. 


16 

THE GAME PLAN 



Wherein is told the famous legend of the origin of chess, which Beremiz Samir, the 
Man Who Counted, told to the caliph of Baghdad al-Mutasim Billab emir of all believers. 

It would be difficult to learn, given the uncertainty of ancient documents, the precise 
period when there lived and reigned in India a prince named Ladava, lord of the province 
of Taligana. Nevertheless, it would be fair to say that this monarch is considered by 
various Hindu historians to be one of the richest and most generous sovereigns of his day. 

“War with, its fatal procession of calamities, ruined the life of King Ladava, engulfing 
his royal pleasures with lamentable tribulations. Because it was his royal duty to protect 
the tranquility of his subjects, our good and generous monarch was forced to seize the 
sword of war and, riding at the head of his small army repulse the sudden and brutal 
attack of the adventurer Varangul who was known as the prince of Calian. 

“The violent clash of the rival forces covered the fields of Dacsina with the bodies of 
the dead and made the holy waters of the Sabdhu River run red with blood. According to 
historians, King Ladava possessed a rare military talent. Before the invasion, he calmly 
planned his strategy; he then carried it out so skillfully that he totally destroyed the 
treacherous trespassers who would destroy the peace of his kingdom. 

“Unfortunately, his triumph over the fanatic Varangui required many hard sacrifices. 
Many young soldiers paid with their lives for the safety of the throne and the dynasty. 
Among the dead left on the battlefield, his chest pierced by an arrow, was Prince 
Adjamir, son of King Ladava, who had sacrificed himself at the height of the battle to 
hold the position that secured victory for his country. 


“When the bloody campaign was over and the boundaries of his kingdom secure, the 
king returned to his sumptuous palace in Andra. However, he strictly forbade the 
traditional, noisy demonstrations with which Hindus typically celebrate their victories. 
After retiring to his private rooms, he would emerge to consult with his ministers or 
Brahmin sages only when a grave problem required a decision in the interest of his 
subjects’, welfare. 

“With the passage of time, the memories of the painful campaign had not died but 
grown much worse, plunging the king into agony and sorrow. What good are your rich 
palaces? Your war elephants, the immense treasures you possess, if you have lost the one 
thing that made life worthwhile? What value could material wealth have in the eyes of an 
inconsolable father who can never forget his lost son? 

“The king could not put out of his mind the ebb and flow of the battle in which 
Adjamir had died. The unfortunate monarch passed hour alter hour tracing in a large 
sandbox the movements of his troops during the assault. One furrow indicated the 
advance of the infantry; on the other side, parallel, another line traced the advance of the 
war elephants. A little beneath, in a symmetrical pattern of circles, was the cavalry under 
the command of an old captain who was said to enjoy the protection of Techandra, 
goddess of the moon. In the middle, the king sketched the position of the enemy columns, 
so badly positioned on the field as a result of the king’s own strategy that they were 
easily, decisively defeated. 

“Having completed this sketch of the battle with all the details that he could 
remember, the king erased it all and started again, as if he gained a certain pleasure in 
reliving the past in all its private agony. 

“In the early morning, when the old Brahmins arrived at the palace to hear the reading 
of the Vedas, the king had already traced and erased in the box of sand the order of battle 
that he could not forget or stop retracing. 

‘“Unhappy king,’ murmured his distressed priests. ‘He works like a slave whom God 
has deprived of reason. Only Dhanoutara strong and merciful can save him. “And the 
Brahmins prayed for their king, burning aromatic roots, begging the eternal guardian of 
the sick to help the king of Taligana. 

“Finally, one day, the king was informed that a young Brahmin, humble and poor, was 
asking for an-audience. He had already asked several times, but the king had always 
refused, claiming that his spirit was not strong enough to receive visitors. But this time he 
agreed to the request and ordered that the young stranger be brought into his presence. 

“After arriving at the throne room, the young Brahmin was questioned, according to 
the ritual, by one of the king’s noblemen. 

“‘Who are you? Where do you come from? What do you wish of him who, by the will 
of Vishnu, is king and ford of Taligana?” 

“‘My name,’ replied the young Brahmin, is Lahur Sessa, and I come from the village 
of Namir, a thirty days’ walk from this handsome city. Word reached the place where I 
live that our king is afflicted with a deep sorrow that he is embittered by the loss of the 
son who was snatched from him by the fortunes of war. It would be a terrible thing, I 
thought, if our noble sovereign shut himself off in his palace, like a blind Brahmin 
surrendering to his own pain. Therefore, I thought it might be useful to invent a game that 



could distract him and open his heart to new pleasures. This is the humble gift that 1 
bring to our King Ladava.’ 

“Like all the great princes of this or any other book of history, the Hindu sovereign 
was afflicted by a great curiosity. When he learned that the young Brahmin was offering 
him a new, unknown game, the” king was eager to see and appraise this gift, without 
delay. 

“Sessa brought before King Ladava a board divided into sixty-four squares of equal 
size. On the board were placed, without obvious care or planning, two sets of game 
pieces, one white and the other black. The shaped figures were symmetrically placed on 
the board, and there were curious rules governing their movements. 

“Sessa patiently explained to the king, the nobles, and the courtiers who gathered 
around, the purpose and essential rules of the game: 

“Each player has eight small pieces, called pawns. They represent the infantry that one 
sends to thwart the enemy. Supporting the advance of the pawns are the elephants of war 
represented by larger, more powerful game pieces. The cavalry, absolutely essential in 
combat, also appears in the game, in the form of two pieces that can jump over the others, 
like horses. And to strengthen the attack, there are two lords of the king; two noble and 
respected warriors. Another piece symbolizes the patriotic spirit pf the people and is 
called the queen. This piece can make many movements and is more efficient and 
powerful than the rest. Completing the set is a piece that alone can do little but becomes 
very strong when it is supported by the others. That piece is the king.’ 

“King Ladava was so interested in the rules of the game that he pursued the inventor 
with questions: 

“’And why is the queen stronger than the king?’ 

‘“She is stronger,’ explained Sessa, ‘because in this game the queen represents the 
spirit of the people. The throne’s greatest strength rests in the exaltation of its subjects. 
How could the king repulse the enemy’s attack without a spirit and commitment to 
sacrifice in those around him, -a spirit that protects the sovereignty of the nation?’ 

“After a few hours, the king, who quickly learned all the rules of the game, defeated 
his nobles in a beautifully played game, 

“From time to time, Sessa intervened respectfully to clarify a problem or to suggest 
another plan of attack or defense. 

“At one moment, the king noted with great surprise that the pieces, after all the 
various moves, were deployed in the exact positions of the battle of Dacsina. 

“‘Observe,’ said the young Brahmin, ‘that in order to win this battle this noble must be 
sacrificed...’ 

“And he indicated precisely the piece that the king had placed at the head of the force 
fighting in the thick of the battle. The wise Sessa thus demonstrated that at times the 
death of a prince was required to ensure the peace and liberty of his people. 

“On hearing these words, King Ladava, his spirit overcome with enthusiasm, said, ‘ 1 
did not believe that a clever human could create something as interesting and instructive 
as this game! By moving these simple pieces, I have just learned that a king is worthless 



without the support and dedication of his subjects and that, at times, the sacrifice of but 
one pawn is worth as much as the sacrifice of a powerful piece in winning a great 
victory. ’ 

“And turning to the young Brahmin, he said, ‘I would like to pay you, my friend, for 
this marvelous gift, which has done so much to relieve my former agonies. Therefore, tell 
me what you desire, within the realm of what I can give, so that I can show you how 
grateful 1 can be to those deserving reward. ’ 

“Sessa appeared unaffected by the king’s generous offer. His serene face indicated 
neither excitement nor joy nor surprise. The apparent indifference of the young Brahmin 
astonished the courtiers. 

‘“Magnificent lord!’ the young man replied with both moderation and pride. ‘I desire 
no greater reward for the gift I have brought you than the satisfaction of knowing that 1 
have relieved the ford of Taligana of his infinite sadness. Thus I have already been 
rewarded, and any other prize would be excessive.’ 

“The good king smiled somewhat disdainfully at this answer, as it suggested 
indifference exceedingly rare among the normally greedy Hindus. Not able to believe in 
the sincerity of the young man’s reply, the king insisted, ‘Your disdain and indifference 
toward material things surprises me, voting man. Modesty, when excessive, is like the 
breeze that extinguishes the light and blinds the old man in the long darkness of the night. 
So that man can overcome the obstacles that life places in his path, he must subordinate 
his spirit to an ambition that leads him to a fixed goal. Therefore, you should not hesitate 
to choose a reward commensurate with the value of your gift to me. Would you like a 
sack of gold? Would you like a chest of jewels? What about a palace? Would you accept 
a province of your own to govern? Take care with your answer, and you shall have your 
reward, by my word of honor! ’ 

“‘After what you have said, a rejection of your offer would be more disobedience than 
discourtesy,” Sessa replied. ‘Thus I will accept a reward for the game I have invented. 
The reward should fit your generosity. Nevertheless, I do not wish either gold or lands or 
palaces. I want my reward in grains of wheat.’ 

“‘Grains of wheat?’ exclaimed the king, not hiding his surprise at such an astonishing 
request. ‘How could I reward you in such insignificant currency?’ 

“‘Nothing could be simpler,’ Sessa explained. ‘You give me one grain of wheat for the 
first square on the board, two for the second, four for the third, eight for the fourth, and so 
on, doubling the amount with each square up to the sixty-fourth and last square on the 
board! I beg you, O King, in accordance with your magnanimous offer, to pay me in 
grains of wheat in the manner I have indicated. ’ 

“Not only the king but all the nobles, Brahmins, and everyone present burst out 
laughing at such a strange request. In fact, the young man’s request amazed all those 
more attached to the material things of life than Sessa was. The young Brahmin, who 
could have acquired a province or a palace, wished only for a few grains of wheat. 

“‘Fool!’ exclaimed the king. ‘Where did you leam such ridiculous disregard for 
wealth? The reward you request is absurd. You must know that a handful of wheat 
contains innumerable grains. With just a few handfuls, I could pay what you ask and 
more following your formula of doubling the number of grains with each square on 



the board. This reward you claim would not even satisfy the hunger of the least village in 
my kingdom for more than a few days. But, all right, I gave my word, and 1 shall give 
you exactly what you have asked for.’ 

The king ordered that the most gifted mathematicians of his court be brought before 
him, and he bade them calculate the amount of wheat due the young Sessa. After a few 
hours of deep study, the wise calculators returned to the throne room to present the king 
with their finished calculations. 

“Interrupting the chess game he was playing, the king asked the mathematicians, 
‘How many grains of wheat must I give the young Sessa in order to comply with his 
request? ’ 

‘“Magnanimous King!’ declared the wisest of the mathematicians. ‘We have 
calculated the number of grains of wheat, and we have reached a sum that is beyond 
human imagination. With the greatest care, we have calculated the number of ceiras 
required to hold the appropriate quantity of wheat, and we have arrived at the following 
conclusion: the wheat that you will have to give to Lahur Sessa is the equivalent of a 
mountain with a diameter at its base the size of the city of Taligana and a height ten times 
greater than that of the Himalayas. If all the fields of India were sown with wheat, in two- 
thousand” centuries you would not harvest what you have promised young Sessa.’ 

“How can one describe the astonishment of the king and his illustrious nobles? The 
Hindu sovereign realized, perhaps for the first time in his life, that he could not possibly 
fulfill his promise. 

“According to the historians of that period, Lahur Sessa, like a good subject, had no 
desire to bring grief to his sovereign. After publicly renouncing his request, thus 
liberating the king from his royal duty, he respectfully spoke to his sovereign: 
“‘Consider, O King, the great truth that the wise Brahmins repeat over and over again: 
the most intelligent men are sometimes blinded not only by the deceptive appearance of 
numbers but also by the false modesty of the truly ambitious. Unhappy is he who 
assumes the burden of a debt whose worth cannot be measured by the simple means of 
his own intelligence. Much wiser is he who praises much and promises little!’ 

“And, after a short pause, he continued, ‘We learn less from the vain science of the 
Brahmins than by direct experience of life, whose lessons are so frequently 
underappreciated! The longer a man lives, the more he is beset by moral troubles. One 
moment sad, the next happy, today fevered, tomorrow tepid, now ambitious, now lazy — 
so one’s composure varies. Only the truly wise man, learned in the laws of the spiritual, 
can raise himself above these-troubles and the tickle whims of mood. ’ 

“These words, so unexpected and so wise, “had a profound effect on the kind’s spirit. 
Forgetting the mountain of wheat he had promised the young Brahmin, the king named 
Sessa his first noble. 

“And Lahur Sessa, entertaining the king with clever games of chess and instructing 
him with wise and prudent counsel, poured his blessings on the people and their king, to 
the greater security of the throne and the greater glory of the nation.” 

Beremiz’s story of the origin of chess enchanted the caliph al-Mutasim. He called the 
chief scribe and ordered that the legend of Sessa be written on special sheets of cotton 
paper and stored in a chest of silver. 



Then the generous sovereign considered whether he should give the Man Who 
Counted a cloak of honor or one hundred pieces of gold. 

“God speaks to the world through the hands of the generous.” 

Such generosity on the part of the ruler of Baghdad delighted everyone. The courtiers 
in the room were friends of the vizier Maluf and the poet Iezid. They listened with 
sympathy and accord to the words of the Man Who Counted. 

Beremiz, after thanking the sovereign for the presents he had given him, withdrew 
from the room. The caliph turned to the consideration of his various duties, listening to 
his ministers of justice and passing wise judgment. 

We left the palace at dusk. It was beginning of the month of Shaban. 


17 

THE APPLES AND ANTS 



The Man Who Counted received many inquiries. Beliefs and superstitions. Numbers 
and figures. The historian and the calculator. The case of the ninety applet. Science and 
charity. 

After the famous day when we first sat with the caliph in his throne room, our life 
changed completely. Beremiz’s fame grew extraordinarily. In the modest inn where we 
lived, the guests showered us with greetings and shows of respect. 

Every day the Man Who Counted received dozens of inquiries. A tax collector wanted 
to know how many ratls there were in an abas, and how many of both there were in a 
cate. Then a doctor wanted Beremiz to explain how to cure levers with a cord knotted 
seven times. More than once, camel drivers and incense sellers came asking the counting 
man how many times a man had to jump over a fire to exorcise a demon. At times, at 
dusk, Turkish soldiers with stem faces came to Beremiz and asked for sure ways to win 
at various games of chance. On other occasion, women, hidden behind heavy veils, came 
to consult with the mathematician, asking which numbers they should write on their left 
forearm for good luck, joy or wealth. 

Beremiz Samir received them all with patience and kindness. He gave explanations to 
some and advice to others. He tried to dispel the superstitions of the ignorant and 


demonstrated that, according to the will of God, there is no relation between numbers and 
the joys, sorrows, and anxieties of the heart. 

In all his consultations, he was guided only by a sense of altruism, with no expectation 
of reward. He rejected any money that was offered him, and when a rich sheik whose 
problems Beremiz had solved insisted on paying him, Beremiz took the sack of money, 
thanked the sheik, and ordered that the money be distributed among the poor of the 
district. 

Once a merchant named Aziz Neman came, clutching a paper filled with numbers and 
complaining about a business partner whom he called “a miserable thief,” a “hideous 
jackal,” and other epithets equally insulting. Beremiz calmed the man. 

“You should beware of reckless judgments,” he said, “for these often blind you to the 
truth. He who looks through a colored glass sees everything in the color of the glass. If 
the glass is red, all appears bloodied. If the glass is yellow, all appears honeyed. Passion 
is like a glass before the eyes of man. If someone pleases us, we are all praise and 
forgiveness. If someone displeases us, however, we judge everything he does harshly.” 

Then he patiently proceeded to examine the accounts on the paper and found there 
various errors that made the sums incorrect. Aziz realized that he had been unfair to his 
partner and was so pleased with the intelligent, thoughtful manner of Beremiz that he 
invited us to stroll through the city with him that night. 

Our companion took us to the Cafe Bazarique in the Plaza of Ottoman. There a 
famous historian, sitting in a room filled with heavy smoke was captivating the patrons 
with his tales. 

We were fortunate to arrive just at the moment that Sheik el-Medah was finishing his 
introductory speech and beginning his story. He was a man of about fifty, with a very 
dark complexion, jet black beard, and sparkling eyes. Like all the tale-tellers of Baghdad, 
he was wearing a large white cloth tied around his head with a cord of camel’s hair, and 
this garb gave him the majesty of an ancient priest. Sitting hi the midst of his rapt 
audience, he spoke in a loud, quavering voice to the accompaniment of a lute and a drum. 
The men in the cafe hung on his every word. The sheik’s gestures were so dramatic, his 
voice so eloquent, his face so expressive that at times it seemed that he had actually 
experienced the adventures that he invented. When he spoke of a long journey, he 
affected the slow rhythm of a tired camel. At other times, “he took on the deep fatigue of 
a Bedouin searching for a drop of water. Sometimes he would let his head and shoulders 
droop as if he were a man completely sunk in despair. 

In the words of the tale-teller, one could see Arabs, Armenians, Egyptians, Persians, 
and bronze nomads of Hejas. How richly he deserved our admiration, this clever, 
intelligent man who, with soulful eyes, projected the deep sentiments hidden behind the 
savage visage! The teller of tales moved from one side to the other, then back to the 
center, then covered his face with his hands, and then threw his arms to the sky, and, just 
at the moment that he tore the air with his words, the musicians raised a bright flurry of 
thundering sound. 

When he finished, the applause was deafening. Then the customers began to speak 
among themselves about the most dramatic episodes in the teller’s tale. 



The merchant Aziz Neman, who seemed to be very popular in this noisy company, 
moved to the center of the room and spoke to the sheik in a deliberate and solemn tone: 
“We have among us tonight, O brother Arabs, the celebrated Beremiz Samir, the Persian 
calculator, secretary of the Vizier Maluf.” 

Hundreds of eyes turned to Beremiz, whose presence did honor, to the patrons of this 
cafe. 

The storyteller, after respectfully saluting the Man Who Counted, spoke in a clear, 
well modulated voice; “Friends! I have told many marvelous stories of kings and genies, 
good and bad. In honor of the illustrious calculator who joins us this evening, I am going 
to tell a story that contains a problem to which a solution has never been found.” 

“Very good, very good!” shouted the audience. 

After evoking the name of Allah — to whom all praise and glory! — the sheik told his 
story. 

“There once lived in Damascus an enterprising peasant who had three daughters. One 
day, the peasant told a qadi, a judge, that his daughters were not only very intelligent but 
blessed with rare skills of the imagination. The qadi a jealous and stingy man, was 
annoyed at hearing a peasant speak so praisingly of his daughters’ talents, in response he 
declared, ‘This is the fifth time that you have told me, in such exaggerated phrases, how 
wise your daughters are. I am going to summon them to my chambers m see for myself 
whether they are as clever as you claim they are.’ 

“The qadi had the three girls brought before him. Then he said to them, ‘Here are 90 
apples for you to sell in the market. Fatima, the oldest, you will take 50; Cunda, you will 
take 30; and Shia, the youngest, you will take 10. If Fatima sells the apples at a price of 7 
to the dinar, you other two will have to sell yours at the same price. And if Fatima sells 
her apples for 3 dinars each, you two will have to do the same. But, no matter what you 
do, each of you must end up with the same amount of money from your different 
numbers of apples.’ 

“‘But can I not give away some of the apples that I have?’ Fatima asked. ‘Under no 
circumstances,’ said the wretched qadi. These are the terms: Fatima must sell 50 apples. 
Cunda must sell 30 apples. And Shia must sell the 10 apples that remain. And all of you 
must sell your apples at the same price, and all of you must earn exactly the same profit 
in the end.’ 

“The dilemma of the three sisters was, of course, an absurd one. How could they 
possibly solve it? Although they all had to sell their apples at the same price, the sale of 
50 apples would vastly exceed the sale of 30 or 10 apples. 

“Because the girls did not know how to resolve the problem, they went to a holy man 
who lived in their district. After filling many pages with calculations, the holy man 
concluded thus: 

“‘Girls, the solution is crystal clear. Sell the 90 apples as the qadi has ordered, and you 
will each receive the same profit.’ 

“The holy man gave the three sisters directions that did not seem to resolve the 
problem of the 90 apples at all. But the sisters went to the market and sold their apple as 
instructed. That is, Fatima sold 50, Cunda sold 30, and Shia sold 10, all at the same 



price — and each received exactly the same amount as profit. That is the end of the story. I 
leave it to the calculator to explain how the holy man solved the problem.” 

Hardly had the storyteller finished than Beremiz spoke to the assembled customers. 

“Problems presented in the form of a story are always extremely interesting because 
the story so beautifully disguises the real problems of mathematical logic. The solution to 
the problem with which the qadi of Damascus tormented the three sisters is the 
following: 

“Fatima starts selling her apples at a price of 7 apples for 1 dinar. She sells 49 of her 
apples at this price, but keeps back 1 . 

“Cunda sells 28 of her apples at this price, but keeps back 2. 

“Shia sells 7 of her apples at this price, but keeps back 3. 

“Then Fatima sells her 1 remaining apple for 3 dinars. In accordance with the rules of 
the qadi, Cunda then sells her 2 remaining apples for 3 dinars each. And Cunda then sells 
her 3 remaining apples for 3 dinars each. 

“Thus: 


FATIMA 

First phase: 

49 apples bring a profit of 7 dinars 

Second phase: 

1 apple brings a profit of 3 dinars 

Total: 

50 apples bring a profit of 10 dinars 

CUNDA 

First phase: 

28 apples bring” a profit of 4 dinars 

Second phase: 

2 apples bring a profit of 6 dinars 

Total: 

30 apples bring a profit of 10 dinars 

SHIA 

First phase: 

7 apples bring a profit of 1 dinar 

Second phase: 

3 apples bring a profit of 9 dinars 

Total: 

10 apples bring a profit of 10 dinars 


“Therefore, each made a profit of 10 dinars; and thus the problem set by the envious 
qadi of Damascus was solved.” 

May Allah condemn the wretched and reward the good! 

Then Sheik el-Medah, overjoyed with the solution Beremiz provided, exclaimed, 
raising his hands to heaven, “By the Second Coming of Muhammad! This young 
counting man is truly a genius! - This is the first man I have met who, without resorting to 
complicated explanations, perfectly solved the problem of the qadi! ” 

As one, the multitude gathered in the cafe joined in the sheik’s exaltations. 

“Bravo! Bravo! Allah praise the wise young man!” 


After silence the noisy crowd, Beremiz continued, “My friends. I must protest that I do 
not deserve the honorable title of wise man. The man is not wise who merely diminishes 
ignorance. What value can the science of man have when compared with the science of 
God?” 

And before anyone could reply, Beremiz began the following story: 

“There was once an ant that, traveling across the face of the earth, came upon a 
mountain of sugar. Very happy with his discovery, he removed one grain of sugar from 
the mountain and took it to his anthill. ‘What is this?’ his neighbors asked. ‘This, the vain 
ant replied, ‘is a mountain of sugar. I found it in my path and decided to bring it home to 
you.’” 

And Beremiz added. With fierceness much out of character with his usual tranquility, 
“This is the wisdom of the proud — to find a mere crumb and call it the Himalayas. 
Science is a great mountain of sugar, and from this mountain we satisfy ourselves with 
mere morsels.” 

And then he said with great determination, “The only science of value to ma nk ind is 
the science of God.” 

A sailor from Yemen asked. “O great calculator, what is the science of God?” 

“The science of God is kindness and generosity.” 

At that moment I remembered the admirable verses spoken” by Telassim in the garden 
of Sheik Iezid when the birds were set free: 


Though I speak with the tongues of men 
and of angels, 
and have not charity, 
lam become as sounding brass, 
or a tinkling cymbal, 
lam nothing 
lam nothing 

At midnight, when we left the cafe, several men offered to accompany us with their 
lanterns through the dark night and the twisted alleys, and soon we were lost. I looked up 
to the heavens, and there, high in the darkness, shining in the midst of the bright caravan 
of stars, was the unmistakable star Sirius. 

Allah! 


18 

A PERILIOUS PEARL 



Which tells of our return to Sheik Iezid’s palace. A meeting of poets and learned men. 
An homage to the maharaja of Lahore. Mathematics in India. The appealing legend of 
“The Pearl of Lilavati. “ The great treatise that Hindus wrote, on mathematics. 

The following day, at an early hour, an Egyptian arrived at our humble inn with a 
letter from the poet Iezid. 

“It is vet very early for “giving classes,” said Beremiz patiently. “I’m afraid my 
pupil may not be prepared.” 

The Egyptian explained to us that the sheik, before the mathematics lesson, wished to 
introduce the Persian to a group of friends, and so it would be convenient for us to arrive 
as early as possible at his palace. 

This time, as a precaution, we took with us three black slaves, strong, determined men, 
for it was very possible that the fearsome and envious Tara Tir might attack us on the 
way, in an attempt to kill Beremiz, whom he saw as his hated rival. 

Nothing untoward happened, and an hour later we arrived at the sumptuous dwelling 
of Sheik Iezid. The Egyptian servant showed us through an endless gallery into a rich 
reception room, blue, with gold friezes. We followed him in silence, not without some 
disquiet on my part at the suddenness of our summons. 

There we found the father of Telassim, surrounded by some poets and learned men. 

“Peace be with you!” 

We exchanged greetings. The master of the house addressed us in the friendliest 
fashion and bade us be seated. We disposed ourselves on soft silken cushions, and a black 
slave brought us fruit, pastries, and rose water. 

I realized that one of the guests seemed to be a foreigner and was dressed with 
exceptional luxury. He wore a tunic of white Genoa silk, tied with a blue sash studded 
with gems, and a splendid dagger encrusted with sapphires and- Tapis lazuli. His turban 
was of pink silk decorated with black threads and precious stones. The glitter of fine rings 
on his delicate fingers highlighted the olive skin of his hand. 



“Noble man of numbers,” said Sheik Iezid, addressing Beremiz. “I know you must be 
surprised by this gathering I have summoned to my modest house. I must tell you, 
however, that its purpose is to pay homage to our illustrious guest. Prince Cluzir el din 
Mubarek Shah, ford of Lahore and Delhi.” 

Beremiz, inclining his head, greeted the young man in the jeweled sash, who was the 
maharaja. 

We were well aware, from the gossip of foreigners at our inn, that the prince had left 
his rich domain in India to fulfill one of the obligations of a good Muslim — namely, to 
make a pilgrimage to Mecca, the Pearl of Islam. He was spending a few days in Baghdad, 
and would leave very soon for the Holy City with his countless servants and aides. 

“We are very eager,” Sheik Iezid went on, “that you help us clear up a question raised 
by Prince Cluzir Shah. What have Hindus contributed to the advancement of 
mathematics, and who are the Indian geometers who have made outstanding 
contributions to the study?” 

“Generous Sheik!” replied Beremiz. “I feel that the task you present me with requires 
both learning and serenity — learning, to know in detail the history of science, and 
serenity, to analyze and assess it with some discerning. However, O Sheik, your least 
desires are as commands to me. I shall recount to this distinguished company, as small 
homage to Prince Cluzir Shah, what little I have learned about the development of 
mathematics in the country of the Ganges.” 

And so he began, “Nine or ten centuries before Muhammad, there lived in” India a 
distinguished Brahmin known as Apastarnba. This sage, in order to instruct priests on the 
building of altars and the design of, temples, wrote a work called the Sulbasutra, which 
contains numerous mathematical examples. It is most unlikely that this work was 
influenced by Pythagorean theory, since the Hindu sage does not follow Greek methods 
of investigation. In his pages, however, there are various theorems and rules of their 
construction. To illustrate the building of an altar, Apastarnba proposes - Drawing a right- 
angled triangle, whose sides measure. 39, 36, and 15 inches respectively. To resolve the 
problem, he applies a principle attributed to Pythagoras the Greek: 

The area of a square drawn on the hypotenuse is equal to the sum of squares drawn on 
the two adjacent sides.” 

And. turning to Sheik Iezid, who was listening attentively, Beremiz spoke as follows: 
“It would be easier to explain this well-known proposition by a diagram.” 

Sheik Iezid signaled to his servants, and in a moment two slaves brought in a large 
sandbox, on the smooth surface of which Beremiz was able to draw figures and outline 
his calculations for the prince of Lahore. Beremiz drew his diagram with a bamboo stick. 

“Here is a right-angled triangle. Its longest side is called the hypotenuse. Let us now 
draw a square on each of the three sides. Thus it will be easy to prove that the large 
square drawn on the hypotenuse is exactly equal in its area to the sum of the other two 
squares, so proving the truth of Pythagoras’ principle. 

The Prince asked whether the same principle was valid for all triangles. Beremiz 
replied gravely, “It is true and constant for all right-angled triangles. I will say, without 
fear of error, that Pythagoras’ law expresses an eternal truth. Even before the sun shone 



on us, even before there was air to breathe, the square of the hypotenuse was equal to the 
sum of the squares on the other two sides.” 



Fascinated by Beremiz’s explanation, the prince addressed the poet Iezid warmly: 
“How wonderful is geometry, my friend! What a remarkable science! In its lessons we 
perceive two things that move even the humblest and most unthinking of men — clarity 
and simplicity.” 

And, lightly touching Beremiz on the shoulder with his left hand, he asked him, “And 
does this Greek proposition appear in the Sulbasutra of Apastamba?” Beremiz did not 
hesitate. 

“Oh yes, my Prince!” he said. “You will find the so-called Pythagorean theorem in the 
Sulbasutra in a somewhat different form. It was through reading Apastamba that priests 
learned how to construct oratories, turning right angles into their equivalent squares.” 

And did other notable works of calculation come out of India? 

“Quite a number,” Beremiz replied. “I would mention the strange work Suna Sidauta, 
remarkable although by an unknown author. It lays down very simply the laws governing 
decimals, and shows the supreme importance of zero to the calculator. No less important 
were the writings of two Brahmin sages much admired today by men of numbers: 
Aryabhata and Brahmagupta. Aryabhata’s treatise is in four parts: on celestial harmonies, 
on time and its measures, on spheres, and on the elements of calculating. Errors in 
Aryabhata’s writings were not rare — they teach, for example, that the volume of a 
pyramid can be found by multiplying half the base by its height.’ 

“And that is not true?” asked the prince. 

“Completely erroneous,” Beremiz replied. “To discover the volume of a pyramid, we 
must multiply not a half but a third of the base area by its height.’” 

At the prince’s side sat a tall, thin man, richly dressed, with red hair showing in his 
gray beard, who did not by his appearance seem Hindu. I thought he was a tiger hunter, 
but I was mistaken. He was a Hindu astrologer who was accompanying the prince on his 
pilgrimage to Mecca. He wore a blue turban, wound around three times, somewhat 
ostentatiously. He was called Sadhu Gang, and he seemed extremely interested in 
Beremiz’s words. 




At one point Sadhu, the astrologer, decided to enter the debate. Speaking in an 
awkward foreign accent, he asked Beremiz, “Is it true that geometry in India was studied 
by a wise man who knew the secrets of the stars and the deepest mysteries of the 
heavens?’ 

After thinking a moment, Beremiz took up his bamboo stick, cleared off the surface of 
the sandbox, and wrote a single name: 

Bhaskara the Learned 

And he said solemnly, “That is the name of India’s most famous geometer. Bhaskara 
knew the secrets of the stars and studied the deepest mysteries of the heavens. He was 
born in Bidom, in the province of Decan, five centuries after Muhammad. His first text 
was called Bijaganita. 

“Bijaganita said the man in the blue turban. “Bija means seed, and ganita in one of 
our ancient dialects, means ‘to count’ or ‘to measure.’” 

“Just so,” nodded Beremiz. “The best translation of that title would be “The Art of 
Counting Seeds.’ Apart from Bijaganita, Bhaskara the Learned wrote another famous 
work, Lilavati, which, as we know, was the name of his daughter.” 

The astrologer in the blue turban broke in, “It is said that there is a legend surrounding 
Lilavati. Do you know it?” 

“Indeed I do,” replied Beremiz, “and if our prince agrees, I could tell it.” 

“With much pleasure,” exclaimed the prince of Lahore. “Let us hear the legend of 
Lilavati! I am sure it will prove most absorbing.” 

At that moment, at a sign from Sheik Iezid, five or six slaves appeared in the room and 
handed around to the guests stuffed pheasants, milk cakes, fruit, and refreshments. When 
we had finished the delicious repast and made our ritual ablutions, the Man Who Counted 
was asked to proceed with the story. 

Beremiz raised his head, looked at all those who were present, and began to speak. 

“In the name of Allah, the Wise and Merciful! Bhaskara the Learned, the famous 
geometer, had a daughter called Lilavati. When she was bom, the astrologer consulted the 
heavens and, by the disposition of the stars, ascertained that she was destined to remain a 
maid for her whole life, passed over by the love of fine young men. Bhaskara did not 
accept such a determination of destiny and consulted the most famous astrologers of the 
day. How could the gentle Lilavati find a husband and make a happy marriage? 

“One astrologer advised Bhaskara to take his daughter to the province of Dravira, 
close to the sea. In Dravira there is a temple carved out of stone with a Buddha figure 
holding a star in his hand. Only in Dravira, the astrologer swore could Lilavati find a 
husband, but the marriage would be a happy one only if the wedding ceremony took 
place on a certain day marked on the cylinder of time. 

“To her pleasant surprise, Lilavati’s hand was sought in marriage by a rich voting 
man, honest, hardworking, and of a high caste. The day was fixed and the hour marked, 
and friends came together to attend the ceremony. 

“Hindus measured and determined the hours of the day with the help of a cylinder 
placed in a vase filled with water. The cylinder, open at the top, had a small hole in the 


center of its base. As the water entered slowly through the hole, the cylinder sank in the 
vase until it filled it completely, at a predetermined hour. 

“Bhaskara placed the cylinder of the hours in position with the greatest care and 
waited for the water to arrive at the level marked. His daughter, impelled by an 
irresistible curiosity common to women, wished to observe the rising of the water in the 
cylinder and leaned over it to watch. One of the pearls on her dress came loose and fell 
into the vase. By mischance, the pearl, forced by the water, blocked the small hole in the 
cylinder, as the astrologer had foretold. The groom and the guests withdrew to fix, after 
consulting the stars, another day for the ceremony. Some weeks thereafter, the young 
Brahmin who had asked Lilavati’s hand in marriage disappeared, and Bhaskara”s 
daughter remained a maid forever. 

“Realizing that it is fruitless to fight against destiny, the wise Bhaskara said to his 
daughter, I will write a book that will perpetuate your name, and you will live in the 
memory of men for a much longer time than the life of the children who would have been 
born from your ill-fated marriage.’ 

“Bhaskara ’s book enjoyed great fame, and the name of Lilavati continues to be 
immortal in the history of mathematics, what mathematicians refer to as Lilavati is a 
methodical demonstration of decimal enumeration and of the arithmetic operations of 
whole numbers. It makes a minute study of the four operations — the problems of 
elevation, of the square, of the cube, and of the extraction of square roots; and it goes on 
to study the cube root of any number Then it tackles fractions with the well-known rule 
of reducing them to a common denominator. 

In enunciating these problems, Bhaskara used an elegant and even romantic style. 
Here is one from his book: 

Much-loved Lilavati, whose eyes are softer than the gentle gazelle ’s, tell me which is 
the number that comes from multiplying 135 by 12. 

“Another interesting problem in the book refers to the calculations surrounding a 
swarm of bees: 

A fifth part of a swarm of bees came to rest on the flower of Kadamba, a third on the 
flower of Silinda. Three times the-difference between these two numbers flew over a 
flower of Krutaja, and one bee alone remained in the air, attracted by the perfume of a 
jasmine and a bloom. Tell me, beautiful girl, how many bees were in the swarm. 

The answer to that is 15. Bhaskara showed in his book that the most complex 
problems can be presented in a lively, even elegant form.” 

And Beremiz continually drawing in the sand, showed the prince of Lahore a number 
of strange problems culled from Lilavati. 

Unhappy Lilavati! 

And saying the name of the unfortunate girl, I remembered these verses of the poet: 


As the ocean surrounds the earth, 
so do you, my lady, 
surround the heart of the world 


with the abyss of your tears. 


19 

SAILOR’S CHOICE 



Prince Cluzir Shah praises the Man Who Counted. Beremiz was the problem of the 
three sailors and discovery the secret of the medallion. The generosity of the maharaja of 
Lahore. 

Beremiz’ s praise of Hindu science, and its place in the history of mathematics, made a 
great impression ort Prince Cluzir Shah. The young sovereign said that he considered the 
Man Who Counted a very wise man, one capable of teaching the algebra of Bhaskara to a 
hundred Brahmins. 

“The story of the unfortunate Lilavati, who lost her groom by one pearl of her gown, 
was enchanting,” he added. “The problems of Bhaskara of which the counting man so 
eloquently spoke indicate the poetic spirit that is so often missing from mathematical 
works. It is unfortunate, however, that the illustrious mathematician did not mention the 
famous problem of the three sailors, which is included in so many books and to which no 
solution has-ever been found.” 

“Magnanimous Prince.” replied Beremiz “I did not mention this problem for the 
simple reason that I have only a vague knowledge of it and its famous difficulty.” 

“I know it quite well,” said the prince, “and would take great pleasure in repeating this 
mathematical problem, which has so preoccupied algebraists.” 

And Prince Cluzir Shah told the following story: 

“A ship that was returning from Serendib with a cargo of spices was struck suddenly 
by a violent storm. The ship would have been destroyed by the furious waves had it not 
been for the bravery of three sailors who, in the midst of the storm, manipulated the sails 
with exceptional skill. 

“Wishing to reward the brave sailors, the ship’s captain gave them a certain quantity 
of coins. The number was between 200 and 300. The coins were placed in a chest so that, 


when the ship reached port the following day, the tax collector could divide the sum 
among the three sailors. 

“But during the night one of the three sailors woke and thought, ‘It would be better if 
I took my part right now. That way 1 would not have to quarrel with my two friends over 
the money.’ Without saying anything to the other two sailors, he got up and found the 
chest. He divided the money into three parts, but they were not perfectly equal. One coin 
was left over. Because of this one miserable coin, he thought, we will certainly quarrel 
tomorrow morning. Better to throw it away. Then he tossed it into the sea and quietly 
returned to his bed. 

“He took his own portion of the reward with him and left the portions of the other two 
in the chest. 

“An hour later, the second sailor had the same idea. He went to the chest and, not 
knowing that one of his fellows had already withdrawn his portion, he divided the money 
into three equal parts. Once .again, there was one coin left over. In order to avoid any 
quarreling in the morning, the second sailor did as the first had and tossed the extra coin 
into the sea. Then he returned to his bed with the portion of money he believed was 
rightfully his. 

“The third sailor had exactly the same idea. Not knowing that his two fellows had 
already acted, he rose in the early hours of the morning, went to the chest, and divided the 
money into three parts: Again there was one coin left over after the division, and again he 
chose to throw it into the sea. Then the third sailor took his one-third portion and happily 
went back to bed, 

“The following morning, when the ship docked, the tax collector found a handful of 
coins in the chest. He divided the coins into three equal parts and gave one to each of the 
three sailors. But again the division was not exact. One coin was left over, which the tax 
collector kept as payment for his services. Of course, none of the three sailors complained 
about the division, because each believed he had already received his just share. 

“So here is the problem: How many coins were in the chest at the beginning? And my 
many coins did each safer receive?” 

The Man Who Counted, seeing that the prince’s story had excited great curiosity 
among the gathered nobles, decided that he should give a complete explanation of the 
problem and its solution. Thus he spoke: 

“The number of coins which, as you said, were between 200 and 300, must have been 
24 1 before they were divided by the first sailor. 

“The first sailor then divided them into three equal parts, throwing one into the sea: 

241 + 3 = 80 coins leaving 1 

“He took his one-third part and then returned to bed, leaving in the chest 

241-(80+l) =160 coins 

“The second sailor then divided the 160 coins left in the chest into three parts with one 
left over, which he threw into the sea: 

160 ^ 3 = 53 coins leaving 1 

“Then he put his one-third part in his pocket and went back to bed, leaving in the chest 



160 - (53 + 1) = 106 coins 

“The third sailor then divided the- 106 coins into three equal parts with one left over, 
which he threw into the sea: 

106 ^ 5 = 35 coins, leaving 1 

“Then he returned to bed with his one-third part, leaving in the chest 

1 06 - (35 + 1) = 70 coins 

“This was the number of coins left when the ship docked and the tax collector, 
following the instructions of the ship’s captain, divided the coins equally in three parts 
with one left over. 

70 -^5 = 23 coins, leaving 1 

“The tax collector thus gave 23 coins to each sailor and kept the one left over for 
himself. Therefore, the division of the original 241 coins went like this: 


First Sailor 80 + 23 103 

Second Sailor 80 + 23 76 

Third Sailor 35 + 23 58 

Tax Collector 1 

The Sea 3 

Total 241 


Having solved the problem, Beremiz fell silent. 

The prince of Lahore took a silver medallion from his pocket and, turning to the Man 
Who Counted, said, “Because of the clear and simple solution you have given to the 
problem of the three sailors, I see that you might also be capable of explaining even more 
intricate numerical problems. 

“This piece,” the prince continued, “was engraved by a religious artist who lived for 
some time in the court of my grandfather. On the piece, the engraver set a puzzle, that, 
until now no magician or astrologer has been able to solve. On one side of the medallion, 
the number 128 is surrounded by seven small rubies. On the other side, divided into four 
parts, are the numbers 

7, 21, 2, and 98 

“As one can see, the sum of the four numbers is 128. But what is the meaning of the 
four parts into which the number 1 28 is divided?” 

Beremiz took the medallion from the prince’s hand. For a time he examined the 
medallion in silence and then spoke as follows: 

“This medallion, O Prince, was engraved by a man well steeped in numerical 
mysticism. The ancients believed that certain numbers had magical powers. The number 
3 was considered divine and 7 sacred. The seven rubies surrounding the number 128 
show that the engraver was preoccupied by the relation between the number 128 and the 
number 7. The number 128, as we know, can be divided into 7 multiplications of 2: 

2x2x2x2x2x2x2 

“The number 128 can also be divided into four parts: 



7, 21, 2, and 98 

which suggests the following properties: add 7 to the first, subtract 7 from the second, 
multiply the third by 7, divide the fourth by 1, and you will get the same result: 

7+ 7 = 14 

21 - 7=14 

2 x 7=14 

98=7=14 

“This medallion must have been used as a talisman, for it contains relations that 
depend on the number 7, which was considered sacred.” 

The prince of Lahore was so enchanted with Beremiz’s explanation that he offered, as 
a reward, not only the medallion but also a sack of gold coins. The prince was generous 
and good. Then we filed into a grand salon, where Sheik Iezid offered his guests a 
splendid banquet. Bit by-bit, Beremiz’s esteem was growing, proving that he was 
destined to occupy a distinguished position much greater than one of his poor 
upbringings could have expected. 

Some of the guests could not hide their disappointment. As for me, I was of no 
consequence. 


20 

THE POWER OF TEN 



Beremiz gives his second mathematics lesson. Numbers and the idea of numbers. 
Figures. Number systems. Decimal numbers. Zero. We hear again the exquisite voice of 
the invisible pupil. Doreid, the grammarian, quoted a poem. 

The repast concluded, the Man Who Counted rose at a sign from Sheik Iezid. It was 
the appointed hour for his second mathematics lesson, and his invisible pupil was waiting 
for her teacher. 

After taking his leave of the prince and the attendant sheiks, Beremiz, accompanied by 
a slave, made his way to the room set aside for the class. I rose also and went with him, 
since I wanted to take full advantage of the permission that had been granted me to at end 
the lessons Beremiz was giving to young Telassim. 


One of the guests,, the grammarian Doreid, a friend of the household, also showed an 
interest in attending; taking leave of the prince, he followed us- He was middle-aged, of a 
cheerful disposition, with a fine, expressive face. 

We passed through an elegant gallery covered in Persian carpets, and. shown the way 
by a startlingly beautiful Circassian slave, we reached the room where Beremiz was to 
give his class. The red carpet that had hidden Telassim some days before had been 
replaced by a blue one with a starred heptagon in its center. — The grammarian Doreid 
and I sat in a corner of the room close to a window opening on the garden. Beremiz 
settled himself, as on the first occasion, in the center of the room on an ample silk 
cushion. By his side, on a small ebony table, lay a copy of the Koran. The Circassian 
slave and another Persian with soft, smiling eyes stationed themselves beside the door. 
The Egyptian slave, Telassim ’s personal guard, leaned against a column. After a prayer, 
Beremiz began. 

“We do not know when the idea of numbers first arose. The researches of philosophers 
go back to a time hidden from us by the clouds of the past. 

“Those who study the evolution of numbers show that, even among primitive men, 
human intelligence had a special faculty that we shall call the sense of number. That 
faculty allows us to know, in a purely visual way, whether a group of things has grown or 
diminished — that is, whether it has undergone numerical change. 

“The sense of number should not be confused with the ability to count. Human 
intelligence alone can achieve the level of abstract on that we call the sense of number, 
while the ability to count can be observed in many animals. Some birds, for example, are 
able to count the eggs they lea e in the nest, distinguishing between two and three, and 
certain wasps can tell the difference between five and ten. 

“Tribesmen of North Africa knew all the colors of the rainbow and named each one, 
but the tribe had no word for color. In the same way many primitive languages have 
words for one, two, three, and so on, but no separate word for number. 

“Where does the idea of number come from? 

“We do not know how to answer that, my lady. In the desert, a Bedouin sees a caravan 
in the distance, moving slowly. The camels approach with their burdens of men and 
goods. How many camels are there? We have to answer that question with a number. 
Forty, perhaps? A hundred? To give an answer, the Bedouin must do something special; 
he must ‘count.’ In order to count, the Bedouin must connect each object in the series 
with a certain symbol: one, two, three, four, and so forth. To arrive at a result of his 
counting — or, in other words, a number — the Bedouin needs to invent a ‘number 
system.’ 

“The oldest number system is the quinary, the system that groups its units in fives. 
Each group of five units is called a quine. Eight units are one quine plus three, written as 
13. It has to be made clear that in this system the figure on the left is worth five times 
what it would be if it were on the right. As mathematicians put it, this system has a base 
of 5. There are traces of this system in the poems of antiquity. 

“The Chaldeans had a number system with a base of 60. In ancient Babylon, the 
symbol 1.5 stood for the number 65. 



“A number of people also used a system with a base of 20. In this system, our number 
90 would be written as 4. 1 — that is to say, four twenties plus ten. 

“Following this, my lady, came the system with a base of 10, much more advantageous 
for working with greater numbers. The origins of that system come from the number of 
fingers on two hands. In certain kinds of trading, we find a distinct preference for a base 
of 12, the system that counts by dozens, half dozens, quarter dozens, and so on. Twelve 
has a considerable advantage over 10 in that it has more divisors. 

“The system with a base of 10, the decimal system, has been universally adopted. 
From the Tuareg who counts on his fingers to the mathematician with his calculators, all 
count in tens. Given the profound differences between peoples such universality is 
surprising: no religion, moral code, form of govermnent, economic plan, philosophical 
structure, language, or alphabet can boast of anything like it. Counting is one of the few 
matters about which men do not differ. They consider it both simple and natural. 

“If we take note, my lady, of savage tribes and of the ways of children, it becomes 
obvious that the fingers are the basis of our-number system. Using our ten fingers, we 
begin to count in tens, and our whole system is founded on groups of ten. 

“Very probably, the shepherd who at evening had to be sure that all his sheep had 
entered the fold had to count beyond the first ten sheep. As the sheep passed him, he 
counted each on a finger, and, after each ten, he let fall a stone. When his count was 
complete, the stones stood for the number of complete hands or tens of sheep. On the 
following day, he could repeat his count by counting the pile of stones. Later on, some 
mind capable of abstraction discovered that this could equally apply to other useful things 
like fruit, wheat, days, distances, and stars. And when, instead of using stones, we made 
distinct and lasting marks, a system of written numbers came into being. 

“All peoples used the decimal system in their spoken language. Other systems were 
forgotten. But the adaptation of that system to written numbers came about very 
gradually. It took several centuries for humanity to find a perfect solution to the problem 
of writing down numbers. To stand for numbers, man had to think up special characters 
called figures or ciphers, each one of which stood for one, two, three, four, five, six, 
seven, eight, or nine. Other extra characters like d, c, and m showed that the 
accompanying number stood for ten, a hundred, a thousand, and so forth. Thus a 
mathematician of antiquity wrote the number 9,765 as 9m7c6d5. The Phoenicians, the 
most assiduous traders of antiquity, used accents instead of letters: 9”’7”6’5. 

“At first, the Greeks did not use this system. Instead, they gave a value to each letter 
of the alphabet, adding an accent to it. The first letter, alpha, was 1 ; the second, beta, was 
2; the third, gamma, was 3; and so on up to 19. Six was an exception and had its own 
sign, sigma- The letters were combined in pairs: 20, 22, and so on. 

“In the Greek system, the number 4,004 was represented by two figures, the number 
2,022 by three, and the number 3,333 by four distinct figures. 

“The Romans showed less imagination, using three characters — I, V. and X — to form 
the first ten numbers and. in combination with them, L (50), C (100), D (500), and M 
(1,000). Numbers written in Roman figures are thus absurdly complicated and were most 
unsuitable for the simplest arithmetic, making even small calculations a torture. Roman 
numbers could be added, but they had to be written one below the other in such a way 



that the figures with the same terminal letter fell in the same column, making it necessary 
to leave spaces between figures. 

“So stood the science of numbers until some four hundred years ago, when a Hindu, 
whose name has been lost, thought up a special sign, the zero, to indicate in a written 
number the absence of any of the decimal numbers. Thanks to this invention, all special 
signs, letters, and accents were rendered unnecessary. There remained only the nine 
numbers and zero. The possibility of writing any number at all by using these ten 
characters alone was the first great miracle brought by the zero. 

“Arab geometers adopted the Hindu’s invention and discovered that, by adding a zero 
to the right of a number, it was automatically raised to a higher decimal order. They made 
the zero into a device for instant multiplication by ten.” 

“Traveling the long and luminous path of science, we must always keep before us the 
wise counsel of the poet and astronomer Omar Khayyam, may Allah praise him! Here is 
what he taught: 

Let your wisdom cause no distress to your neighbor. Keep a close *watch over 
yourself and never give yourself up to rage. If you wish for peace, smile at the fate that 
wounds you. Do ill to no one. 

“Here, evoking a famous poet, 1 end these few remarks on the origin of numbers and 
figures. If Allah is willing, next time we shall consider the principal operations of 
numbers, along with their properties.” 

Beremiz fell silent. The second class had concluded. 

Give me, O Lord the strength for my love to bear fruit and be of use. 

Give me the strength never to look down on poverty or bend my knee to insolence. 
Give me the strength to raise my spirit to the Heights, above the small matters of the day. 

Give me the strength to bow with love before thee. 

I am no more than a fragment of cloud wandering useless in the sky, O glorious Sun! 

If it be thy desire or pleasure, take my nothingness, paint it in a thousand colors, 
irradiate it in gold, make it waver in the wind and spread itself across the sky in a 
thousand wonders. . . . 

And afterwards, if it be your desire to end this play with the fall of night, I shall 
disappear, dissolving in the darkness, or perhaps in the smile of the dawn, in the freshness 
of a transparent purity. 

“Wonderful!” stammered Doreid, the grammarian. 

“Yes,” I said to him. “Geometry’ is wonderful.” 

“I didn’t mean geometry!” he protested. “I did not come here to listen to that endless 
business about numbers and figures. That does not interest me. What I meant is that 
Telassim’s voice is wonderful. . .” 

And since I was looking at him with astonishment, at his bluntness, he added 
maliciously, “I was hoping that during the class, the girl would show her face. They say 
that she is as pretty as the fourth moon in the month of Ramadan, a real flower of Islam.” 

And he arose, singing in a low voice: 



If thou art idle or uncared for, letting the pitcher float on the water, come, come to me. 

The grass is greening on the hill, and the flowers on the woods now are opening. 

Thy thoughts shall fly from thy dark eyes as birds fly from their nests, and thy veil 
shall Fall at thy feet. 

Come, o come to me! 

We departed somewhat wistfully from the light-filled room. I noticed that Beremiz 
was not wearing on his finger the ring he had won in the inn on the day of our arrival. 
Could he have lost that fine jewel? 

The Circassian slave was looking about vigilantly, as it she feared the spell of some 
invisible genie. 


21 

THE WRITING ON THE WALL 



I begin my work of copying texts on medicine. The invisible pupil makes great advance 
in her studies. Beremiz is called in to resolve a complicated problem. King Mazim and 
the prisons of Khorasan. Sanadik, the smuggler. A poem, a problem, and a legend. King 
Mazim ’s judgment. 

Our life in the splendid city of the caliphs was becoming busier every day. Vizier 
Maluf set me to copying two books by the philosopher Rhazes, books containing a great 
deal of medical knowledge! In them I read important observations on the treatment for 
scarlet fever, the curing of childhood diseases, diseases of the kidneys, and a thousand 
other maladies that afflict men. Involved as I was in this work, I could no longer attend 
Beremiz’s classes in Sheik Iezid’s palace. 

From what I heard from my friend, the invisible pupil had made great advances in the 
last weeks. Now she had mastered the four numerical operations, as well as the first three 
books of Euclid, and could solve fractions with the denominators 1, 2, and 3. 

One day, at the end of the afternoon, we were about to begin our modest meal, 
consisting of meat pies, honey, and olives, when we heard in the street a great tumult of 



horses and the shouts, orders, and oaths of Turkish soldiers. Startled, I rose to my feet. 
What was happening? I had the impression that the inn had been surrounded by troops 
and that some violent action” by-the ill-tempered police chief was about to occur. The 
sudden uproar did not upset Beremiz. Completely oblivious, he went on drawing with a 
carbon stick geometric figures on a wooden tablet. How extraordinary he was! If Asrail, 
the angel of death, had suddenly materialized bearing the inevitable sentence, he would 
have continued impassively drawing curves and angles and studying the properties of 
figures and numbers. 

Into the little inn burst old Salim, accompanied by two black slaves and a camel 
driver, all of them excited, as if something terrible had happened. 

“For Allah’s sake!” I cried out impatiently. “Don’t disturb Beremiz! What is all this 
tumult about? Has there been a revolt in Baghdad? Has the Mosque of Suleiman fallen 
down?” - 

“Master,” stammered old Salim, “an escort of Turkish soldiers has just arrived.” 

“By Allah, what escort, Salim?” 

“The escort of the great Vizier Maluf. The soldiers have orders to bring Beremiz 
Samir to him immediately.” 

“But why so much noise?” I exclaimed. “This is not so unusual. Naturally enough, the 
vizier, our great friend and protector, wishes urgently to solve some mathematical 
problem and needs the help of our wise friend,” 

My predictions were as sure as the calculations of Beremiz. Moments later, 
accompanied by the officers of the escort, we arrived at the palace of Vizier Maluf. 

We found the vizier in his reception room, surrounded by three aides. He held in his 
hand a page filled with numbers and calculations. What new problem could have arisen 
to unset the caliph’s worthy adviser so profoundly? 

“This is very serious,” said the vizier to Beremiz. “I find myself confronted by one of 
the most difficult problems I have ever faced in my life. Let me tell you in detail how it 
arose, for only with your help can we possibly find a solution.” 

And the vizier told us the following: 

“The day before yesterday, a few hours before our noble caliph was to leave for Basra 
for a stay of three weeks, there was a terrible fire in the prison. The prisoners shut up in 
their cells underwent a long torment of unspeakable agony. Our generous sovereign 
immediately decided to cut in half the sentences of all the prisoners. At the beginning, we 
thought nothing of it, for it appeared quite simple to carry out the king’s command to the 
letter. The following day, however, when the caravan of the prince of believers was 
already far away, we discovered that his last-minute orders raised an extremely delicate 
problem, which seemed to have no ideal solution. 

“Among those sentenced to prison is a smuggler from Basra, Sanadik, who has already 
served tour years of a life sentence. This man’s sentence is to be rut in half. But since he 
was sentenced to prison for the rest of his life, now under the law his sentence must be 
reduced to half his remaining life. But we have no idea how long he will live, so how can 
we divide an incalculable time?” 



After some moments of thought, Beremiz spoke, choosing his words with great care. 

“Your problem seems to me an extremely delicate one, since it involves both 
mathematics and the interpretation of the law, and is as much a matter of human justice 
as of numbers. I cannot apply to it any rigorous analysis until I have visited the cell of the 
condemned man Sanadik. It may be that the x, the unknown in the life of Sanadik, has 
already been decided by destiny on the wall of his cell. 

“What you say seems very strange indeed,” remarked the vizier. “I cannot see any 
connection between the outpourings with which madmen and prisoners cover the prison 
walls and the solution of so delicate a problem.” 

“My lord!” exclaimed Beremiz. “On prison walls are to be found many interesting 
writings, fonnulas, verses, and inscriptions that affect our spirits and lead us to feelings 
of mercy. It so happens that on one occasion, King Mazim, ruler of the rich province of 
Khorasan, was infonned that a convict had written magic words on the wall of his cell. 
King Mazim summoned a diligent scribe and ordered him to copy down all the letters, 
numbers, verses, and writings that he could find on the gloomy walls of the prison. It 
took many weeks for him to fulfill the king’s strange command. Finallv, after working 
with great patience, he brought him pages covered with symbols, unintelligible words, 
nonsensical drawings, and blasphemies and meaningless numbers. Was there any way to 
decipher or translate the incomprehensible writings that filled those pages? One of the 
country’s sages, called in by the king, declared, ‘Your Majesty, those pages contain 
curses, heresies, cabalistic words, legends, even a mathematical problem.’ 

“The king answered. ‘Curses and heresies do not interest me. The words of the Cabala 
leave me indifferent. I don’t believe in the secret power or letters or mysteries concealed 
in human symbols. I am interested, however, in the poems and the legends, since these 
are noble” utterances in which men can find comfort, lessons for the ignorant, or 
warnings to the powerful. 

“‘A condemned man’s despair does not give rise to inspiration,’ said the wise man. 

“‘Even so, I want to see these writings,’ replied the king. 

“So the wise man picked up at random a page that the scribe had copied, and read: 

‘Happiness is difficult since we lack the materials that make us happy. Do not speak of 
happiness to those less fortunate. When one does not have that which one loves, one must 
love that which one has. ’ 

“The king remained silent as if lost in deep thought, and the wise man, to distract him, 
went on, ‘Here is one problem scratched on the cell wall of a condemned man: “ 

Arrange ten soldiers in five rows in such a way that each row has four soldiers.’ 

“This problem which looks impossible has a very simple solution, as is indicated by 
this drawing, which shows five rows with four soldiers in each one. 

“The wise man continued reading: 

They tell a story of how young Tzu-chang went one day to the great Confucius and 
asked him, ‘How many times, O illustrious” one, should a judge reflect before passing 
sentence?’ 

“Today once; tomorrow ten times,’ replied Confucius. 



Prince Tzu-chang was startled at hearing these words and did not understand. 



Once should be enough, explained the master patiently, for the judge, after examining 
the case, to decide on a pardon. A judge, however, ought to think ten times before passing 
a sentence of condemnation. 

And he concluded with this piece of profound wisdom: ‘He who hesitates in granting a 
pardon may be making a grave mistake, but he who condemns without hesitating makes a 
much greater mistake in the eyes of God.’ 

“King Mazim was filled with admiration when he realized that on the damp walls of 
prison cells were to be found such treasures written by wretched prisoners, so many 
things of beauty and interest. Obviously, among those who watched the bitter days pass 
from the depths of their cells, there were people of culture and intelligence. So the king 
ordered all sentences to be revised and discovered that many of them involved cases of 
blatant injustice. In the light of what the scribe had revealed, innocent prisoners were 
immediately freed and many judicial errors were corrected.” 

“Be that as it may,” replied Vizier Maluf, “but it is possible that, in the prisons of 
Baghdad, you will not find geometric figures or moral legends or poems. Nevertheless, 
wish to see where your inquiries will lead. And so I give you leave to visit the prison.” 


22 

HALF AND HALF 



Of all that happened in the course of our visit to the Baghdad prison. How Beremiz 
solved the problem of halving the years of life remaining to Sanadik. The instan t of time. 
Conditional liberty. Beremiz explains the basis of a sentence. 

The great prison of Baghdad looked like a Persian or Chinese fortress. Entering we 
crossed a small patio in the center of which stood the famous Well of Hope. That was 
where a condemned man, hearing .sentence passed on him gave up forever all hope of 
salvation. No one could imagine the suffering and the misery of all — those who lived 
deep in the dungeons of that magnificent Arab city. 

The cell of the unfortunate Sanadik lay in the deepest part of the prison. We were led 
down to the cellars by a jailor and accompanied by two guards. A Nubian slave, a giant 
of a man, carried the huge torch whose light showed us the recesses of the prison. 

Going down a narrow corridor that was barely wide enough for a man to pass, we 
descended a dark, dank staircase. Deep in the cellars was the small cell where Sanadik 
was shut up. Not a single ray of light penetrated that darkness. The heavy, fetid air was 
difficult to breathe without nausea. The floor was covered with a laver of putrid mud, and 
within the four walls there was not even a Bunk for the condemned man to stretch out. 

By the light of the torch carried by the great Nubian, we saw the unfortunate Sanadik, 
half naked, with a thick, tangled beard and long hair falling over his shoulders, crouched 
on a flagstone, his hands and feet fastened to irons. 

Beremiz watched him in an attentive silence. It seemed hard to believe that the 
unfortunate man could have clung to life throughout four years in that wretched; inhuman 
situation. 

The cell walls, stained and dripping, were covered with writings and with figures, 
bizarre signs of many generations of prisoners. Beremiz examined them closely, reading 
and translating with great concentration, stopping every now and again to make long, 
laborious calculations. How could the counting man decide, on the basis of these curses 
and blasphemies, how many years of life were left to Sanadik? 


Great was my relief at leaving behind the gloomy prison where the wretched prisoners 
were tortured. When we reached the sumptuous reception room, Vizier Maluf appeared, 
surrounded by courtiers, secretaries, various sheiks, and wise men of the court. They 
were all waiting for Beremiz*s arrival, wishing to know the formula the Man Who 
Counted would use to solve the problem of halving a life sentence. 

“We await you, O counting man.” said the vizier affably, “and I beg you to give us a 
solution without any delay. We are most eager to comply with the orders of our great 
Emir.” 

Beremiz bowed respectfully, made his salaams, and spoke as follows: 

“Sanadik of Basra, the smuggler, captured four years ago at the frontier, was 
sentenced to life imprisonment. That sentence has just been cut in half by the wise and 
just decree of our gracious caliph, ruler of believers, servant of Allah on earth. ... 

“Let us give the value of A to the period of Sanadik’s life that begins with his capture 
and his life sentence. The latter condemned Sanadik to x years in prison, that is, to 
imprisonment for life. Now, by virtue of the royal decree, that sentence is reduced by 
half. It is important to state, if we divide the time that x stands for into separate periods 
that for each period in prison there must be an equal period of freedom.” 

“Exactly!” exclaimed the vizier keenly. “I follow your reasoning very well. 1 ' 

“Now, as Sanadik has already served four years, it seems clear that he must have an 
equal period of freedom, that is, four years. In fact, let us imagine that a friendly 
magician could foresee the precise number of years left to Sanadik and could say to us, 
‘This man had only eight years of life left to him when he was imprisoned.’ In that case, x 
would be equal to eight, that is to say, Sanadik had been condemned to eight years in 
prison, now reduced to four. But since Sanadik has already served four years he has in 
fact completed his sentence and must be considered a free man. If the smuggler by the 
will of destiny were to live more than eight years, if x were greater than eight, his life 
would be made up of three periods: one of four years in prison already served f another of 
four years of freedom, and a third, which also would have to be divided into two parts, 
imprisonment and freedom. This makes it easy to conclude that whatever value x, the 
unknown, may have, the condemned man must immediately be set free to enjoy four 
years of freedom. He has absolute right to this period of liberty, as I have shown, in 
accordance with the law. 

“Once this period is over, he must return to prison and remain there for a period equal 
to half the rest of his life. It would be easiest perhaps to imprison him for a year and then 
grant him freedom for the following year. Thanks to the caliph’s command, he would be 
in prison for one year and free for one year, in that way enjoying the benefits of the 
caliph’s mercy. Such a solution, however, would be a precise one “only if the condemned 
man were to die on the very last day of one of his periods of freedom. 

“Let us imagine that Sanadik, after a year in prison, is set free and dies in the fourth 
month of his freedom. Of that part of his life — one year and four months — he would 
have spent one year locked up and four months at liberty. That would not be correct; an 
error of calculation, for his sentence would not have been cut by half. 



“It would be simpler perhaps to imprison Sanadik for a month and set him free for the 
following month. That solution, however, could lead to a similar error if he, after 
spending a month in prison, did not enjoy a whole month of freedom. 

“It seems, you will say, that the best solution in the end would be to imprison Sanadik 
for one day and set him free the next for an equal period of freedom, continuing thus until 
the end of his life. Yet that solution would still not satisfy the demands of mathematical 
exactitude, because Sanadik might die a few hours after, a day in prison. Keeping him 
imprisoned for an hour and then setting him free, and going on in that way until the last 
hour of his life, would be a solution only if Sanadik were to die in the last minute of one 
of his hours of freedom. 

Otherwise, his sentence would not have been reduced by half, as the caliph’s decree 
commands. 

“The exact mathematical solution is as follows: Keep Sanadik in prison for an instant 
of time and set him free the following instant. It is necessary, however, that his prison 
time, one instant, be so small as to be indivisible. The same applies to the period of 
liberty that follows. 

“In reality, such a solution is impossible. How can you lock up a man for one 
indivisible instant and set him free the following instant? So that idea has to be put aside 
as impossible. I can see, O Vizier, only one way of solving the problem. Let Sanadik be 
given conditional liberty under the eye of the law. It is the only way of allowing him to 
serve his sentence and to be free at the same time.” 

The vizier ordered that Beremiz’s suggestion be immediately implemented, and that 
very day the imprisoned Sanadik was granted conditional liberty, a judgment that Arab 
magistrates adopted often from then on in passing wise sentence. The following day I 
asked Beremiz what details and calculations he had managed to gather from the prison 
wall during our famous visit, and what had led him to come up with such an original 
solution to the problem. This was his answer: - 

“Only someone who has been even for a brief moment within the gloomy walls of a 
dungeon can kn ow how to solve those problems in which numbers form a terrible part of 
human misery.” 


23 

ALL IS RELATIVE 


What happened in the course of an honored visit to us. The words of Prince Cluzir 
Shah. A princely invitation. Beremiz solves a new problem. The raja ’s pearls. A 
cabalistic number. Our departure for India is set. 

The humble district in which we lived was enjoying a glorious morning when Beremiz 
received an unexpected visit from Prince Cluzir Shah. The lavish retinue crowded down 
our street, bringing the curious out onto the rooftops and balconies. Old men, women, and 
children gaped at the spectacle in dumb astonishment. First came thirty horsemen 
mounted on superb Arab chargers with gold trappings and velvet cloths edged in silver. 
They wore white turbans and helmets that shone in the sunlight, cloaks and tunics of silk, 
and scimitars hanging “from belts of worked leather. They carried before them 
standards showing the prince’s shield, a white elephant on a blue background. After them 
came archers and scouts, also mounted. 

Bringing up the rear was the all-powerful prince, accompanied by two secretaries, 
three doctors, and ten pages. He wore a scarlet tunic decorated with rows of pearls. 
Sapphires and rubies sparkled in his turban. When old Salim saw the magnificent 
assembly from the inn, he almost went mad. He cast himself to the floor and began to 
shout, “What is this? Where am I?” 

I sent a water carrier to lead my poor old friend to the patio until he recovered his 
caim. The main room of the inn was too small for the illustrious company. Beremiz, 
somewhat awed by the gracious visit, went down to the patio to receive his visitors. 

Prince Cluzir Shah, entering with his retinue, greeted the Man Who Counted with a 
friendly salaam and said to him, “A poor wise man is he who seeks out the rich, but a 
nobler rich man he who seeks out wise men.” 

“I realize, my ford,” replied Beremiz, “that your words are inspired by deep feelings of 
friendship. The insignificant knowledge that 1 have acquired is as nothing before your 
generous heart.” 

“My visit is dictated more by my own desires than by love of science,” replied the 
prince. “Since I first had the honor of listening to you in the house of Sheik Iezid, I have 
considered offering you some suitable position in my court. I wish to appoint you my 
secretary or, even better, master of the Delhi Observatory. Will you accept? We shall 
leave in a few weeks for Mecca, and from there we shall go on directly to India.” 


“My generous Prince!” replied Beremiz, “unfortunately I cannot leave Baghdad at this 
moment. I am bound to this city by a serious obligation. I shall be able to leave only 
when the daughter of the illustrious sheik Iezid has mastered the beauties of geometry.” 

The maharajah smiled and replied, “If your refusal is because of that obligation, I 
think that there is a way of fulfilling it. Sheik Iezid told me that, thanks to her progress, 
young Telassim in a few months will be ready to teach even wise men the famous 
problem of the rajah’s pearls.” 

I sensed that our noble visitor’s words surprised Beremiz, for he seemed somewhat 
confused. 

“I would be delighted,” the prince went on, “to understand this difficult problem, 
which eludes many calculators and which was first set by one of my distinguished 
ancestors.” 

Accordingly, Beremiz in compliance with his wishes spoke on the problem in his slow 
and steady manner. 

“It is less a problem than an arithmetical curiosity,” he said. “The situation is as 
follows: A rajah on his death left to his daughters a certain number of pearls with 
instructions that they be divided up in the following way: his eldest daughter was to have 
one pearl and a seventh of those that were left. His second daughter was to have two 
pearls and a seventh of those that were left. His third daughter was to have three pearls 
and a seventh of those that were left. And so on. The youngest daughters went before the 
judge complaining that this complicated system was extremely unfair. The judge, who, as 
tradition has it was skilled in solving problems, replied at once that the claimants who 
were mistaken that the proposed division was just, and that each of the daughters would 
receive the same number of pearls: — 

“How many pearls were there? How many daughters had the rajah? 

“The solution to this problem is really not at all difficult. Observe. 

“There were 36 pearls in all, and the rajah had six daughters. 

The first daughter received one pearl and one-seventh of the 35 pearls remaining, that 
is, 5. She received 6 pearls in all, and 30 were left. 

“The second daughter received 2 pearls and one-seventh of the 28 remaining pearls, 
that is, 4. In all, she received 6 pearls, and 24 were left. 

“The third daughter received 3 pearls and one-seventh of the 2 1 remaining pearls, that 
is, 3. In all, she received 6 pearls, and 18 were left. 

“The fourth daughter received four pearls and one-seventh of the 14 that were left, that 
is, 2. In all, she received 6 pearls, and 12 were left. 

“The fifth daughter received 5 pearls and one-seventh of the 7 that were left, that is, 1 . 
She received 6 pearls in all, leaving 6, which became the share of the sixth and youngest 
daughter.” 

And Beremiz concluded, “As you see the problem, although ingenious, is not at all 
difficult. No subtlety is required to reach its solution.” 

At that moment, the prince’s attention was caught by a number written five times on 
the wall of the room. The number was 142,857. 


• • •••• ••••• 


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Graphic solution to the famous problem of the rajah’s pearls. 

“What is the significance of that number?” he asked. “That is one of the oddest 
numbers in all of mathematics,” Beremiz replied. 

“There are extraordinary coincidences in its relations with its multiples. 

“Let us multiply it be two: 

142,857 x 2=285,714 

“Notice that the numbers in the product are the same as those in the given number but 
in a different order. The 14 that was on the left has moved to the right. 

“Let us multiply now by three: 

142,857 x3 = 428,571 

“Once more, notice how curious the answer is. Its numbers are the same ones again, 
but in a different order. The 1 that was at the left has moved to the right; 
the other numbers stay where they were. 

“When we multiply by four, although the numbers shift position, they remain in the 
same order: 

142,857x4 = 571,428 

“The same thing happens when we multiply it by five: 

142,857x5= 714,285 

“Watch what happens when we multiply by six: 

142,857 x6 = 857,142 

“In this case, the two groups of three numbers have changed places. “Something quite 
different happens when we multiply by seven: 

142,857 x 7= 999,999 

“Now, let us multiply the number by eight: 

142,857x8 = 1,142,856 

“All the numbers appear in the answer except 7. The 7 of our first number is now in 
two parts, 6 and 1 , with 6 on the right and 1 on the left. 

“Now, let us multiply our number by nine: 

1 42,857x9 = 1,285,713 

“Look closely at the answer. The only number that is missing is 4. What has 
happened to it? It seems to have separated into two parts, 1 on the left and 3 on the right. 


“We can further-show the oddness of the number 142,857 when we multiple by 11, 
12, 13, 14, 15, 16, 17, 18, and so on 

“All this has caused the number 142,857 to be considered one of the most mysterious 
numbers in all of mathematics. I was taught this by the dervish No-Elim ...” 

“No-Elim?” the prince cried, lighting up. “Did you really know that wisest of men?” 

“I knew him very well, My Prince,” replied Beremiz. “It was from him that I learned 
all the principles that 1 use today in my mathematical studies.” 

“The great No-Elim was a friend of my father’s,” the prince explained. “Sometime 
later, after losing a son in a cruel and unjust war, he withdrew from the city and never 
came back. I made many attempts to locate him but could find no trace of his 
whereabouts. I came to the conclusion that perhaps he had died in the desert, eaten by 
panthers. Can you possibly tell me where he is to be found?” 

“When I departed from Baghdad,” Beremiz replied, “I left him in Khoi, in Persia, with 
three friends.” 

“Then, when we return from Mecca, we shall go to the city of Khoi to seek out that 
great master,” the prince replied. “I wish to bring him to my palace. Can you possibly 
help us in that ambitious task, O Beremiz Samir?” 

“My ford,” Beremiz replied, “if it is a question of assisting in any way the man who 
was my guide and master, I shall come with you, if needs be, all the way to India.” 

And that was how, because of the number 142,857, the matter of our voyage to India, 
the land of the rajahs, was settled. That number truly is a magical one. 


24 

EUREKA 



The ill-tempered Tara Tir, The epitaph of Diophantus. Hiero ’s problem. Beremiz frees 
himself from a dangerous enemy. A letter from Captain Hassan. The cube roots of 8 and 
27. A passion for calculus. The death of Archimedes. 


The threatening presence of Tara Tir was troubling my spirit. The ill-tempered sheik, 
who had been out of Baghdad for some time, had been seen the preceding night in the 
company of assassins, prowling in our street. He was no doubt preparing an ambush for 
the unsuspecting Beremiz, who, “busy with his studies, had no sense of the danger that 
was following him like a dark shadow. 

I spoke to him about Tara Tir and reminded him of the careful warning Sheik Iezid 
had given us. 

“Such fear is groundless,” he replied, paying no attention to my warning. “I do not 
believe in these threats. What interests me at the moment is the solution of a problem that 
lies in the epitaph on the tomb” of “the famous Greek geometer Diophantus. 

“The stone on the grave of Diophantus reveals his age by means of an 
arithmetical artifice, quite wonderful to contemplate, as follows: 

The gods granted him childhood for a sixth of his life, and a twelfth for his 
adolescence. A barren marriage took tip a seventh of his life. Five years passed, and then 
a child was born to him. No sooner had this child reached half the age of its father than it 
died. Diophantus lived for four more years, drowning his pain in the study of numbers, 
and then gave up his life. 

“A study of that text will reveal to you that his age was eighty-four. It is possible that 
Diophantus was so busy solving arithmetic problems of indeterminacy for so much of his 
long life that he never thought of finding a solution to the problem by King Micro, since 
it does not appear in his works.” 

“What problem is that?” I asked him. And he told me the following story: 

“Hiero, of Syracuse, delivered to his goldsmiths a quantity of gold to be made into a 
crown he wished to offer up to Jupiter. When the king received the finished crown, he 
verified that-it weighed the same as the gold he had sent, but the color led him to think 
that some silver might have been mixed with the gold. He took his doubts to the geometer 
Archimedes. 

“Archimedes, having proved that gold loses fifty-two thousandths of its weight in 
water whereas silver loses ninety-nine thousandths, weighed the crown in water and 
discovered that the difference in its weight indicated the presence of silver mixed with the 
gold. 

“They say that Archimedes took a long time over Hiero’s problem. One day in his 
bath, he hit on the means of solving it and, excited, jumped out and ran to the palace 
crying, ‘Eureka! Eureka!’ which means ‘I found it! I found it!” 

While we were talking, a visitor appeared, Captain Hassan Maurique, head of the 
sultan’s guard. He was a big man, very easy and obliging. He had heard of the case of the 
thirty-five camels and, from then on, never ceased to praise the skills of the Man Who 
Counted. Every Friday on his way back from the mosque, he paid us a visit. 

“I never thought,” he declared, “that mathematics was so prodigious. I was thrilled by 
your solution to the problem of the camels.” 

I took him up to a balcony that overlooked the street, while Beremiz was still busy, 
and I spoke to him of the danger that threatened us from the hateful Tara Tir. 



“There he is,” I said, pointing beside the fountain. “Those men with him are dangerous 
assassins. Given the least chance, they will fall on us. Tara Tir harbors a deep resentment 
against Beremiz, and I am afraid, since he is a violent and angry man, that he will take 
revenge. I have seen him spying on us a number of times.” 

“What are you telling me?” exclaimed Captain Hassan. “I could not have imagined 
that such a thing might happen. How could a thug like that disturb the life of a wise man 
like the Man Who Counted? By the Prophet, I’m going to put an end to this matter at 
once.” 

When he had gone, I went back to my room and lay down, smoking quietly for a 
while. 

However violent Tara Tir might be. Captain Hassan was no one to trifle with; and he 
went into action on our behalf. An hour later, I received the following letter from him: 

Everything is resolved. The three assassins have been summarily executed. Tara Tir 
has been given eight lashes and has paid a fine of twenty-seven pieces of gold. He has 
been told to leave the city immediately, and I have sent him to Damascus under guard. 

I showed the captain’s letter to Beremiz. Thanks to this news, we could now live in 
Baghdad in peace. “Very interesting,” remarked Beremiz. “It is really quite curious. This 
letter makes me remember a strange numerical relationship between the numbers 8 and 
27.” 

I looked surprised at his remark, but he continued, “Excluding the number 1, 8 and 27 
are the only numbers equal to the sum of their cubes. Look: 

8 3 = 512 
27 3 = 19,683 

The sum of the numbers in 5 12 is 8, and the sum of the numbers in 19,683 is 27.” 

“You are quite astonishing, my friend!” I exclaimed. “Busy with your cubes, you have 
forgotten all about being threatened by dangerous assassins.” 

“Mathematics, O man of Baghdad, takes our attention so much that at times we lose 
ourselves and forget the dangers that surround us. Do you remember how the great 
geometer Archimedes died? When the city of Syracuse was taken by storm by the forces 
of Marcellus, the Roman general, Archimedes was wholly absorbed in a problem whose 
solution he was working out by drawing in the sand, oblivious to wars and death. The 
pursuit of truth was all that interested him. A Roman soldier found him and ordered him 
to come before Marcellus. The sage told him to wait a moment until he had finished the 
problem he was working on. The soldier insisted and seized him roughly by the arm. ‘Be 
careful! Watch where you’re stepping!’ cried the sage to the soldier. ‘Don’t erase that 
drawing!’ Annoyed at not being instantly obeyed, the Roman struck him, putting an end 
to the life of the wisest man of the day. 

“Marcellus, who had given strict orders that Archimedes’ life was to be saved, did not 
hide his sorrow at the death of Archimedes. Over his tomb, he had a stone erected on 
which was engraved a circle inside a triangle, a figure in memory of one of the theorems 
of the famous geometer.” 



Beremiz finished and came over to me, placing a hand on my shoulder. “Don’t you 
think, my friend, that it would be proper to include the wise man of Syracuse among 
those who are martyrs to geometry?” 

What could I say to him? 

The tragic end of Archimedes reminded me of the treacherous and envious Tara Tir. 

Were we really free of that ill-tempered salt vendor? Might he not later return from his 
exile in Damascus to cause us more harm? 

Standing close to the window with his arms crossed, Beremiz was watching the men 
coming from and going to the market with a somewhat sad expression on his face. I 
decided to intrude on his thoughts and take him out of his melancholy. 

“What is this?” I asked him. “Are you sad? Do you feel a longing for your own 
country, or are you simply planning new calculations? Is it mathematics or nostalgia?” 

“My friend of Baghdad,” he replied, “nostalgia and calculations are not unrelated. One 
of our most inspired poets has put it as follows: 


Nostalgia can be calculated 
Also using numbers 
It is distance multiplied 
By a factor of love. 

I do not think, however, that nostalgia, once reduced to a formula, can be measured in 
numbers. When I was a child, I heard my mother many times singing as follows: 

Nostalgia is an old song. 

Nostalgia is someone ’s shadow. 

Only time will carry it off. 

When it carries me off as well. 


25 

THE QUESTIONS COMMENCE 



Beremiz is summoned to the palace once again. A strange surprise. A difficult match: 
one against seven. The return of the mysterious ring. Beremiz is presented with a blue 
carpet. Verses that stir a full heart. 

The first night after Ramadan, arriving at the caliph’s palace, we were informed by an 
old scribe who worked with me that the sovereign was preparing a strange surprise for 
our friend Beremiz. 

A portentous event lay ahead of us. The Man Who Counted was going to have to 
compete before an audience with seven mathematicians, three of whom had arrived from 
Cairo the day before. What was to be done? With that challenge ahead, I tried to 
encourage Beremiz, telling him that he must have absolute confidence in his powers, 
which had been proved so many times. He reminded me of a proverb of his master No- 
Elim: “He who has no confidence in himself does not deserve the confidence of others.” 

Filled with a heavy apprehension, we went into the palace. 

The huge reception hall was ablaze with light and filled with courtiers and 
distinguished sheiks. To the right of the caliph sat the young prince Cluzir Shah, guest of 
honor, accompanied by eight Hindu doctors wearing rich robes of golden velvet and 
strange turbans from Kashmir. To the left of the throne were seated the viziers, the poets, 
the judges, and the most distinguished members of Baghdad society. On a dais, on 
several silk cushions, sat the seven sages who were going to question the Man Who 
Counted. At a sign from the caliph, Sheik Nuredin Zarur took Beremiz by the arm and led 
him solemnly to a kind of rostrum set up in the middle of the splendid hall. 

The faces of those present were alive with expectation, although not all of them 
wished success to the Man Who Counted. 

A huge black slave sounded a heavy silver pong three times. All the turbans bowed. 
The strange ceremony was about to begin. My thoughts, I must confess, were in a whirl. 

A high priest held the Holy Book and read in a slow and steady voice a prayer from 
the Koran: 



In the name of Allah, the wise and merciful, Creator of all worlds. 

We praise thee, O Lord, and beg thy divine aid. Lead us by the strait way, 
the way of those chosen and blessed by thee. 

When the last word had echoed through the galleries of the palace, the king took two 
steps forward, stopped, and said, “Our friend and ally. Prince Cluzier Shah, ford of 
Lahore and Delhi, has asked me to propose to the learned men of his retinue tile chance 
to witness the wisdom and skill of the Persian mathematician, secretary to Vizier Ibrahim 
Maluf. It would be ungracious not to attend to that request on the part of our illustrious 
guest. And so, seven of the wisest and best-known sages of Islam are going to present 
Beremiz, the Man Who Counted, with a series of questions related to the science of 
numbers. If Beremiz can answer these questions, he shall receive, I promise, a reward 
that will make him one of the most envied men in Baghdad.” 

At that moment, we saw the poet Iezid approach the caliph. “Lord of all believers!” 
said the sheik. “I have here something that belongs to Beremiz, a ring that was found in 
my house by one of the slaves. I wish to return it to him before he undertakes this most 
important test to which he is to be submitted. It may be that it is some kind of charm, and 
I do not wish to deprive him of any supernatural help it may give him.” 

And after a brief pause, the noble Iezid continued, “My well-beloved daughter 
Telassim, the true treasure among the treasures in my life, asked me to allow her to offer 
to the Persian mathematician, her master in the science of numbers, this rug on which she 
has embroidered a border. This rug, if Your Eminence will allow it is to lie placed under 
the cushion set aside for Beremiz when he is put to the test by the seven most famous 
sages of Islam.” 

The caliph ordered the ring and the rug to be taken immediately to the Man Who 
Counted. Sheik Iezid himself, cordial and friendly as always, handed Beremiz the small 
box containing the ring. Then, at a signal from the sheik, a young slave appeared carrying 
a small blue rug that was laid under Beremiz’s green cushion. 

“All that is a spell, a charm for good luck,” said a low voice behind me, the voice of a 
thin, old man in a blue tunic. “That young Persian knows a lot about magic. That blue rue 
seems to me somewhat mysterious.” 

How could they understand, the majority of those present, that Beremiz’s aptitude for 
calculation was the fruit of his pure intelligence? The unlettered, when something is 
beyond their understanding, always attribute what they do not understand to magical 
powers. However, the intelligence and culture of the sheiks in attendance was sufficiently 
high for them to understand that what was taking place was a matter of pure intelligence. 
Beremiz was to be put to the test by those most skilled in a discipline in which we Arabs 
have always excelled. Would he manage to meet the test? 

Beremiz seemed deeply moved at receiving the ring and the carpet. Even at a distance 
I could see at that moment that something was affecting him deeply. When he opened the 
little box, his bright eyes misted over. I discovered later that along with the ring the 
gentle Telassim had enclosed a slip of paper on which Beremiz read, “Courage. Trust in 
God. I pray for you.” And the carpet was there really something magical about it, as the 
old man in the blue tunic had suggested? 



It was not magic. That rug, which in the eyes of the sheiks and the wise men was no 
more than a small gift, contained, written in Kufic characters that Beremiz alone knew 
how to read and understand some verses that touched his heart. Those verses, which I 
later translated, had been, embroidered by Telassim to look like arabesques alone the 
borders: 

I love you, my dear one. Forgive me my love. 

You consoled me, a bird strayed in its flight. 

When-my heart was touched, it lay unveiled and open to the weather. 

Enfold it in your mercy, my dear one, and forgive me my love. 

If you cannot love me, my dear one, forgive me my pain. 

I will return to my song. I will stay seated in the darkness. 

I will cover with my hand my naked modesty. 

Was Sheik Iezid aware of that double message of love? I do not know quite why that 
idea came to me so strongly. Only later, as I have said, did Beremiz tell me the secret. 

Only Allah knows the truth! 

A profound silence fell over the whole sumptuous company. In the rich reception hall 
of the caliph’s palace, it was about to begin, this extraordinary contest 'that had never 
before taken place under the skies of Islam. 

Allah! 


26 

ONE FOR THE BOOK 



Our encounter with a famous theologian. The problem of the life to come. Every 
Muslim must know the Holy Book. How many words are there in the Koran? How many 
letters? A Deception by Beremiz. 

The wise man appointed to begin the questioning solemnly got to his feet. He was in 
his eighties and inspired in me a deep respect. Like a prophet, he had a long white beard, 


which reached to his broad chest. “Who is that fine old man?” I asked in a low voice a 
doctor with a thin and tanned face who was sitting next to me. 

“He is the celebrated sage Mohadeb Ibhage Abner Rama,” he answered. “They say 
that he knows more than fifteen thousand maxims on the Koran. He is a professor of 
theology and rhetoric.” 

The sage Mohadeb pronounced his .words in a strange manner, syllable by syllable, as 
if detennined to measure the sound of his own voice. 

“I am going to question you. O counting man, on a matter of prime importance to a 
Muslim. Before he studies Euclid or Pythagoras, he of the Islamic faith must have a 
profound knowledge of his religion, for life is inconceivable if truth and faith are 
separated. He who does not dwell on the life to come and the salvation of his soul and 
does not know the precepts of Allah and the commandments does not deserve to be called 
a wise man. So I want you to give us, without any hesitation, fifteen numerical references 
from the Koran, the Book of Allah. Among those-fifteen examples you must include 

1 . The number of chapters in the Koran. 

2. The exact number of verses. 

3. The number of words. 

4. The number of letters in the Holy Book. 

5. The exact number of prophets mentioned in its pages.” 

He continued, his voice deepening, “And apart from the five instances I have asked 
you for, I want you to give us ten more that have to do with numbers. Pray proceed.” 

A deep silence followed his words, as everyone waited for Beremiz to speak. With 
remarkable calm, the young counting man replied as follows: 

“O wise and venerable Father! The Koran consists of 114 chapters, 70 of them 
dictated in Mecca -and 44-of them in Medina. These are-divided into 611 sections and 
contain 6,236 verses, 7 in the first chapter and 8 in the last. The longest chanter is the 
second, with 280 verses. In the Koran, there are 46,439 words and 323,670 letters, each 
one of which contains ten special virtues. Our Holy Book mentions the names of twenty- 
five prophets. Jesus, the son of Man’, is referred to nineteen times. The names of five 
animals are taken as epigraphs to 5 chapters: the cow, the bee, the ant, the spider, and the 
elephant. Chapter 102 is entitled The Reply of the Numbers.’ That chapter is 
distinguished by the warning it gives in its 5 verses to those involved in sterile arguments 
about numbers, which have no importance at all in the spiritual progress of men.” 

At this point, Beremiz paused briefly and then added, “In answer to your question, 
these are numerical references in the Book of Allah. In the reply I have just given you 
there is one error that I hasten to point out Instead of fifteen, I have given you sixteen.” 

“By Allah!” exclaimed the old man in the blue tunic seated behind me. “How can any 
man know so many numbers, so many things, by heart? It is amazing! He knows even 
how many letters there are in the Koran!” 

“He studies a great deal,” muttered his neighbor, a fat man with a scar on his chin. “He 
studies a great deal, and he remembers everything. I have heard that from a number of 
people.” 



“Remembering isn’t everything,” whispered the old man. “For example, I can’t even 
remember the ages of my cousins.” 

I was extremely annoyed by all these whisperings around me. But the fact is that 
Mohadeb confirmed all the references Beremiz had supplied, even the number of letters 
in the Holy Book. 

They told me that Mohadeb, the-theologian, was a man who chose to live in poverty, 
and it must have been true. Allah deprives many wise men of riches, since riches and 
wisdom seldom occur together. 

Beremiz had brilliantly overcome the first challenge offered him in that fearsome 
debate, but six more were still to come. 

.’’May it be Allah’s will,” I thought, “that the rest will go as this has gone, and that all 
will end well.” 


27 

HISTORY IN THE MAKING 



A wise historian questions Beremiz. The geometer who could not see the sky. Greek 
mathematics. Praise for Eratosthenes. 

The first challenge having been met in detail, the second wise man took up the 
questioning of Beremiz. He was a famous historian who had given classes for twenty 
years in Cordoba and who later, for political reasons, moved to Cairo, where he resided 
under the protection of the caliph. He was a short man with a bronzed face and an 
elliptical beard. His eyes looked dead, lifeless. Here is’ what he said to the Man Who 
Counted: 

“In the name of Allah, the Wise and Merciful! They are mistaken, those who think that 
the worth of a mathematician lies in the skill with which he calculates — or applies the 
banal rules of calculation. In my view, the true mathematician is he who has a thorough 
knowledge of the progress of mathematics through the centuries. To study the history of 



mathematics is to pay homage to those geniuses who have elevated and dignified 
previous civilizations through their intelligence and have been able to reveal some of the 
most profound mysteries of nature, managing through science to better our wretched 
human condition. Throughout the pages of history, we honor our glorious ancestors who 
formed the science of mathematics, and we keep alive the names of the works they left 
behind them. And so I wish to pose to the Man Who Counted a question on an interesting 
case in the history of mathematics. What was the name of the famous geometer who 
committed suicide because he could not see the sky?” 

Beremiz thought for a moment and answered, “His name was Eratosthenes, the 
mathematician of Cyrenaica, who was first educated in Alexandria and later in the School 
of Athens, where he learned Plato’s doctrines. Eratosthenes was appointed to direct the 
great library in the University of Alexandria, a post he occupied until the end of his days. 
Besides possessing an enviable depth of scientific and literary knowledge, which put him 
among the wisest men of his day, Eratosthenes was a poet, an orator, a philosopher, and, 
even more, an all-around athlete. It is enough to mention that he was a winner of the 
pentathlon, the fivefold contest in the Olympic Games. Greece then enjoyed a golden age 
in science and letters. It was the homeland of the epic: poets, who declaimed their verses 
to a musical accompaniment at the-banquets and gatherings of kings and great leaders. 

“It should be pointed out that, among the most celebrated and cultivated of the Greeks, 
Eratosthenes was considered an extraordinary man who hurled the javelin, wrote poems, 
defeated the great runners, and solved astronomical problems. Various works of his have 
come down to posterity. He presented King Ptolemy III of Egypt with a table on which 
the prime numbers were etched on a metal plate, with those numbers with multiples 
marked with a small hole. And so they give the name of Eratosthenes’ sieve to the 
process the wise astronomer used to draw up his table. 

“Because of an affliction to his eyes, which he caught on the banks of the Nile in the 
course of a journey, Eratosthenes went blind. He who had cultivated astronomy with 
passion was prevented from seeing the sky or admiring the incomparable beauty of the 
firmament on starry nights. The blue light of Sirius could never penetrate that black cloud 
that covered his eyes. Overcome by such misfortune, and not able to bear the burden of 
blindness, the sage committed suicide by allowing himself to die of hunger, shut up in his 
library.” 

The wise historian with the dead eyes turned toward the caliph and declared, after a 
brief silence, “I consider myself well satisfied with the brilliant historical exposition 
delivered by the Persian calculator. The only famous geometer who committed suicide 
was indeed Eratosthenes the Greek poet, astronomer, and athlete, close friend of the most 
famous Archimedes of Syracuse. Allah be praised!” 

“By the beauteous fountain of paradise!” cried the caliph enthusiastically. “How-many 
things I have just learned! How many things we are ignorant of! That -famous Greek, 
who studied the stars, wrote poems and performed athletic feats, is worthy of our deepest 
admiration. From now on, whenever I look a the sky on starry nights, whenever I see the 
star Sirius, I shall think of the tragic end of that wise man who wrote the poem of his 
death surrounded by a treasure of books he was not able to read.” 

And, touching the shoulder of the prince, he added, “Now let us see if the third wise 
man will be able to defeat our counting man!” 



28 

FALSE HOPES 



The memorable content continues. The third wise man questions Beremiz. False 
induction. Beremiz shows that a false principle may he suggested by true examples. 

The third wise man who was to question Beremiz was the famous astronomer Abul 
Hassan Ali, from Alcala, who had come to Baghdad at the caliph’s special invitation. He 
was tall and Bony, with a wrinkled face, and he wore on his right wrist a broad gold 
bracelet that, according to rumor, had engraved on it the twelve signs of the zodiac. After 
saluting the king and the nobles, the astronomer turned to Beremiz. His deep voice 
seemed to resound in the hall. 

“The two replies that you have just delivered show, that you have a good grounding. 
You speak of Greek science and the close details of our Holy Book with the same ability 
and mastery. In the science of mathematics, however, the most interesting part is that 
reasoning that leads to the truth. A collection of facts is as far from forming a body of 
knowledge as a mirage in the desert is from being a real oasis. Knowledge must observe 
facts and, from them, deduce its laws. With the help of these laws we can deal with other 
facts or improve the conditions of life. All that is true. But how do we arrive at the truth? 
The following question arises: 

“Is it possible in mathematics to arrive at a false rule from true facts? I wish to hear 
your reply, O calculator, and to have it illustrated by a simple example.” 

Beremiz waited a moment, deep in thought, and then he gathered himself and spoke as 
follows: “Let us suppose that a mathematician out of curiosity wishes to determine the 
square root of a four-figure number. We know that the square root of a number is another 
number, which, multiplied by itself, gives an answer equal to the original number. That is 
a mathematical axiom. 

“Let us suppose further that the mathematician, picking three numbers for his 
experiment, chooses the following: 2,025, 3,025, and 9,801. 

“Let us begin with the number 2,025. Making the appropriate calculations, we 
discover that the square root of that number is 45, which is to say that 45 x 45 equals 
2,025. But we can also show that 45 can be obtained-by adding 20 and 25, which are 


parts of the number 2,025, divided in the middle. The same can be demonstrated with the 
number 3.025, whose square root is 55. The number 55 can also be arrived at by adding 
30 and 25, both parts of the number 3,025. The same is true of the number 9,801. Its 
square root is 99, that is to say 98 plus 1. Given these three examples, a careless 
mathematician might feel inclined to pronounce the following rule: 

“To calculate the square root of a four-figure number, divide the number in the middle 
into two separate numbers, and add these numbers. The answer will be the square root of 
the given number. 

“That rule, which is obviously erroneous, was arrived at from three real examples. It-is 
impossible in mathematics to arrive at the truth by simple observation, but even so, it is 
important to take special care to avoid such false induction.” 

The astronomer Abul Hassan, obviously satisfied with Beremiz’s reply, declared that 
he had never heard such a simple and interesting explanation of false induction in 
mathematics. 

At a signal from the caliph, the fourth wise man took his turn and got ready to ask his 
question. His name was Jabal ibn-Wafrid, a poet, a philosopher, and an astrologer. In his 
native Toledo, he had become well known as a teller of stories. I shall never forget his 
venerable presence or his serene and kindly gaze. He moved to the” edge of the dais and, 
addressing the Man Who Counted, began, “In order for you to understand my question, 1 
must first tell you an old Persian legend.” 

“Tell it then, O eloquent sage!” said the caliph. “We are eager to hear your wise 
words, which are as drops of gold to us who listen.” 

The wise man of Toledo, in a voice as firm and steady as the progress of a caravan, 
told the following story. 


29 

SINGLE HANDED SUCCESS 



We hear an old Persian legend. The material and the spiritual. Problems human and 
superhuman. The most famous multiplication of all. The Sultan fiercely chastises 
intolerance among the sheiks of Islam. 



A powerful king who governed Persia and the great plains of Iran heard a certain 
Dervish once declare that the true wise man must know and distinguish the spiritual and 
the material in, life. That king was called Astor and was known familiarly as the Serene. 

One day he summoned the three wisest men in all of Persia, gave to each of “them two 
silver dinars, and spoke to them as follows: “In this palace there are three identical 
rooms, all of them completely empty. Each one of you is commissioned to fill up one of 
these rooms, but in performing this task you must not spend any more than you have just 
received.” 

The problem seemed a truly difficult one. Each wise man had to fill up an empty 
room, spending no more than the insignificant sum of two dinars. The wise men set out to 
fulfill the difficult commission set them by King Astor. 

Some time later, they returned to the throne room. The king, eager to hear their 
solutions to the problem, questioned them in turn. 

The first spoke as follows: “My ford, I have spent the two dinars, and the room is now 
completely full. My solution was a practical one, I bought a number of sacks of hay and 
with them I filled the whole room from floor to ceiling.” 

“Very good!” exclaimed King Astor. “Your solution was truly imaginative. In my 
opinion, you are well aware of the material part of life, and from that vantage point you 
are well able to take on the problems that life presents to you.” 

The second wise man, after bowing to the king, gave his answer as follows: “In 
carrying out my task, I spent only half a dinar. Let me explain. I bought a candle and lit it 
in the empty room. Now, O King, you may see for yourself. The room is completely 
filled, filled with light.” 

“Bravo!” exclaimed the king. “Your solution is quite brilliant. Light represents .the 
spiritual part of life. Your spirit, it seems to me, is well able to face the problems of 
existence from the spiritual point of view.” 

Then the third wise man spoke as follows: “O King of the four corners of the world! At 
first I thought of leaving the room exactly as I found it. It would have been easy to say 
that the room was not empty since, obviously, it was filled with air and darkness. I did 
not, however, wish to seem guilty of laziness and trickery, so I decided to act, as did my 
companions. I took a handful of hay from the first room, set it on fire from the candle, 
and extinguished it so that the room filled up entirely with smoke. As you might suppose, 
this cost me nothing. The sum you gave me is intact, and the room is full, full of smoke.” 

“Admirable!” exclaimed Astor the Serene. “You are the wisest of men in Persia, and 
perhaps in the whole world, since you know how to combine the material with the 
spiritual in achieving perfection.” 

The sage of Toledo finished his story and, turning toward Beremiz, addressed him in a 
friendly manner: “My wish is, O calculator that you prove, as did the third sage in my 
story, that you are able so to unite the material and the spiritual, and to solve not just 
human problems but problems of the spirit. My question then is as follows: Which is the 
famous act of multiplication, which all histories mention and all men of culture know 
well, which uses only one factor?” 



That question took the illustrious gathering by surprise. Some of those present did not 
conceal their impatience. A judge at my side grumbled in irritation, ‘That question is 
outrageous!” 

Beremiz, after a moment’s thought, replied as follows: “The only multiplication using 
a single factor, known to all historians and men of culture, is the multiplication of loaves 
and fishes performed by Jesus, the son of Mary. In that multiplication there is only one 
factor: the miraculous power of the will of God.” 

“An excellent reply!” said the sage of Toledo. “It is the best answer I have ever heard, 
and the Man Who Counted has solved my problem irrefutably. Praise be to Allah!” 

Some of the more intolerant among the faithful looked at one another in astonishment, 
and there was much whispering. The caliph interrupted in a loud voice: “Silence, all of 
you! It is for us to venerate Jesus, the son of Mary, whose name is mentioned nineteen 
times in the Holy Book of Allah.” 

He turned to face the fifth wise man and said in a friendly voice, “We arc waiting for 
your question now, Sheik Nascif Rahal. You are next.” 

At the king’s command, the fifth wise man pot to his feet. He was a short fat man with 
white hair. Instead of a turban, he wore a small green cap. He was well known in 
Baghdad since he gave classes in the mosque and elucidated to scholars obscure points in 
the sayings of the Prophet. I had seen him two or three times coming out of the 
bathhouse. He spoke nervously, in a somewhat aggressive manner. 

“The value of any sage can be measured only by the power of his imagination. 
Numbers selected by chance, historical deeds remembered in detail, have only a 
momentary interest; after a time they are forgotten. How many of you remember the 
number of letters in the Koran? There are numbers, names, words, even whole books that 
are doomed to be forgotten irretrievably. Knowledge alone does not make a man wise. 
And so 1 am going to test the worth of this Persian calculator who appears before us by 
asking- him a question that cannot be answered simply — by memory or ability alone. I 
would like Beremiz Samir to tell us a story, a simple fable, in which there should appear 
a division of three by three that is suggested but not carried out, and another division of 
three by two that is carried out but leaves no remainder.” 

“A fine idea!” whispered the old man in the blue tunic. “We are going to get away 
from calculations that nobody understands and hear a story instead.” 

“But that story will have numbers, I’m sure of it,” grumbled the doctor beside me in a 
low voice. “You will see, my friend. Everything comes down to calculations, numbers, 
and problems.” 

“I hope not,” said the old man. 

I was somewhat startled and taken aback by the demands made by the fifth wise man. 
How could Beremiz invent in a moment a story that encompassed a division suggested 
but not realized or, even more difficult, a division of three by two leaving no remainder, 
when logic decrees that three divided by two must leave a remainder of one? But I put 
my anxieties to one side and trusted in the imagination of my friend and in the goodwill 
of Allah. 



The Man Who Counted, after running through his memory for a few moments, began 
to tell the following story. 


The Man Who Counted tells us a story. The tiger suggests how to divide three by 
three. The jackal proposed the division of three by two. How to arrive at a quotient in the 
mathematics of the strong. The sheik in the green cap praises Beremiz. 

In the name of Allah, the Wise and Merciful! 

On one occasion, the lion, the tiger, and the jackal left the dark cave where they lived 
and set out on a friendly pilgrimage to wander through the world in search of some region 
rich-in flocks of tender sheep. 

In the middle of the great jungle, the fearsome lion, who naturally led the group, felt 
weariness in his rear paws and throwing back his huge head, let loose a roar so fierce that 
the nearest trees trembled. 

Startled, the tiger and the jackal looked at each other. That terrifying roar with which 
their dangerous leader disturbed the silence of the forest meant to the other two animals, 
loosely translated, “I am hungry!” 

“I can understand your impatience,” said the jackal anxiously to the lion. “But I can 
assure you that in this jungle there is a secret path that no one knows, and it will lead us 
quickly to a small village, almost a ruin, where there is abundant hunting in easy reach of 
our claws ...” 

“Then let us go, jackal!” roared the lion. “Lead me to this wonderful place!” At 
evening, fed by the jackal, the travelers reached the top of a low mountain, from where 
they could see a broad, green plain. In the middle of the plain, they saw three gentle 
animals grazing, unaware” of any danger: a sheep, a pig, and a rabbit. 

At the sight of such easy and sure prey, the lion shook his abundant mane with 
obvious satisfaction and, his eyes shining with greed, turned to the tiger and said, in a 
seemingly friendly tone, “Now, my fine tiger! I see there three splendid and tasty 
mouthfuls: a sheep, a pig, and a rabbit. You, who are an expert, are charged with dividing 


30 

THREE OF A KIND 



them among the three of us. Do it justly and equably — divide these three animals among 
three hunters in a brotherly fashion.” 

Flattered at the invitation, the vain tiger, after protesting his inadequacy with roars of 
false modesty, answered as follows: “O King, the division you have generously proposed 
is quite simple and can be done with relative ease. The sheep, which is the choicest and 
tastiest, could satisfy the hunger of a whole pride of desert lions, and so it is yours alone, 
yours absolutely. And that pig skinny, dirty, and sad; not equal to one leg of that fat sheep 
will be for me, since I am modest and content with little. And finally, that small and 
miserable rabbit with little flesh, unworthy of the palate of a king, will fall to our friend 
the jackal as a reward for the valuable directions he gave us.” 

“Idiot! Egoist!” roared the lion, terrifying in his fury. “Who taught you to make 
divisions like that? Imbecile! Who has ever seen three divided by three giving a result 
like that?” 

And raising his paw, he swiped the head of the unsuspecting tiger so fiercely that he 
fell dead a few feet away. Then, turning to the jackal, who had witnessed that tragic 
division of three by three with some horror, the lion spoke as follows: “Now, my dear 
jackal, I have always had the highest regard for your intelligence. I know you are a most 
ingenious and clever animal, and I know no other who can resolve the most difficult 
problems as cleverly. And so I charge you to make this division, such a simple and trivial 
one, which the stupid tiger, as you have just seen, could not solve satisfactorily. Take a 
look, friend jackal, at these three mouthwatering animals: the sheep, the pig, and the 
rabbit. There are two of us and three meals to divide. So do your division — 1 wish to 
know which is my exact share.” 

“I am no more than your humble and poor servant,” sniveled the jackal meekly. “I 
must blindly obey my orders. Like a wise calculator, I am going to divide these three 
animals by two, a simple division! The most mathematically sure and fair division is the 
following: that splendid sheep, a feed worthy of a king, is for your royal mouth alone, 
since you are unquestionably the king of all animals. That appetizing pig, whose gentle 
grunts you can hear from here, must also be destined for your royal palate, since those 
who know say that the flesh “of a pig gives both strength and energy to lions. And the 
skittish rabbit with its large ears is also for you as a savory bite, since kings traditionally, 
at the finest banquets, must always enjoy the most delicate fare.” 

“Incomparable jackal!” exclaimed the lion, enchanted with the division he had just 
heard. “How wise and hannonious your words always are! Who taught you that 
wonderful trick of dividing three by two so surely and perfectly?” 

“It was the justice you just dealt out a moment ago to the tiger for not knowing how to 
divide three by two when one of the two is a lion and the other a mere jackal- In the- 
mathematics of the strong, I always say, the quotient is always clear, while to the weak 
must fall only the remainder.” 

And from that day on, suggesting such divisions, inspired by meanness, the ambitious 
jackal decided that he could live in tranquility only as a parasite, receiving only the 
leftovers from the lion’s feasts. 

But he was wrong. 



After two or three weeks, the lion, angry and hungry, tired of the jackal’s servility, 
ended up killing him as be had the tiger. 

And the moral is that the truth must always be told, a thousand and one times, for the 
punishment of God is closer to the sinner than are his own eyelids. 

“And here, O most wise judge!” said Beremiz in conclusion, “is the simplest of fables 
in which “occur two divisions. The first was a division of three by three, suggested but 
not realized, the second, of three by two, realized with no remainder.” 

A deep silence followed the words of the Man Who Counted. All those present 
awaited with lively interest the verdict of the grave wise man. 

Sheik Nascif Rahal, after adjusting his green cap nervously and stroking his beard, 
gave his judgment with seeming reservation: “The story you have told complied perfectly 
with my demands. I confess that it was new to me, and, in my view, it is a worthy tale. 
Aesop the Greek himself could not have done it better. That is my opinion.” 

Beremiz’s story, approved by the sheik in the green cap, pleased all the viziers and 
noblemen present Prince Cluzir Shah, the king’s guest, spoke in a loud voice to the whole 
company: 

“The story we have just heard contains a moral lesson. Wretched adulators who fawn 
in the courts, who crawls on the carpets of the powerful, may in the beginning gain 
something by their servility, but in the end they are always punished, since God’s 
punishment is always close at hand. I shall tell it to all my friends and acquaintances 
when I return to my own country.” 

The caliph, too, thought Beremiz’s story quite wonderful and said also that that 
notable division of three by three ought to be preserved in his archives, since the story, 
because of its moral conclusion, deserved to be written in gold letters on the transparent 
wings of the white butterfly of the Caucasus. 

Immediately, the sixth wise man stepped forward. 

He was from Cordoba, in Spain, where he had lived for fifteen years before having to 
flee for having incurred the wrath of his sovereign. He was middle-aged, with a round 
face and an open, humorous expression. His admirers said that he was most adept at 
writing humorous and satirical poems directed against tyrants. For six years he had 
worked in Yemen as a simple guide. 

“Emir of the world,” he began, addressing the caliph. “I have just heard with true 
satisfaction the splendid fable about the division of three by two. That story contains, in 
my view, a great lesson and a profound truth, a truth as clear as the sun at noon. I must 
confess that moral precepts come alive when they are presented in the form of stories or 
fables. I know a story that has no divisions, square roots, or fractions, but that contains a 
problem in logic that can be solved only by means of once mathematical reasoning. I 
shall tell it in the form of a story, and we shall see how our excellent calculator shall 
solve the problem contained in it.” And the wise man of Cordoba told the following 
story. 



31 

IN BLACK AND WHITE 



The wise man of Cordoba tell a story. The three suitors of Dahize. The puzzle of the 
five discs. How Beremiz explained the reasoning of a clever suitor. 

Macudo, the famous Arab historian, in the twenty-two volumes of his writings, speaks 
of the seven seas, the great rivers, famous elephants, the stars, the mountains, the 
different kings of China, and a thousand other things, but he does not even mention “the 
name of Dahize, the only daughter of King Cassim the Uncertain. It does not matter. In 
spite of that, Dahize will never be forgotten, since in Arab manuscripts her name appears 
in more than 400,000 verses in which hundreds of poets rhapsodize over her beauty. The 
ink used in describing her eyes alone would, if it were turned into oil, be enough to light 
up the city of Cairo for half a century. You may think I am exaggerating. But, my 
brothers, I am not, since exaggeration is a form of lying. But let me get to my story. 

“When Princess Dahize was eighteen years and twenty-seven days old, her hand in 
marriage was sought by three princes whose names have passed into legend: Aradin, 
Benefir, and Comozan. 

“King Cassim was uncertain. Of the three rich suitors, how could he choose the one 
who should marry his daughter? If he were to do so, it could have the following fatal 
result: he, the king, would gain a son-in-law, but the two unsuccessful suitors would 
become his bitter enemies. It was a hard decision for a sensitive and cautious king who 
only wanted to live in peace with his people and his neighbors. He asked Princess 
Dahize, but she declared only that she would marry the one who was most intelligent. 

“Her decision pleased King Cassim, for he saw a simple solution to what seemed an 
impossible choice. He summoned five of the wisest men in his court and told them to put 
the three princes through a rigorous test to see which of the three was the most intelligent. 

“When they had done so, the wise men reported to the king that all three princes were 
indeed most intelligent. They were well versed in mathematics, literature, astronomy, and 
physics. They could solve difficult chess problems, the subtleties of geometry, and all 
kinds of complex enigmas. ‘We do “not see any way,’ said the wise men; “of making” a 
clear decision in favor of one of them. 

“After this distressing failure, the king decided to consult a dervish who had a 
reputation for knowing much about magic and the occult. 

“The dervish addressed himself to the king. ‘I know only one way that will allow us to 
decide which prince is the most intelligent of the three — the test of the five disks.’ 

“‘Then let us do it!” exclaimed the king. 


“The three princes were summoned to the palace, and the dervish, showing them five 
simple wooden disks, said to them, ‘Here are five disks, two of them black and three of 
them white. They are all the same size and weight and are different only in color.’ 

“Next, a page carefully bound the eyes of the three princes so that they could-see 
nothing. The old dervish then picked three disks at random and fastened one each to the 
backs of the three suitors, saying as he did so, ‘Each one of you has on his back a disk 
whose color you do not know. You are to be questioned in turn. The one who discovers 
the color of the disk he is wearing will be declared the winner and will receive the hand 
of the beautiful Dahize in marriage. The first one questioned can look at the disks of the 
other two. The second can see only the disk of the third, and the third must make his 
reply seeing none of the others. The one who gives the correct answer must, in order to 
prove that he was not simply guessing, justify his answer by clear reasoning. Now, who 
wants to go first?’ 

“‘Let me be first,’ said Prince Comozan promptly. 

“The page removed the bandage from his eyes, and Prince Comozan saw the disks on 
the backs of his two rivals. The dervish took him aside to hear his answer, but it was 
wrong. Declaring himself beaten, he withdrew. He had seen the two disks on the hacks of 
the other princes and still not been able to determine the color of his own disk. 

“‘Prince Comozan has failed,’ said the king in a loud voice, to inform the other two. 

“Then let me be next,’ said Prince Benefir. Once his eyes were uncovered, the second 
prince saw the disk worn by the third on his back. He motioned to the dervish and 
whispered his reply to him. The dervish shook his head. The second prince was also 
mistaken and was given leave to withdraw immediately. Only one was left, Prince 
Aradin. 

“When the kin” announced that-the second suitor had also failed, he approached with 
his eyes still bandaged and announced in a loud voice the correct color of the disk on his 
back.” 

When the story was finished, the wise man pf Cordoba turned to Beremiz and said, “In 
making his answer. Prince Aradin reasoned in such a way as to reach with complete 
certainty the solution to the problem of the five disks and to win the hand of the beautiful 
Dahize. Now, I wish you to tell me, first, what his reply was and, second, how he could 
be so “sure of the color of his own disk.” 

Beremiz, his head down, thought for some time; then, looking up, he gave the 
following explanation in a firm, clear voice. 

“Prince Aradin, the hero of your strange story, said to King Cassim, “My disk is 
white,’ and in so saying he knew that his answer was true. What, then, was the reasoning 
that led him to this conclusion? He considered what the first two suitors must have seen. 

“The first prince, Comozan, saw the two disks of his rivals but, nevertheless, gave a 
wrong answer. Why did he err? He erred because he was uncertain. But if he had seen 
two black disks he could not have made a mistake or felt any doubt and would have said 
to the king, ‘I see that my rivals each wears a black disk, and since there are only two 
black disks, mine must be white.’ 



“But Comozan’s answer was wrong, so consequently the disks he saw were not both 
black. If they were not both black, two possibilities remain. Either they were both white, 
or one was white and the other black. If Comozan saw two white disks, Aradin reasoned, 
then the disk on my back must be white. But if Comozan saw one white disk and one 
black, which of us wore the black disk? If it were I, Aradin reasoned, Bene fir would have 
known the answer. 

“In effect, Bencfir would have reasoned as follow: I see that the third prince is 
wearing a black disk. If mine were also black, then Comozan, the first prince, seeing two 
black disks, could not have made a mistake; but since he was wrong, my disk must be 
white. But the second prince was also wrong, and so he must also have been uncertain. 
His uncertainty must have come, Aradin reasoned, from seeing on my back not a black 
disk but a white one. So Aradin concluded as follows: according to the second 
hypothesis, my disk then must be white. 

“That was how Aradin reasoned,” stated Beremiz, “in resolving the problem with 
absolute certainty and being able to say, ‘My disk is white.’” 

The wise man of Cordoba then addressed the caliph, declaring that Beremiz’s solution 
to the problem of the five disks was correct and quite brilliant. His simple and clear 
reasoning was impeccable, and the wise man said he was sure that all those present had 
understood the problem-and would be able to recount it afterwards at any caravan halt in 
the desert. 

A Yemenite sheik, seated in front of me on a red cushion, a dark, evil-faced man, 
wearing many jewels, murmured to a friend at his side, “Do you hear that, Captain 
Sayeg? That man from Cordoba .says that we have all understood that story of the white 
disks and the black disks. I doubt it very much. As for me, I didn’t understand a single 
word of it. Only a mad dervish would have thought of putting black and white disks on 
the backs of the three suitors, don’t you thi nk ? Wouldn’t it have been more practical to 
hold a camel race in the desert? There would have been a clear winner, and the matter 
would be perfectly resolved, no?” Captain Sayeg did not reply, and seemed to ignore the 
stupid Yemenite who wanted to solve a problem of love by a camel race in the desert. 

The caliph declared affably that Beremiz had come through the sixth proof of the 
contest. 

Would our friend, the Man Who Counted, be successful in the seventh and last round? 
Would he surmount it with the same brilliance? Only Allah kn ows the truth! 

Finally, everything seemed to be going as we had wished. 


32 

A BALANCING ACT 



Beremiz is questioned by a Lebanese astronomer. The problem of the lightest pearl. 
The astronomer quoted a poem in praise of Beremiz. 

His name was Mohildin Ihaia Banabixacar, geometer, astronomer, and one of the most 
notable names” in all of Islam, the seventh and last sage to face Beremiz. He had been 
born in Lebanon, and his name was inscribed on five mosques and his books were read 
even by Christians. It would be impossible to find under the skies of Islam anyone of 
more lucid intelligence or more substantial knowledge. 

The learned Banabixacar, his voice clear and precise, spoke as follows: “I am truly 
delighted with everything I have heard up until now. The distinguished Persian 
mathematician has just demonstrated over and over again his unquestionable talent. I 
should like, as my part in this brilliant contest, to present him with an interesting problem 
I learned when I was young from a Buddhist priest well versed in the science of 
numbers.” 

The caliph, showing a lively interest, declared, “Let us hear it then, Arab brother! 

We shall listen to your problem with the greatest of pleasure. I hope that our young 
Persian, who until now has shown himself insuperable hi the realms of calculation, will 
know how to solve the puzzle of the old Buddhist.” 

The Lebanese sage, seeing that his words had caught the attention of the king and all 
those present, spoke as follows, looking keenly at the Man Who Counted: “My problem 
should properly be called The Problem of the Lightest Pearl.” 

He went on, “A merchant of Benares, in India, had in his possession eight pearls 
identical in shape, size, and color. Of these eight pearls, seven were the same weight, 
while the eighth weighed slightly less than the others. How could the merchant discover 
which pearl was lighter, using a scale but making only two weighings and not using any 
weights? That is the problem, and may Allah lead you, O Counting man, to a simple and 
perfect solution.” 


Hearing the problem so stated, a white-haired sheik wearing a gold collar, sitting by 
Captain Sayeg, murmured in a low voice, “What an elegant problem! This Lebanese is a 
man of genius. Praise be to Lebanon, the country of cedars!” 

Beremiz Samir, after his customary pause for thought, spoke in a slow and firm voice: 
“It does not seem to me too difficult, the problem of the Buddhist. A path of clear 
reasoning can surely lead us to a solution.” 

“Let us see. There are eight identical pearls, identical in shape, color, size, and 
brilliance. We have been assured that one of those eight pearls is lighter than the others, 
while the other seven all weigh the same. The only way of discovering which pearl 
weighs less than the other is to use a scale, a finely calibrated scale with long arms and 
light plates. One thing more: the scale must be precise. If I took the pearls two by two and 
placed them on the scale, one in each pan, it would naturally be easy to discover 
eventually the light pearl. But supposing it were one of the last pair, I would have been 
obliged to make four weighings, and the problem demands that I discover the light pearl 
in only two. The simplest solution seems to me to be the following; 

“Let us divide the pearls in three groups: A, B, and C. Group A will consist of three 
pearls, group B also of three, and group C of the remaining two. In only two weighings, I 
shall find out which is the light pearl, given that the other seven weigh exactly the same. 

“Let us put the groups A and B on the scale, one in each pan. Two things can happen: 
either groups A and B weigh the same, or one weighs more than the other. 

“In the first case, where A and B weigh the same, we can be sure that the light pearl 
belongs to neither of the two groups, therefore it must be one of the two pearls that make 
up group C. We shall then weigh these two pearls, one in each pan, and the scale, at this 
second weighing, will indicate which one is lighter. 

“In the second case, where group A weighs less than group B or vice versa, it will be 
clear that the light pearl is in one of the groups. From that group let us take any two 
pearls and put them in the pan of the scale for a second weighing. If the scale balances 
equally, and then the third pearl from the group, the one we have put aside, is the light 
pearl if, however, the balance is unequal, then the light pearl must be in the pan that rises. 

“And so, the problem posed by the illustrious Buddhist priest is solved, and I present 
my solution to our distinguished guest, the sage from Lebanon. 

Banabixacar, the astronomer, agreed that the solution offered by Beremiz was indeed 
impeccable, and he spoke in the following manner: “Only a true mathematician could 
produce such perfect reasoning. The solution I have just listened to is a true poem in its 
beauty and simplicity.” 

And, in homage to the Man Who Counted, the Lebanese astronomer quoted the 
following verses from Omar Khayyam, a Persian poet of the highest Quality and also a 
celebrated mathematician: 

If you have kept one rose of love close to your heart. . . 

If you have addressed your humble prayer to one just and supreme God. . . 

If raising your cup, you one day sing your praise of life . . . 

You have not lived in vain ... 



Much moved, Beremiz bowed his thanks for this .homage and placed his right hand on 
his heart. 


33 

EYE TO EYE 



The offer that Caliph al-Mutasim made to the Man Who Counted. Beremiz will not 
accept either gold, goods, or palaces. He asked instead for a hand. The problem of the 
black eyes and the blue eyes. Beremiz discovers by reasoning the color of the five slaves. 

When Beremiz had finished solving the problem set him by the wise man of Lebanon, 
the sultan, after conferring in a low voice with two of his counselors, addressed him. 

“By the answers you have given to all the questions, you have shown yourself most 
deserving of the reward I promised you. And so I give you a choice: Do you wish to 
receive twenty thousand gold dinars, or would you rather have a palace in Baghdad? 
Would you like to be governor of a province, or would you prefer the post of vizier in my 
court.” 

“O generous King!” replied Beremiz, deeply moved. “I seek neither riches nor titles 
nor honors nor gifts, since I know that such things are worth nothing. I am not drawn by 
the prestige that such things would give me, since my spirit does not seek the ephemeral 
glories of worldly possessions. If, however, you wish to make me the envy of all 
Muslims, as you previously said, my request is as follows: 

I wish to marry young Telassim, daughter of Sheik Iezid Abul Hamid.” 


This surprising request by the Man Who Counted caused an indescribable uproar. 
From all the comments that I heard around me, I realized that all those present were 
convinced that Beremiz was completely mad. 

“That man is crazy,” murmured the thin old man in the blue tunic behind me. “Crazy, I 
say. He rejects wealth, he turns his back on fame, and all to marry a young girl he has 
never seen!” 

“He is indeed delirious,” addled the man with the scar. “Yes, delirious. He asked for 
the hand of a girl who may very well detest him. By Allah!” 

“Could it be the spell of the blue rug?” asked Captain Sayeg in a low voice, .not 
without malice. “Has the blue rug bewitched him?” 

“Blue rug, indeed!” exclaimed the old man. “There is no spell for winning the heart of 
a woman.” 

I listened to ail those remarks while pretending to be thinking of something else. At 
Beremiz’s request, the caliph frowned and looked grave. He summoned Sheik Iezid, and 
the two of them whispered together for a few moments. What would be the result of their 
consultations? Would the sheik agree to the betrothal of his daughter? 

After a time, the caliph spoke as follows, in the midst of a deep silence: “I shall not 
oppose your happy marriage with the beautiful Telassim, O Beremiz. My esteemed friend 
here, Sheik Iezid, whom I have just consulted, will accept you as a son-in-law. I realize 
that you are a man of character, well educated, and deeply religious. Now, it is true that 
the beautiful Telassim was promised to a sheik of Damascus who is now fighting in 
Spain. But if she herself wishes to alter the course of her life, I will not stand in her way. 
It is written! The arrow, once in flight, cries happily, “I am free, I am free!” but in truth it 
deceives itself, for its destiny has been appointed by the aim of the marksman. Such is the 
case with the young “Flower of Islam! Telassim rejects a rich and noble sheik who 
tomorrow could be vizier or governor, and accepts for her husband a simple Persian 
calculator. It is written! May it be what Allah wishes!” 

The caliph paused and then continued forcefully, “Nevertheless, I lay down one 
condition. Before those gathered here, you must solve a strange problem set by a dervish 
from Cairo. If you solve it correctly, you may marry Telassim. If not, you must give up 
forever this wild dream and you will receive nothing from me. Do you accept the 
conditions?” 

“O Master of all believers!” replied Beremiz firmly. “Only tell me the problem for me 
to solve it. . .” 

And so the caliph spoke: “The problem, put very simply, is the following: I own five 
beautiful slave girls, recently purchased from a Mongol prince. Two of those young 
enchantress’s have black eyes; the other three, blue eyes. The two have a truthful answer 
to any question, whereas the three with black eyes always give a truthful answer, whereas 
the three with blue eyes are bom liars and never answer with the truth. In a few moments 
the five of them will be brought here, all of their faces covered by a heavy veil, which 
will make it impossible for you to see their faces. You must discover, with no room for 
error, which of them have black eyes and which blue eyes. You may question three of the 
five slaves, one question to each one. From the three answers, you must solve the 



problem and explain the precise reasoning that led you to your answer. Your Questions 
should be quite simple ones, well within the compass of these slaves to answer.” 

Some moments later, watched curiously by everyone present, the five slave girls 
appeared in the reception hall, their faces covered with black veils, like phantoms of the 
desert. 

“Here they are,” said the emir somewhat proudly. “Two of them as I told you have 
black eyes and speak only the truth. The other three have blue eyes and always lie.” 

“What a scandal!” muttered the thin old man, “Imagine my bad luck! My uncle’s 
daughter has black eyes, very black — yet she lies all day long!” 

His remark seemed to me out of order. It was a very serious moment, no time for 
jokes. Luckily, no one paid any attention to the nasty words of the impertinent old man. 
Beremiz realized that he had reached a decisive moment, perhaps the high point of his 
whole life. The problem that the caliph of Baghdad had presented him with was original 
and difficult and could be full of pitfalls. He was free to question three of the slave girls. 
But bow would their replies indicate the color of their eyes? Which of them should he 
question? How would he know the eye colors of the two he could not question? 

There was only one certainty, that the two with black eyes always spoke the truth and 
the other three invariably lied. But would that be enough? When Beremiz questioned 
them, the question had to be quite natural, well within the reach of the girl questioned. 
But how could he be sure of her reply, whether it was true or false? It was really very 
difficult indeed. 

The five veiled slave girls lined up in the middle of the sumptuous hall in complete 
silence. The sheiks and viziers waited for the solution to this singular problem set by their 
king with a lively interest. The Man Who Counted approached the first slave girl on the 
right of the row, at the end, and asked her quietly, 

“What color are your eyes?” 

The girl replied in a language that was apparently Chinese, a language that nobody 
present could understand. It made no sense to me. Hearing her, the caliph ordered that the 
other replies be in Arabic, simple and precise. 

This unexpected setback made things more difficult for Beremiz. He had only two 
questions left, and the answer to his first question had been completely lost. 

This did not seem to upset him, however, as he approached the second slave and asked 
her, “What was the reply that your companion just gave?” 

The second slave answered, “She said, ‘My eyes are blue.’ That reply cleared up 
nothing. Had the second slave told the truth, or was she Tying? And the first? Was that 
her real reply? 

The third slave girl, in the center of the row, was” questioned next by Beremiz. . 
“What color are the eyes of those two girls 1 have just questioned?” 

And the third girl, the last to be questioned, replied as’ follows: “The first girl has 
black eyes and the second blue eyes.” 

Beremiz paused a moment and then calmly approached the throne, speaking as 
follows. “Lord of all believers. Shadow of Allah on earth, I have a solution to your 



problem, arrived at through strict logic. The first slave girl on the right has black eyes. 
The second has blue eyes. The third has Mack eyes, and the other two have blue eyes.” 

At this the five slave girls lifted their veils, uncovering smiling faces. Great sighs 
arose throughout die reception hall. Beremiz in his faultless intelligence had exactly 
detennined the color of all their eyes. 

“All praise to the Prophet!” cried the king. “This problem has been set to hundreds of 
wise men, poets, and scribes, and at last this modest Persian is the only one who has been 
able to solve it. How did you come by your answer? Show us how you came to be so 
certain of your solution.” 

The Man Who Counted gave the following explanation: 

“When I asked the first question — what is the color of your eyes — I knew, that the 
slave girl’s answer had to be ‘My eyes are black,’ because if she had black eyes she 
would have to tell the truth, or if she had blue eyes, she would have to lie. So there could 
be only one reply, namely. ‘My eyes are black.’ I expected that answer; but when the girl 
replied to me in an unknown language, she helped me enormously. Claiming not to have 
understood, I asked the second slave girl, ‘What was the reply your companion just 
gave?’ and received the second answer. She said: my eyes are blue.’ That reply proved to 
me that the second girl was lying, since, as I have already shown, it could not have been 
the reply of the first girl. Consequently, if the second slave was lying, she had blue eyes. 
This was an important point, O King, in solving the problem. Of the five slave girls, there 
was at least one whom I had identified with mathematical certainty, namely, the second. 
She had lied, and so she had blue eyes. 

“My third and last question I asked of the girl in the center of the row. ‘What color are 
the eyes of those two girls I have just questioned?’ She gave me the following reply: The 
first girl has black eyes and the second blue eyes.’ Since I knew already that the second 
did have blue eyes, what conclusion was I to come to over the reply of the third girl? 
Very simple. The third girl was not lying, since she was confirming what I already knew, 
namely, that the second girl had blue eyes. Her reply also told me that the first slave girl 
had black eyes. Since the third girl was not lying, her words spoke the truth, and therefore 
she too had black eyes. From there, it was simple to deduce that the two other girls, by 
exclusion, had blue eyes. 

“I can assure you, O King, that in this problem, although no equations or algebraic 
symbols appear, the perfect solution can be reached only by the rigorous logic of pure 
mathematics.” 

So was resolved the caliph’s problem. But for Beremiz there soon awaited another, 
more difficult problem, that of Telassim, the treasure he had dreamed of in Baghdad. 

All praise to Allah, who created woman, love, and mathematics! 



34 

OF LIFE AND LOVE 



I bring to a close the story of Beremiz, the Man Who Counted. 

In the third moon of Rhegeb, in the year 1258, hordes of Tartars and Mongols, under 
the command of a grandson of Genghis Khan, attacked Baghdad. 

Sheik Iezid died in the fighting close to the Suleiman Bridge. The caliph al-Mutasim 
was taken prisoner and beheaded by the Mongols. The city was sacked and cruelly 
leveled. The splendid city, which for five hundred years had been a center for arts and 
letters and the sciences, was reduced to a heap of ruins. 

By good fortune, I was not on the scene of that crime perpetrated on civilization by 
these savage conquerors. Three years before, when the generous Prince Cluzir Shah 
died — May Allah give him peace! — I had gone to Constantinople with Telassim and 
Beremiz. 

Every week I visit him, and, at times, I am envious of the happy state in which he lives 
with his wife and his three sons. When I see Telassim, I am reminded of the words of the 
poet: 

Sing, birds, your purest songs! 

Shine, 0 Sun, with your sweetest light! 

Loose your arrow, God of love! 

Blessed be your love, my lady! Great your joy! 

There is no question that, of all problems, the one Beremiz solved best was that of life 
and love. 

Here I end, without numbers or formulas, the story of the Man Who Counted. 

End