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The Art of Mathemagics
Neville de Mestre

Australia has just been fortunate tohave had Professor Art Benjamin fromHarvey Mudd College (http://www.math.
hmc.edu/faculty/benjamin/) visiting thecountry for the five months from late January to early July. During that time Art wason sabbatical leave at UNSW but managed
to present his Mathemagics show in Sydney,Melbourne, Adelaide, Hobart and Brisbane.
He also appeared on two television shows
on Channel 9 and participated in numerous
radio interviews on the ABC.
I attended his show in Brisbane on Friday 8 July just before he left for his homein the U.S. In case you inadvertently missed
one of his splendid offerings, here is a summary of the mathematical ideas contained in
his show which appeals to the general public everywhere. Incidentally, Art has performed at Disneyland and off-Broadway, so
he is one of the few active mathematicians
who are taking the message of the importance of mathematics to the public.
The Brisbane show was sponsored by theQueensland University of Technology, theUniversity of Queensland and Queensland
Association of Mathematics Teachers. Itwas held in The Hall at the recently-openedQUT Creative Industries Precinct at theKelvin Grove campus of QUT. The audience of about 120 consisted of mathematics
teachers and lecturers, university students,plus families consisting of parents and youngchildren.
Art Benjamin is a true performer, andappeared in a dinner suit, sporting a bow
tie with numbers on it. The show beganwith Art performing his invisible card trick
to warm us all up to his capabilities. I still
can’t figure out how he does this one, butknow that it must be using either sleight ofhand or sleight of mathematics!

Art Benjamin (photo: Anne White)

Next Art demonstrated how he could
beat any calculator in squaring either 2-digit
or 3-digit numbers in his head. With 4digit numbers he finished in a draw withthe calculators. However, with squaring 5digit numbers in his head, the more advanced calculators (those that could record
10 digit numbers accurately) were in front
by a whisker.
Next he used the number 6084 and asked
four members of the audience with calculators to each choose any secret 3-digit number and obtain the product of their chosen
number and 6084. They then were asked toeach tell Art nearly all the digits in their answer in any order, but to keep one digit back.
To the amazement of the audience, he thentold them the missing digit. Mathematicians will recognize the use of the methodof “casting out 9s” to achieve this “magic”
feat.
Next he wrote down the first 60 digits ofthe value of π from memory, and emphasisedthat he knew more, but it would take sometime to do this, and 60 was probably enoughNeville de Mestre

to demonstrate the point that he was goingto reveal to us all later, namely, how he manages to memorize vast amounts of numbers.
He then mesmerized the audience withhis ability to name the day of the week whenanyone was born given the day, month andyear of their birth.
Continuing with the birthday theme, asmall girl named Georgina was chosen fromthe audience and gave her birthday as 29
September 1994. Art drew up a 4 × 4 gridon his board and placed the numbers 29, 9,9, 4 in the top row. He then added numbers
quickly to the remaining 12 cells to create a
special magic square for Georgina. The total of the top row was 51, and this was thesame for all rows, all columns, the central
2 × 2 cells, the four corner 2 × 2 cells andthe diagonals.
The emphasis now shifted from displaying his abilities to how he carried these calculations out. It was mainly to do withmemorizing numbers using phonetics to construct words that would represent a stringof numbers. However the squaring of numbers had algebra embedded in it through theidentity
A2 = (A + d)(A − d) + d2

Although this squaring method has been
known for some time before Art was born,
he said that he chanced upon it independently and indirectly as a Grade 8 student.The day of one’s birth was performedby associating specific 1-digit numbers witheach year and month respectively, adding
these to the number of the day in the monthand using congruences modulo 7.
Art invited questions from the audience,
and although the formal presentation finished after an hour, he was still at the board
some 45 minutes later with 25 of the audience still clustered about him while hefielded questions on π, cycloids, Pascal’s triangle, e, and patterns.
I was fortunate to spend some time withArt before his show, and here is a little
mathematical gem that he shared with me
and I pass it on for readers to think about.
Write down any two positive integers andadd them to obtain a third integer. Keepadding these terms in Fibonacci style until you reach the tenth term, for example,
T7 = T5 + T6 . Calculate the ratio T10 /T9 .
The first digits will always be 1.61. . . andapproaches the Golden Ratio if we consider
the ratio of later consecutive terms. Why isthis so?

School of Information Technology, Bond University, Gold Coast, Queensland, QLD 4229
E-mail: neville de mestre@staff.bond.edu.au
September 23, 2005.

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