nyuuzyou/wikis huggingface dataset:
Henry Ernest Dudeney/Puzzles and Curious Problems/10 - Mental Arithmetic/Solution

== by : $10$ ==
;Mental Arithmetic
== Solution ==
Buying a box of $100$ is actually $5 \oldpence$ more than buying the cigars individually.
One presumes that the box itself must have an intrinsic worth of $5 \oldpence$.
The rule for calculating the price of $100$ of something costing $n \oldpence$ is:
:reduce $n \oldpence$ to farthings to get $4 n$ farthings
:double this amount to get $8 n$.
:add $8 n \shillings$ to $4 n \oldpence$ to get the price of $100$.
== Proof ==
We have:
:$100 \times 7 \tfrac 3 4 = 700 + 3 \times 25 = 775$
But:
:$775 \oldpence = 64 \shillings 7 \oldpence$
which is $5 \oldpence$ less than buying the whole box.
One presumes that the box itself may have an intrinsic worth of $5 \oldpence$.
Let $n$ be the number of (old) pennies an item costs.
Hence its cost in farthings is $4 n$.
Then $100$ of them cost $100 n \oldpence$ 
This is $400 n$ farthings.
This is:
:$\dfrac {400 n} {48} \shillings = {\dfrac {384 n} {48} \shillings} + 16 n \times {\tfrac 1 4 \oldpence}$
That is:
:$100 n \oldpence = {8 n \shillings} + {4 n \oldpence}$
== Sources ==
* : Solutions: $10$. -- Mental Arithmetic
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/Solutions
Henry Ernest Dudeney/Puzzles and Curious Problems/110 - An Absolute Skeleton

== by : $110$ ==
;An Absolute Skeleton
:Here is a good skeleton puzzle.
:The only conditions are:
::$(1)$ No digit appears twice in any row of figures except the dividend.
::$(2)$ If $2$ be added to the last figure in the quotient it equals the last but one,
::::and if $2$ be added to the third figure from the end it gives the last figure but $3$ in the quotient.
::::That is to say, the quotient might end in, say, $9742$, or in $3186$.
 ********
 ------------
 ***)***********
 ***
 ---
 ***
 ***
 ----
 ****
 ****
 -----
 ***
 ***
 ----
 ****
 ****
 -----
 ****
 ****
 -----
 ****
 ****
 -----
 ****
 ****
 ----
:We have only succeeded in finding a single solution.
prime.mover reports:
:A previous owner of this copy of open before me on my study desk has pencilled in a second solution.
=== Click here for solutions ===
== Sources ==
* : Arithmetical and Algebraical Problems: Digital Puzzles: $110$. -- An Absolute Skeleton
* : Arithmetical and Algebraical Problems: Skeleton Puzzles: $147$. An Absolute Skeleton
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/110 - An Absolute Skeleton
Category:Henry Ernest Dudeney/Puzzles and Curious Problems
Category:Skeleton Puzzles
Henry Ernest Dudeney/Puzzles and Curious Problems/110 - An Absolute Skeleton/Mistake

== Source Work ==
:
:Arithmetical and Algebraical Problems
::Digital Puzzles
:::$110$ -- An Absolute Skeleton
:
:Arithmetical and Algebraical Problems
::Skeleton Puzzles
:::$147$ -- An Absolute Skeleton
== Mistake ==
:It can soon be discovered that the divisor must be $312$,
::that $9$ cannot be in the quotient because $9$ times the divisor contains a repeated figure.
:We therefore know that the quotient contains all the figures $1$ to $8$ once, and the rest is comparatively easy.
But of course his first deduction is not necessarily the case, as we have a second solution, where the divisor is $310$.
== Sources ==
* : Solutions: $110$. -- An Absolute Skeleton
* : Answers: $147$. An Absolute Skeleton
Category:Mistakes/Puzzles and Curious Problems
Category:Mistakes/536 Puzzles & Curious Problems
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/110 - An Absolute Skeleton
Henry Ernest Dudeney/Puzzles and Curious Problems/110 - An Absolute Skeleton/Solution

== by : $110$ ==
;An Absolute Skeleton
== Solution ==
=== Solution 1 ===
=== Solution 2 ===
== Proof ==
=== Declarations ===
=== Initial Deductions ===
== Sources ==
* : Solutions: $110$. -- An Absolute Skeleton
* : Answers: $147$. An Absolute Skeleton
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/Solutions
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/110 - An Absolute Skeleton
Henry Ernest Dudeney/Puzzles and Curious Problems/110 - An Absolute Skeleton/Solution/Declarations

== by : $110$ ==
;An Absolute Skeleton
== Declarations ==
This section declares the variables which are to be used during the deduction of the solution to this skeleton puzzle.
Let $D$ denote the divisor.
Let $Q$ denote the quotient.
Let $N$ denote the dividend.
Let $q_1$ to $q_8$ denote the digits of $Q$ which are calculated at each stage of the long division process in turn.
Let $n_1$ to $n_8$ denote the partial dividends which are subject to the $1$st to $8$th division operations respectively.
Let $j_1$ to $j_8$ denote the least significant digits of $n_1$ to $n_8$ as they are brought down from $N$ at each stage of the long division process in turn.
Let $p_1$ to $p_8$ denote the partial products generated by the $1$st to $8$th division operations respectively: $p_k = q_k D$
Let $d_1$ to $d_8$ denote the differences between the partial dividends and partial products: $d_k = n_k - p_k$.
By the mechanics of a long division, we have throughout that:
:$n_k = 10 d_{k - 1} + j_k$
for $k \ge 2$.
Hence we can refer to elements of the structure of this long division as follows:
 ******** --> Q
 ------------ ---
 ***)*********** --> D ) N
 *** --> p_1
 ---
 *** --> n_2
 *** --> p_2
 ----
 **** --> n_3
 **** --> p_3
 -----
 *** --> n_4
 *** --> p_4
 ----
 **** --> n_5
 **** --> p_5
 -----
 **** --> n_6
 **** --> p_6
 -----
 **** --> n_7
 **** --> p_7
 -----
 **** --> n_8
 **** --> p_8
 ----
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/110 - An Absolute Skeleton
Henry Ernest Dudeney/Puzzles and Curious Problems/110 - An Absolute Skeleton/Solution 1

== by : $110$ ==
;An Absolute Skeleton
== Solution ==
 32418675
 ------------
 312)10114626600
 936
 ---
 754
 624
 ----
 1306
 1248
 -----
 582
 312
 ----
 2706
 2496
 -----
 2106
 1872
 -----
 2340
 2184
 -----
 1560
 1560
 ----
== Proof ==
From the initial deductions, we have determined that:
:$(1): \quad$ It is possible that there is a solution in which the divisor is $312$
:$(2): \quad$ If this is the case, then the quotient is one of:
:::$32417586$
:::$32418675$
:::$32816475$
:::$32817564$
Thus the dividend in each case will be:
:$10114286832$
:$10114626600$
:$10238740200$
:$10239079968$
It remains to investigate each one.
Hence we set up a long division and check the partial dividends in each case.
 32417586
 ------------
 312)10114286832
 936
 ---
 754
 624
 ----
 1302
 1248
 -----
 548
 312
 ----
 2366
 ****
and we need go no further.
 32418675
 ------------
 312)10114626600
 936
 ---
 754
 624
 ----
 1306
 1248
 -----
 582
 312
 ----
 2706
 2496
 -----
 2106
 1872
 -----
 2340
 2184
 -----
 1560
 1560
 ----
and we have found a solution.
Let us try the remaining solutions:
 32816475
 ------------
 312)10238740200
 936
 ---
 878
 ***
and we need not continue with this one.
Finally:
 32817564
 ------------
 312)10239079968
 936
 ---
 879
 624
 ----
 2550
 ****
and again it is seen that this also does not fit the criteria.
Hence the only solution with $D = 312$ is:
:$\dfrac {10114626600} {312} = 32418675$
== Sources ==
* : Solutions: $110$. -- An Absolute Skeleton
* : Answers: $147$. An Absolute Skeleton
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/110 - An Absolute Skeleton
Henry Ernest Dudeney/Puzzles and Curious Problems/110 - An Absolute Skeleton/Solution 2

== by : $110$ ==
;An Absolute Skeleton
== Solution ==
 32819764
 ------------
 310)10174126840
 930
 ---
 874
 620
 ----
 2541
 2480
 -----
 612
 310
 ----
 3026
 2790
 -----
 2368
 2170
 -----
 1984
 1860
 -----
 1240
 1240
 ----
== Proof ==
From the initial deductions, we have determined that:
:$(1): \quad$ It is possible that there is a solution in which the divisor is $310$
:$(2): \quad$ If this is the case, then the quotient is one of:
:::$32418697$
:::$32419786$
:::$32816497$
:::$32819764$
Thus the dividend in each case will be:
:$10049796070$
:$10050133660$
:$10173114070$
:$10174126840$
It remains to investigate each one.
Hence we set up a long division and check the partial dividends in each case.
 32418697
 ------------
 310)10049796070
 930
 ---
 749
 620
 ----
 1297
 1240
 -----
 579
 310
 ----
 2696
 ****
and we need go no further.
 32419786
 ------------
 310)10050133660
 930
 ---
 750
 620
 ----
 1301
 1240
 -----
 613
 310
 ----
 3033
 ****
 -----
and we need go no further.
 32816497
 ------------
 310)10173114070
 930
 ---
 873
 620
 ----
 2531
 2480
 -----
 511
 ***
 ----
and we need go no further.
Finally:
 32819764
 ------------
 310)10174126840
 930
 ---
 874
 620
 ----
 2541
 2480
 -----
 612
 310
 ----
 3026
 2790
 -----
 2368
 2170
 -----
 1984
 1860
 -----
 1240
 1240
 ----
and we have found a solution.
Hence the only solution with $D = 310$ is:
:$\dfrac {10174126840} {310} = 32819764$
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/110 - An Absolute Skeleton
Henry Ernest Dudeney/Puzzles and Curious Problems/111 - Odds and Evens

== by : $111$ ==
;Odds and Evens
:Every asterisk and letter represents a figure,
::and "$\mathtt O$" stands for an odd figure ($1$, $3$, $5$, $7$, or $9$)
::while "$\mathtt E$" represents an even figure ($2$, $4$, $6$, $8$, or $0$).
:Can you construct an arrangement complying with these conditions?
:There are $6$ solutions.
:Can you find one, or all of them?
 *******
 ----------
 ***)*********
 OE*
 -----
 ****
 OO**
 -----
 ***
 EE*
 ----
 ***
 EO*
 ----
 ****
 EE**
 -----
 ***
 OO*
 ---
=== Click here for solution ===
== Historical Note ==
== Sources ==
* : Arithmetical and Algebraical Problems: Digital Puzzles: $111$. -- Odds and Evens
* : Arithmetical and Algebraical Problems: Skeleton Puzzles: $148$. Odds and Evens
Category:Henry Ernest Dudeney/Puzzles and Curious Problems
Category:Skeleton Puzzles
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/111 - Odds and Evens
Henry Ernest Dudeney/Puzzles and Curious Problems/111 - Odds and Evens/Historical Note

== Historical Note on by : $111$ - Odds and Evens ==
This puzzle was originally submitted to by a person identified merely as the Rev. E. F. O.
== Sources ==
* : Arithmetical and Algebraical Problems: Digital Puzzles: $111$. -- Odds and Evens
* : Arithmetical and Algebraical Problems: Skeleton Puzzles: $148$. Odds and Evens
Category:Historical Notes
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/111 - Odds and Evens
Henry Ernest Dudeney/Puzzles and Curious Problems/111 - Odds and Evens/Solution

== by : $111$ ==
;Odds and Evens
== Solution ==
=== Solution $1$ ===
=== Solution $2$ ===
=== Solution $3$ ===
=== Solution $4$ ===
=== Solution $5$ ===
=== Solution $6$ ===
== Proof ==
=== Declarations ===
=== Initial Deductions ===
== Sources ==
* : Solutions: $111$. -- Odds and Evens
* : Answers: $148$. Odds and Evens
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/Solutions
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/111 - Odds and Evens
Henry Ernest Dudeney/Puzzles and Curious Problems/111 - Odds and Evens/Solution/Declarations

== by : $111$ ==
;Odds and Evens
== Declarations ==
This section declares the variables which are to be used during the deduction of the solution to this skeleton puzzle.
Let $D$ denote the divisor.
Let $Q$ denote the quotient.
Let $N$ denote the dividend.
Let $q_1$ to $q_7$ denote the digits of $Q$ which are calculated at each stage of the long division process in turn.
Let $n_1$ to $n_7$ denote the partial dividends which are subject to the $1$st to $7$th division operations respectively.
:Note that $n_2 = 0$, but has been retained for consistency of numbering.
Let $j_1$ to $j_7$ denote the least significant digits of $n_1$ to $n_7$ as they are brought down from $N$ at each stage of the long division process in turn.
Let $p_1$ to $p_7$ denote the partial products generated by the $1$st to $7$th division operations respectively: $p_k = q_k D$
:Again note that $p_2 = 0$, but has been retained for consistency of numbering.
Let $d_1$ to $d_7$ denote the differences between the partial dividends and partial products: $d_k = n_k - p_k$.
By the mechanics of a long division, we have throughout that:
:$n_k = 10 d_{k - 1} + j_k$
for $k \ge 2$.
Hence we can refer to elements of the structure of this long division as follows:
 ******* --> Q
 ---------- ---
 ***)********* --> D ) N
 OE* --> p_1
 -----
 **** --> n_3
 OO** --> p_3
 -----
 *** --> n_4
 EE* --> p_4
 ----
 *** --> n_5
 EO* --> p_5
 ----
 **** --> n_6
 EE** --> p_6
 -----
 *** --> n_7
 OO* --> p_7
 ---
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/111 - Odds and Evens
Henry Ernest Dudeney/Puzzles and Curious Problems/111 - Odds and Evens/Solution 1

== by : $111$ ==
;Odds and Evens
== Solution ==
 3071294
 ----------
 249)764752206
 747
 -----
 1775
 1743
 -----
 322
 249
 ----
 732
 498
 ----
 2340
 2241
 -----
 996
 996
 ---
== Proof ==
== Sources ==
* : Solutions: $111$. -- Odds and Evens
* : Answers: $148$. Odds and Evens
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/111 - Odds and Evens
Henry Ernest Dudeney/Puzzles and Curious Problems/111 - Odds and Evens/Solution 2

== by : $111$ ==
;Odds and Evens
== Solution ==
 3081294
 ----------
 249)767242206
 747
 -----
 2024
 1992
 -----
 322
 249
 ----
 732
 498
 ----
 2340
 2241
 -----
 996
 996
 ---
== Proof ==
== Sources ==
* : Solutions: $111$. -- Odds and Evens
* : Answers: $148$. Odds and Evens
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/111 - Odds and Evens
Henry Ernest Dudeney/Puzzles and Curious Problems/111 - Odds and Evens/Solution 3

== by : $111$ ==
;Odds and Evens
== Solution ==
 4081293
 ----------
 245)999916785
 980
 -----
 1991
 1960
 -----
 316
 245
 ----
 717
 490
 ----
 2278
 2205
 -----
 735
 735
 ---
== Proof ==
== Sources ==
* : Solutions: $111$. -- Odds and Evens
* : Answers: $148$. Odds and Evens
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/111 - Odds and Evens
Henry Ernest Dudeney/Puzzles and Curious Problems/111 - Odds and Evens/Solution 4

== by : $111$ ==
;Odds and Evens
== Solution ==
 4071293
 ----------
 245)997466785
 980
 -----
 1746
 1715
 -----
 316
 245
 ----
 717
 490
 ----
 2278
 2205
 -----
 735
 735
 ---
== Proof ==
== Sources ==
* : Solutions: $111$. -- Odds and Evens
* : Answers: $148$. Odds and Evens
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/111 - Odds and Evens
Henry Ernest Dudeney/Puzzles and Curious Problems/111 - Odds and Evens/Solution 5

== by : $111$ ==
;Odds and Evens
== Solution ==
 3081294
 ----------
 248)764160912
 744
 -----
 2016
 1984
 -----
 320
 248
 ----
 729
 496
 ----
 2331
 2232
 -----
 992
 992
 ---
== Proof ==
== Sources ==
* : Solutions: $111$. -- Odds and Evens
* : Answers: $148$. Odds and Evens
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/111 - Odds and Evens
Henry Ernest Dudeney/Puzzles and Curious Problems/111 - Odds and Evens/Solution 6

== by : $111$ ==
;Odds and Evens
== Solution ==
 3071294
 ----------
 248)761680912
 744
 -----
 1768
 1736
 -----
 320
 248
 ----
 729
 496
 ----
 2331
 2232
 -----
 992
 992
 ---
== Proof ==
== Sources ==
* : Solutions: $111$. -- Odds and Evens
* : Answers: $148$. Odds and Evens
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/111 - Odds and Evens
Henry Ernest Dudeney/Puzzles and Curious Problems/112 - Simple Division

== by : $112$ ==
;Simple Division
:There is a simple division sum.
:Can you restore it by substituting a figure for every asterisk, without altering or removing the sevens?
:If you start out with the assumption that all the sevens are given, and that you must not use another,
::you will attempt an impossibility, though the proof is difficult;
::but when you are told that though no additional sevens may be used in divisor, dividend or quotient,
::any number of extra sevens may be used in the working.
:It is comparatively easy.
 **7**
 -----------
 ****7*)**7*******
 ******
 -------
 *****7*
 *******
 -------
 *7****
 *7****
 -------
 *******
 ***7**
 --------
 ******
 ******
 ------
=== Click here for solution ===
== Sources ==
* : Arithmetical and Algebraical Problems: Digital Puzzles: $112$. -- Simple Division
* : Arithmetical and Algebraical Problems: Skeleton Puzzles: $149$. Simple Division
Category:Henry Ernest Dudeney/Puzzles and Curious Problems
Category:Skeleton Puzzles
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/112 - Simple Division
Henry Ernest Dudeney/Puzzles and Curious Problems/112 - Simple Division/Mistake

== Source Work ==
:
:Arithmetical and Algebraical Problems
::Digital Puzzles
:::$112$ -- Simple Division
:
:Arithmetical and Algebraical Problems
::Skeleton Puzzles
:::$149$ -- Simple Division
== Mistake ==
:Divide $4,971,636,104$ by $124,972$, and the quotient is $39,782$.
There are three further solutions which did not find:
=== Solution $2$ ===
=== Solution $3$ ===
=== Solution $4$ ===
== Sources ==
* : Solutions: $112$. -- Simple Division
* : Answers: $149$. Simple Division
Category:Mistakes/Puzzles and Curious Problems
Category:Mistakes/536 Puzzles & Curious Problems
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/112 - Simple Division
Henry Ernest Dudeney/Puzzles and Curious Problems/112 - Simple Division/Solution 1

== by : $112$ ==
;Simple Division
== Solution ==
 39782
 -----------
 124972)4971636104
 374916
 -------
 1222476
 1124748
 -------
 977281
 874804
 -------
 1024770
 999776
 --------
 249944
 249944
 ------
== Proof ==
== Sources ==
* : Solutions: $112$. -- Simple Division
* : Answers: $149$. Simple Division
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/112 - Simple Division
Henry Ernest Dudeney/Puzzles and Curious Problems/112 - Simple Division/Solution 2

== by : $112$ ==
;Simple Division
== Solution ==
 19781
 -----------
 124974)2472110694
 124974
 -------
 1222370
 1124766
 -------
 976046
 874818
 -------
 1012289
 999792
 --------
 124974
 124974
 ------
== Proof ==
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/112 - Simple Division
Henry Ernest Dudeney/Puzzles and Curious Problems/112 - Simple Division/Solution 3

== by : $112$ ==
;Simple Division
== Solution ==
 19782
 -----------
 124972)2472195104
 124972
 -------
 1222476
 1124748
 -------
 977281
 874804
 -------
 1024770
 999776
 --------
 249944
 249944
 ------
== Proof ==
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/112 - Simple Division
Henry Ernest Dudeney/Puzzles and Curious Problems/112 - Simple Division/Solution 4

== by : $112$ ==
;Simple Division
== Solution ==
$124 \, 974 \times 39 \, 781 4 \, 971 \, 590 \, 694$
 39781
 -----------
 124974)4971590694
 374922
 -------
 1222370
 1124766
 -------
 976046
 874818
 -------
 1012289
 999792
 --------
 124974
 124794
 ------
== Proof ==
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/112 - Simple Division
Henry Ernest Dudeney/Puzzles and Curious Problems/112 - Simple Division/Solution 5

== by : $112$ ==
;Simple Division
== Solution ==
 points out that if additional $7$s were to be allowed in the dividend, a further solution could be found:
 59782
 -----------
 124972)7471076104
 624860
 -------
 1222476
 1124748
 -------
 977281
 874804
 ------
 1024770
 999776
 -------
 249944
 249944
 ------
== Proof ==
== Sources ==
* : Solutions: $112$. -- Simple Division
* : Answers: $149$. Simple Division
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/112 - Simple Division
Henry Ernest Dudeney/Puzzles and Curious Problems/113 - A Complete Skeleton

== by : $113$ ==
;A Complete Skeleton
:It will be remembered that a skeleton puzzle, where the figures are represented by stars,
::has not been constructed without at least one figure, or some added condition, being used.
:Perhaps the following comes a little nearer to the ideal,
::though there are two division sums and not one,
::and they are related by the fact that the six-figure quotient of the first happens to be the dividend of the second.
:There appears to be only one solution.
 ******
 ----------
 ***)*********
 ***
 ----
 ****
 ***
 -----
 ***
 ***
 ----
 ****
 ****
 ----
 *****
 -------
 **)******
 **
 ----
 ***
 **
 ----
 ***
 ***
 ----
 ***
 ***
 ---
=== Click here for solution ===
== Historical Note ==
== Sources ==
* : Arithmetical and Algebraical Problems: Digital Puzzles: $113$. -- A Complete Skeleton
* : Arithmetical and Algebraical Problems: Skeleton Puzzles: $150$. A Complete Skeleton
Category:Henry Ernest Dudeney/Puzzles and Curious Problems
Category:Skeleton Puzzles
Henry Ernest Dudeney/Puzzles and Curious Problems/113 - A Complete Skeleton/Historical Note

== Historical Noteon by : $113$ - A Complete Skeleton ==
 attributes this skeleton puzzle to a certain W. J. W., whose identity has not been ascertained.
== Sources ==
* : Arithmetical and Algebraical Problems: Digital Puzzles: $113$. -- A Complete Skeleton
* : Arithmetical and Algebraical Problems: Skeleton Puzzles: $150$. A Complete Skeleton
Category:Historical Notes
Henry Ernest Dudeney/Puzzles and Curious Problems/113 - A Complete Skeleton/Solution

== by : $113$ ==
;A Complete Skeleton
== Solution ==
 300324
 ----------
 333)100007892
 999
 ----
 1078
 999
 -----
 799
 666
 ----
 1332
 1332
 ----
 10356
 -------
 29)300324
 29
 ----
 103
 87
 ----
 162
 145
 ----
 174
 174
 ---
== Proof ==
Let us first consider the first division.
Let $d$ denote the divisor.
Let $q$ denote the quotient.
Let $n$ denote the dividend.
Let $n_1$ to $n_4$ denote the partial dividends which are subject to the $1$st to $4$th division operations respectively.
Let $p_1$ to $p_4$ denote the partial products generated by the $1$st to $4$th division operations respectively.
We see that the difference $n_2 - p_2$ is a $2$-digit number.
Therefore $n_2$ must begin with $1$ and $p_2$ with $9$.
So the difference $n_1 - p_1$ is exactly $1$.
Since $n_1$ has $4$ digits and $p_1$ has $3$, we have:
:$n_1 = 1000$
:$p_1 = 999$
$d$ is a $3$-digit divisor of $999$.
Since $p_4$ has $4$ digits, $d$ cannot be $111$.
Since $n_3$ has only $3$ digits and $n_3 - p_3 \ge 10$, $p_3 \ne 999$ and $d \ne 999$.
Hence we get $d = 333$.
Now $p_1 = p_2 = 999$ and thus $q$ must start with $3003$.
Since $p_3$ has $3$ digits but is not $999$, the tens digit of $q$ must be $1$ or $2$.
Therefore $300 \, 310 \le q \le 300 \, 329$.
Now we turn our attention to the second division.
Since $n_2$ has $3$ digits, $p_2$ has only $2$ digits and so does $n_2 - p_2$, $n_2$ must begin with a $1$.
Thus $p_1 = 30 - 1 = 29$.
Since $29$ is prime, the divisor must be $29$.
Using this information, we now know that the quotient $q$ of the first division is divisible by $29$.
Among the numbers $300 \, 310 \le q \le 300 \, 329$, the only multiple of $29$ is:
:$q = 300 \, 324$
As we already found the divisors of both divisions, the rest of the numbers can be filled in accordingly.
== Historical Note ==
== Sources ==
* : Solutions: $113$. -- A Complete Skeleton
* : Answers: $150$. A Complete Skeleton
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/Solutions
Henry Ernest Dudeney/Puzzles and Curious Problems/114 - Elementary Arithmetic

== by : $114$ ==
;Elementary Arithmetic
:If a quarter of twenty is four, what would a third of ten be?
=== Click here for solution ===
== Historical Note ==
== Sources ==
* : Arithmetical and Algebraical Problems: Various Arithmetical and Algebraical Problems: $114$. -- Elementary Arithmetic
* : Arithmetical and Algebraical Problems: Miscellaneous Puzzles: $189$. Elementary Arithmetic
Category:Henry Ernest Dudeney/Puzzles and Curious Problems
Henry Ernest Dudeney/Puzzles and Curious Problems/114 - Elementary Arithmetic/Historical Note

== Historical Note on Henry Ernest Dudeney: Puzzles and Curious Problems $114$ - Elementary Arithmetic ==
:This is the kind of question that was very popular in Venice and elsewhere about the middle of the sixteenth century.
:, generally known as "Tartaglia" (the stammerer) was largely responsible for the invention.
== Sources ==
* : Arithmetical and Algebraical Problems: Various Arithmetical and Algebraical Problems: $114$. -- Elementary Arithmetic
* : Arithmetical and Algebraical Problems: Miscellaneous Puzzles: $189$. Elementary Arithmetic
Category:Historical Notes