Henry Ernest Dudeney/Puzzles and Curious Problems/Arithmetical and Algebraical Problems/Digital Puzzles
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Henry Ernest Dudeney: Puzzles and Curious Problems: Arithmetical and Algebraical Problems
$83$ - Three Different Digits
- Find all $3$-digit numbers with distinct digits that are divisible by the square of the sum of those digits.
$84$ - Find the Cube
- A number increased by its cube is $592 \, 788$.
- What is that number?
$85$ - Squares and Triangulars
- What is the third lowest number that is both a triangular and a square?
- $1$ and $36$ are the two lowest which fulfil the conditions.
- What is the next number?
$86$ - Digits and Cubes
$87$ - Reversing the Digits
- What $9$-digit number, when multiplied by $123\, 456 \, 789$, gives a product ending in $987 \, 654 \, 321$?
$88$ - Digital Progression
- If you arrange the nine digits in three numbers thus, $147$, $258$, $369$,
- they have a common difference of $111$ and are therefore in arithmetic progression.
- Can you find $4$ ways of rearranging the $9$ digits so that in each case the number shall have a common difference,
- and the middle number be in every case the same?
$89$ - Forming Whole Numbers
- Can the reader give the sum of all the whole numbers that can be formed with the four figures $1$, $2$, $3$, $4$?
- That is, the addition of all such numbers as $1234$, $1423$, $4312$, etc.
$90$ - Summing the Digits
$91$ - Squaring the Digits
- Take $9$ counters numbered $1$ to $9$, and place them in a row: $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$.
- It is required in as few exchanges of pairs as possible to convert this into a square number.
$92$ - Digits and Squares
- $(1) \quad$ What is the smallest square number, and
- $(2) \quad$ what is the largest square number
- that contains all the ten digits ($1$ to $9$ and $0$) once, and once only?
$93$ - Figures for Letters
Solve this cryptarithm:
A B C D x E F G H I ----------------- A C G E F H I B D
Each of the letters $A$ to $I$ represents one of the digits from $1$ to $9$ inclusive.
$94$ - Simple Multiplication
* * * * * * * * * * x 2 ------------------- * * * * * * * * * *
- Substitute for the $*$ symbol each of the $10$ digits in each row,
- so arranged as to form a correct multiplication operation.
- $0$ is not to appear at the beginning or end of either answer.
$95$ - Beeswax
- The word BEESWAX represents a number in a criminal's secret code,
- but the police had no clue until they discovered among his papers the following sum:
E A S E B S B S X B P W W K S E T Q ------------------ K P E P W E K K Q
- The detectives assumed that it was an addition sum, and utterly failed to solve it.
- Then one man hit on the brilliant idea that perhaps it was a case of subtraction.
- This proved to be correct, and by substituting a different figure for each letter, so that it worked out correctly,
- they obtained the secret code.
- What number does BEESWAX represent?
$96$ - Wrong to Right
- Solve this cryptarithm:
W R O N G + W R O N G ----------- R I G H T
- Each letter represents a different digit, and no $0$ is allowed.
- There are several different ways of doing this.
$97$ - Letter Multiplication
- In this little multiplication sum the five letters represent $5$ different digits.
- What are the actual figures?
- There is no $0$.
S E A M x T ----------- M E A T S
$98$ - Digital Money
- Every letter in the following multiplication represents one of the digits, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, all different.
- What is the value obtained if $K = 8$?
A B C x K ----------- DE FG H
$99$ - The Conspirators' Code
- Two conspirators had a secret code.
- Their letters sometimes contained little arithmetical sums related to some quite plausible discussion,
- and having an entirely innocent appearance.
- But in their code each of the ten digits represented a different letter of the alphabet.
- Thus, on one occasion, there was a little sum in simple addition which, when the letters were substituted for the figures, read as follows:
F L Y F O R + Y O U R ----------- L I F E
- It will be found an interesting puzzle to reconstruct the addition sum with the help of the clue that $I$ and $O$ stand for the figures $1$ and $0$ respectively.
$100$ - Digital Squares
- Find a number which, together with its square, shall contain all the $9$ digits once, and once only, the $0$ disallowed.
- Thus, if the square of $378$ happened to be $152 \, 694$, it would be a perfect solution.
- But unfortunately the actual square is $142 \, 884$, which gives us repeated $4$s and $8$s, and omits the $6$, $5$, and $9$.
$101$ - Finding a Square
- Here are six numbers:
- $4 \, 784 \, 887$, $2 \, 494 \, 651$, $8 \, 595 \, 087$, $1 \, 385 \, 287$, $9 \, 042 \, 451$, $9 \, 406 \, 087$
- It is known that three of these numbers added together will form a square.
- Which are they?
$102$ - Juggling with Digits
- Arrange the ten digits in three arithmetical sums,
- employing three of the four operations of addition, subtraction, multiplication and division,
- and using no signs except the ordinary ones implying those operations.
- Here is an example to make it quite clear:
- $3 + 4 = 7$; $9 - 8 = 1$; $30 \div 6 = 5$.
- But this is not correct, because $2$ is omitted, and $3$ is repeated.
$103$ - Expressing Twenty-Four
- In a book published in America was the following:
- "Write $24$ with three equal digits, none of which is $8$.
- (There are two solutions to this problem.)"
- Of course, the answers given are $22 + 2 = 24$, and $3^3 - 3 = 24$.
- Readers who are familiar with the old "Four Fours" puzzle, and others of the same class,
- will ask why there are supposed to be only these solutions.
- With which of the remaining digits is a solution equally possible?
$104$ - Letter-Figure Puzzle
\(\text {(0)}: \quad\) | \(\ds A \times B\) | \(=\) | \(\ds B\) | |||||||||||
\(\text {(1)}: \quad\) | \(\ds B \times C\) | \(=\) | \(\ds A C\) | |||||||||||
\(\text {(2)}: \quad\) | \(\ds C \times D\) | \(=\) | \(\ds B C\) | |||||||||||
\(\text {(3)}: \quad\) | \(\ds D \times E\) | \(=\) | \(\ds C H\) | |||||||||||
\(\text {(4)}: \quad\) | \(\ds E \times F\) | \(=\) | \(\ds D K\) | |||||||||||
\(\text {(5)}: \quad\) | \(\ds F \times H\) | \(=\) | \(\ds C J\) | |||||||||||
\(\text {(6)}: \quad\) | \(\ds H \times J\) | \(=\) | \(\ds K J\) | |||||||||||
\(\text {(7)}: \quad\) | \(\ds J \times K\) | \(=\) | \(\ds E\) | |||||||||||
\(\text {(8)}: \quad\) | \(\ds K \times L\) | \(=\) | \(\ds L\) | |||||||||||
\(\text {(9)}: \quad\) | \(\ds A \times L\) | \(=\) | \(\ds L\) |
- Every letter represents a different digit, and, of course, $A C$, $B C$ etc., are two-figure numbers.
- Can you find the values in figures of all the letters?
$105$ - Equal Fractions
- Can you construct three ordinary vulgar fractions
- (say, $\tfrac 1 2$, $\tfrac 1 3$, or $\tfrac 1 4$, or anything up to $\tfrac 1 9$ inclusive)
- all of the same value, using in every group all the nine digits once, and once only?
- The fractions may be formed in one of the following ways:
- $\dfrac a b = \dfrac c d = \dfrac {e f} {g h j}$, or $\dfrac a b = \dfrac c {d e} = \dfrac {f g} {h j}$.
- We have only found five cases, but the fifth contains a simple little trick that may escape the reader.
$106$ - Digits and Primes
- Using the $9$ digits once, and once only,
- can you find prime numbers that will add up to the smallest total possible?
$107$ - A Square of Digits
$\qquad \begin{array}{|c|c|c|} \hline 2 & 1 & 8 \\ \hline 4 & 3 & 9 \\ \hline 6 & 5 & 7 \\ \hline \end{array} \qquad \begin{array}{|c|c|c|} \hline 2 & 7 & 3 \\ \hline 5 & 4 & 6 \\ \hline 8 & 1 & 9 \\ \hline \end{array} \qquad \begin{array}{|c|c|c|} \hline 3 & 2 & 7 \\ \hline 6 & 5 & 4 \\ \hline 9 & 8 & 1 \\ \hline \end{array}$
- The $9$ digits may be arranged in a square in many ways,
- so that the numbers formed in the first row and second row will sum to the third row.
- We give $3$ examples, and it will be found that the difference between the first total, $657$, and the second total, $819$,
- is the same as the difference between the second, $819$, and the third, $981$ --
- that is, $162$.
- Now, can you form $8$ such squares, every one containing the $9$ digits,
- so that the common difference between the $8$ totals is throughout the same?
$108$ - The Nine Digits
- It will be found that $32 \, 547 \, 891$ multiplied by $6$ (thus using all the $9$ digits once, and once only)
- gives the product $195 \, 287 \, 346$ (also containing all the $9$ digits once, and once only).
- Can you find another number to be multiplied by $6$ under the same conditions?
$109$ - Perfect Squares
- Find $4$ numbers such that the sum of every two and the sum of all four may be perfect squares.
$110$ - An Absolute Skeleton
- Here is a good skeleton puzzle.
- The only conditions are:
******** ------------ ***)*********** *** --- *** *** ---- **** **** ----- *** *** ---- **** **** ----- **** **** ----- **** **** ----- **** **** ----
$111$ - Odds and Evens
- Every asterisk and letter represents a figure,
- 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* ---
$112$ - 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,
- It is comparatively easy.
**7** ----------- ****7*)**7******* ****** ------- *****7* ******* ------- *7**** *7**** ------- ******* ***7** -------- ****** ****** ------
$113$ - 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,
- There appears to be only one solution.
****** ---------- ***)********* *** ---- **** *** ----- *** *** ---- **** **** ----
***** ------- **)****** ** ---- *** ** ---- *** *** ---- *** *** ---