## Alphametics

Cryptarithms are puzzles in which letters or symbols are substituted for the digits in an arithmetical calculation. Algebraic expressions might be regarded as cryptarithms of a sort, but algebra is not generally considered to be mathematically recreational. Cryptarithms have existed for centuries, and it is doubtful if it will ever be known when such puzzles were first devised. If a cryptarithm utilizes letters in place of the digits, and these letters form sensible words or phrases, the puzzle is termed an alphametic. J. A. H. Hunter coined the term in 1955.

• The classic one.
```  S E N D
M O R E
---------
M O N E Y
```
Does someone realy need a solution for this one?
Author: H.E. Dudeney in the July 1924 issue of Strand Magazine 68 (1924) 97, 214
Ref: Geoffrey Mott-Smith, Mathematical Puzzles, 1954, Problem 114
Ref: W. R. Ransom, One hundred Mathematical Curiosities, 1955, p120
Ref: Longley-Cook, New Math Puzzle Book, 1970, p53
Ref: E. P. McCravy, Mathematics Magazine 45 (1972) 48-49, number of solutions for base >= 10
Ref: Heinrich Hemme, Das Hexeneinmaleins, 2000, Problem 16, p16

• I receive you loud and clear
```    T H I S
I S A
G R E A T
T I M E
-----------
W A S T E R
```
Ref: Unique solulion can be found in: AMM (1965) p316 E-1681

• Planets
```  S A T U R N
U R A N U S
N E P T U N E
P L U T O
-------------
P L A N E T S
```
Author: Willy Engren Ref: JoRM 12 (1979) 133-134

• Planets
```      M A R S
V E N U S
S A T U R N
U R A N U S
-------------
N E P T U N E
```
Author: Willy Engren Ref: JoRM 13 (1980/81) 293

• Health
```H E A R T  +  E A R S  +  N O S E  +  T H R O A T  =  H E A L T H
```
Author: Richard I. Hess Ref: JoRM 21 (1989) 61 Alpha 1680

• Hurray
```H U R R A Y  +  H U Z Z A H  =  P U Z Z L E S
```
Author: Richard I. Hess Ref: JoRM 21 (1989) 137 Alpha 1698

• Authority Figures
```F A T H E R  +  M O T H E R  =  P A R E N T
```
Author: David J. Porter Ref: JoRM 21:4 Alpha 1743

• Greenhouse Effect
```W I N T E R + I S + W I N D I E R + S U M M E R + I S = S U N N I E R
```
Author: Brian Barwell Ref: JoRM 21:3 Alpha 1809

• Rhyme's Pertner
```R E A S O N  =  (I T)(I S)  +  T H E R E
```
Author: Robert B. Israel Ref: JoRM 21:3 Alpha 1731, (explained solution)

• Funny Farm
```M A D  *  M A N  =  A S Y L U M
```
Author: Robert B. Israel Ref: JoRM 21:4 Alpha 1751, (explained solution)

```A L F R E D  /  E  =  N E U M A N
A L F R E D  *  E  =  N E U M A N
```
These are two base-9 problems.

• Eve double talked
```eve/did = .talktalktalk...
```
There are 2 solutions here. One of them with a reduced fraction.
Ref: Martin Gardner, Mathematical Circus, 1979, Chap. 11
Ref: Zweisteins Zahlenlogeleien, Insel it 1510, Problem 31

```  T H E S E
T E A S E
T I R E D
-----------
R E A D E R
```

```  A D A M
A N D
E V E
---------
M O V E D
```
Ref: J. A. H. Hunter, Entertaining Mathematical Teasers, 1983, p93

```S Q U A R E
D A N C E
-----------
D A N C E R
```
This problem has two solutions: one as an addition problem, the other as subtraction.

```P I E R R E  +  E L L I O T T  =  T R U D E A U
```
Ref: Dick Hess, Puzzles from around the world, 1997, Problem 6

• fruits
```A P P L E  +  G R A P E  +  P L U M  =  B A N A N A
```
Ref: Dick Hess, Puzzles from around the world, 1997, Problem 13
```A P P L E  +  L E M O N  =  B A N A N A
```
Author: Col. G. L. Sicherman

• peace
```W E  +  W A N T  +  N O  +  N E W  +  A T O M I C  =  W E A P O N
```
Ref: Dick Hess, Puzzles from around the world, 1997, Problem 16

• a president from USA
```P E A N U T  +  T E E T H  =  C A R T E R
```
Solve this alphametic in base 10 and base 9.
Ref: Dick Hess, Puzzles from around the world, 1997, Problem 43

• The Digital Atom Smasher
```sqrt(ATOM)  =  A + TO + M
```
this has two solutions
Ref: Angela Fox Dunn, Mathematical Bafflers, 1980, p39

• 13
```T E R R I B L E  +  N U M B E R  =  T H I R T E E N
```
Author: Steven Kahan

• Right and Wrong
```Z E R O E S
O N E S
-----------
B I N A R Y
```
Author: Peter Macdonald, 1977
Ref: JoRM 10 (1977) 155

• Right and Wrong
```W R O N G
W R O N G
---------
R I G H T
```
this has two solutions under the restriction of O = zero.
Ref: Angela Fox Dunn, Mathematical Bafflers, 1980, p103

• palindromic
```N O R A  *  L  =  A R O N
```
Ref: Martin Gardner, Mathematical Magic Show, 1977, Chap 15, Problem 2

• Scrabble
```A L P H A B E T
L E T T E R S
---------------
S C R A B B L E
```
Ref: G. Brandreth, The Puzzle Mountain, 1981, p61

• Scrabble
```  N A I L O R
L I N G E L
-------------
F A C E O F F
```
Ref: rec.puzzles

• an easy one
```A  *  (W E B T V)  =  D D D D D D
```
Ref: rec.puzzles

• Mathematicians
```  G A U S S
R I E S E
-----------
E U K L I D
```
Ref: Zweistein; 99 Logeleien von Zweistein, 1969, Problem 55
The values of G and R are exchangeable.

• trigonometry (French)
```S I N ²  +  C O S ²  =  U N I T E
```
Ref: rec.puzzles

• how many drinks do you need?
```ALCOHOL + ALCOHOL + ... + ALCOHOL = HANGOVER
```
how many drinks do you need?
Ref: rec.puzzles

• Chinese calendar
```H A P P Y  -  T I G E R  =  Y E A R
```
Solve this under the conditions that
1. TIGER being the third in the order of 12 animals (rat, ox, tiger, rabbit, dragon, snake, horse, ram, monkey, cook, dog, boar), the number represented by TIGER divided by 12 gives a remainder 3 and
2. as there are 10 possible digits to fill in the 9 letters that appear in this alphametic problem, there is bound to be one digit missing. However, the missing digit turns out to be the remainder if the number represented by YEAR is divided by 12.
Author: Liang-shin Hahn (1986)
Ref: Liang-shin Hahn; Complex Numbers and Geometry, MAA, 1994, p186

• Diamond
```B O B B Y  +  L O V E D  =  J E W E L S
```
Ref: Ravi Narula, Brain Teasers, 1976, Jaiko Publ. House, p97

• He or She
`H E  *  H E  =  S H E`
Author: M. E. Larsen
Ref: J. Lehmann, Mathe mit Pfiff, Manz Verlag 1977, p71 (P11: AB*AB=CAB)

• The ABC
```B A  *  C B A  =  D C B A
```
Ref: Walter Lietzmann, Lustiges und Merkwürdiges von Zahlen und Formen, 11.Ed, p192

• Aphis
```T H I S  *  I S  =  A P H I S
```
This one has three solutions.
Ref: JoRM 3:1 (Jan 1970) 49, Problem 68 (Solution of Underwood Dudley)

• Puzzleland (Kindergarten) Algebra
```  J G D C H
I F A B E
-----------
B I B D E B
```
Ref: Sam Loyd, Cyclopedia of Puzzles, 1914, p238

• Elements of Nature
```E A R T H  +  A I R  +  F I R E  +  W A T E R  =  N A T U R E
```
Author: Herman Nijon
Ref: JoRM 9 (1977) 207

• square
```T W O  *  T W O  =  S Q U A R E
```
Ref: H. E. Dudeney, Strand 78 (1929) 91, 208

• Hawaii
```NIIHAU ± KAUAI ± OAHU ± MOLOKAI ± LANAI ± MAUI ± HAWAI = 0
```
Ref: Donald E. Knuth; The Art of Computer Programming, Section 7.2.1.2 Exc. 26.

• directions
```N O R T H / S O U T H  =  E A S T / W E S T
```
Ref: Nob Yoshigahara; JoRM 27 (1995) 137

• Prime factors
```A B C B A  =  D  *  B E  *  B F F A
```
The factors shall be prime numbers.
Ref: Ken Russell, MENSA Magazine Oct. 1996, p28
The Solution

• A Facetious Division
```A H H A A H / J O K E  =  H A
```
Ref: Ch. W. Trigg, Mathematical Quickies, 1985, Problem 19.

## Double True

• what is the sum
```    F I V E
F I V E
N I N E
E L E V E N
-----------
T H I R T Y
```
Ref: Ken Russell, MENSA Magazine Oct. 1996, p28

• what is the sum (french)
```  V I N G T
C I N Q
C I N Q
-----------
T R E N T E
```
Author: Alan Wayne
Ref: M. Gardner, Sci. Am. (Dec. 1975), p116

• what is the sum (german)
```  E I N
E I N
E I N
E I N
-------
V I E R
```
Author: Alan Wayne
Ref: M. Gardner, Sci. Am. (Dec. 1975), p116
The Solution

• Das stimmt sogar (german)
```  S E C H S
S E C H S
-----------
Z W O E L F
```
Ref: Zweisteins Zahlenlogeleien, Insel it 1510, Problem 48

• (spanish)
```U N O  +  U N O  +  T R E E  =  C I N C O
```
This one has several solutions. The Solution

• Dr. Matrix
```S I X  +  S I X  +  S I X  =  N I N E  +  N I N E
```
Author: Alan Wayne
Ref: M. Gardner, Sci. Am. (Sep. 1975),
M. Gardner, The Magic Numbers of Dr. Matrix, 1985
Chap. 18: The King James Bible
The Solution

• Dr. Matrix in New York
```F O R T Y  +  T E N  +  T E N  =  S I X T Y
```
Ref: Longley-Cook, New Math Puzzle Book, 1970, p54
Ref: M. Gardner, The Magic Numbers of Dr. Matrix, 1985
Chap. 1: New York
Note: ELEVEN + TWO is an anagram of TWELVE + ONE.
The Solution

• true and false
```T W O  *  T W O  =  T H R E E
```
Author: H.E. Dudeney in the July 1924 issue of Strand Magazine

• 0123
```T H R E E  =  T W O  +  O N E  +  Z E R O
```
Ref: Richard L. Breisch; Recreational Math. Magazine 12 (Dec. 1962) 24

## General Bases

• base eleven
```  U N I T E D
S T A T E S
-------------
A M E R I C A
```
Ref: L. A. Graham; The surprise Attack in Mathematical Problems, 1968, Problem 9
Ref: L. A. Graham; Mathematik aus dem Hinterhalt, 1981, Problem 9

• Fair Fares
```F A R E S  =  F E E ²
```
Solve this alphametic in base 6.
Ref: Ch. W. Trigg, Mathematical Quickies, 1985, Problem 235.

• Number system unknown
```K Y O T O
K Y O T O
K Y O T O
---------
T O K Y O
```
Which number system has a solution? Author: V. Dubrovsky, A. Shvetsov
Ref: Quantum, Brainteaser B144 Quantum CyberTeaser May/June 1995 or Quantum CyberTeaser May/June 1995 (local)

• venusean arithmetic
```U V  +  U V  =  U W U
```
Author: Harry L. Nelson
Ref: Martin Gardner, Mathematical Magic Show, Alfred A. Knopf (1977) New York, Chap. 8 Finger Arithmetic, Fig. 32

• unique base
Examples of problems which are solvable only in one base b.
```X  +  X  +  X  =  X X
```
Solution: 3 X = b X + X. Thus 2 X = b X and as X not equal 0 we have b = 2. Now X = 1.
```X Y  +  X X  =  X Y X
```
Solution: The last digit shows Y = 0. Dropping the last digit we again get 2 X = b X.

• commutator
```A B  -  B A  =  A
```
Ref: J. Lehmann, Mathe mit Pfiff, Manz Verlag 1977, p71 (P15: AB-BA=A)

## Arithmetical Restorations

• A prime cryptarithm
```    X X X
X X
----- *
X X X X
X X X X
---------
X X X X X
```
In this remarkable cryptarithm, each digit is a prime (2, 3, 5 or 7). No letters or digits are provided as clues; but there is only one solution.
Ref: B. A. Kordemsky, The Moscow Puzzles, 1972, Problem 273
Ref: Ch. W. Trigg, Mathematical Quickies, 1985, Problem 123
Ref: Zweisteins Zahlenlogeleien, Insel it 1510, Problem 20
Ref: Walter Lietzmann, Lustiges und Merkwürdiges von Zahlen und Formen, 11.Ed, p130
The following variation has other solutions.
```X X X  *  X X  =  X X X X X
```

• 4 digits
```      x x x
x 2 x
----- *
x x x
x x x x
x 8 x
-----------
x x 9 x 2 x
```

Ref: B. A. Kordemsky, The Moscow Puzzles, 1972, Problem 272 b

• 3 digits
```    x x x
x x
----- *
x x x
x x 4
---------
x x x 0 1
```

Author: Arthur Haas
Ref: AMM 40 (Dec. 1933) 607-608 E-37, explained solution

• An atomar square
```A T O M  *  A T O M  =  x x x x A T O M
```
Ref: Sphinx - Nov 1933 #167 - Page 167 (by M. Pigeolet) saying TOCK TOCK = xxxxTOCK.
Ref: B. A. Kordemsky, The Moscow Puzzles, 1972, Problem 272 g
Ref: W. Engel (Ed.), Mathematische Olympiade-Aufgaben, 1979, A.2.42
Ref: M. Gardner, The Magic Numbers of Dr. Matrix, (1985) Chapter 4: Squaresville
Automorphic numbers: n^2 ends with n in base d. base 10: ...8212890625, ...1787109376.
A related problem is
`H E  *  H E  =  S H E`

• Cryptic Multiplication
```  O E E
E E
------- *
E O E E
E O E
-------
O O E E
```
All even digits are replaced with E and all odd with O. What was the multiplication?
Ref: Ch. W. Trigg, Mathematical Quickies, 1985, Problem 233.
Ref: Zweisteins Zahlenlogeleien, Insel it 1510, Problem 21
The Solution

• Even Odd Multiplication
```    E E O
O O
--------- *
E O E O
E O O
---------
O O O O O
```
All even digits are replaced with E and all odd with O. What was the multiplication?
Author: Fitch Cheney
Ref: M. Gardner; Scientific American 207 (Oct 1962) 132, (Nov 1962) 162
Ref: H. Hemme; Die Quadrate des Teufels, (2003) Problem 98.

• Even Odd Multiplication
```    E O E E
E O E
----------- *
O O E E
O E O E E
E O O E
-----------
E O O O E E
```
All even digits are replaced with E and all odd with O. What was the multiplication?
Ref: Zweistein, Logeleien fuer Kenner, 1974, Problem 21
Ref: D. St. P. Barnard, O. Botsch; Hirnverzwirner, 1975, Problem 92

• the wrong printer
```        ? ? ? ?
? ? 7 ?
---------  *
? ? ? ? ?
? ? ? ?
? ? ? ?
---------------
? ? ? ? ? ? ? ?
```
Only digit 7 is printed.
Ref: Zeit Magazin, Nr 35, 22. Aug. 97, p42, Logelei

• the wrong typewriter
```    ? ? 3
? ? 3
-----  *
3 ? ?
? 3 ?
? ? 3
---------
? ? ? ? ?
```
Only digit 3 is printed.
Ref: Zweisteins Zahlenlogeleien, Insel it 1510, Problem 26

• this is too hard
```    T H I S
I S
---------  *
x x T O O
H A R D x
-----------
x x x x x x
```
Ref: Joseph S. Madachy, Mathematics on Vacation, 1966, p192

• puzzle world
```              P U Z Z L E
x x x x x x x
-------------  *
x x x x x x x
x W x x x x x
x x O x x x x
x W O R L D x
x x x x L x x
x x x x x D x
x x x x x x x
-------------------------
x P x U x Z x Z x L x E x
```
Ref: Puzzle World, Ed. Nob Yoshigahara and Richard Bozulich, Ishi Press, Summer 1992, Pilot Edition.

• Est modus in rebus
```              E S T M O D U S
I N R E B U S
-----------------  *
x x x x x x x x x
x x x x x x x x x
x x x x x x x x
x x x x x x x x x
x x x x x x x x x
x x x x x x x x x
x x x x x x x x x
-----------------------------
x x x x x x x x I N R E B U S
```
Each of the ten digits corresponds to at least one letter, so there is precisely one case where two different letters are replaced by the same digit. Fred. Schuh admits that the puzzle is far from easy.
Ref: Fred. Schuh, The Master Book of Mathematical Recreations, Dover Publ. 1968, p315-317

• Ancient
```x x x ) x x x x x x ( x x x
x 0 x x
-------
x x x x
x 5 0 x
-------
x x x
x 4 x
=====
```
Ref: American Mathematical Monthly (1921) p37
Ref: Rouse Ball, Mathematical Recreations and Essays, Chapter I
Ref: Zweisteins Zahlenlogeleien, Insel it 1510, Vorwort

• ```x x ) x x x x 3 x ( x x x x.x x
x x
---
4 x x
x x x
-----
x x
x x
---
x x
x x
---
x x x
x x x
=====
```
Ref: Zweistein; 88 neue Logeleien von Zweistein, 1983, Problem 64

• The three 5
```x x x ) 5 x x x x x x x x ( x x x x x x x
x x x
-----
x x x x
x x x
-----
5 x x
x x x
-----
x 5 x x
x x x x
=======
```
Ref: Zweistein; 88 neue Logeleien von Zweistein, 1983, Problem 19

• The seven 7
```x x x x 7 x ) x x 7 x x x x x x x ( x x 7 x x
x x x x x x
-----------
x x x x x 7 x
x x x x x x x
-------------
x 7 x x x x
x 7 x x x x
-----------
x x x x x x x
x x x x 7 x x
-------------
x x x x x x
x x x x x x
===========
```
Author: W. E. H. Berwick
Ref: W. E. H. Berwick; School World, 8 (July and October 1906) 280+320
Ref: Heinrich Dörrie; Triumpf der Mathematik, Sect 4 (p9-13)
Ref: Rouse Ball, Mathematical Recreations and Essays, Chapter I

• The seven 7
```x x x x 7 x x ) x x x x x x 7 x x x x x x x x x ( x x x 7 x x x x x x
x x x x x x x
-------------
x x x x x x x x
x x x x x x x x
---------------
x x x x x x x x
x x x x x x x
---------------
x x x x x x x
x x x x x x x
-------------
x x x x x x 7 x
x x x 7 x x x
---------------
x x x x x x x x
x x x x x x x x
---------------
x x x x x x x x
x x x x x x x
---------------
x 7 x x x x x
x x 7 x x x x
-------------
x x x x x x x
x x x x x x x
-------------
x
```
Ref: Fred. Schuh, The Master Book of Mathematical Recreations, Dover Publ. 1968, p318-320

• The lonesome 7
```x x x ) x x x x x x x x ( x 7 x x x
x x x x
-------
x x x
x x x
-----
x x x x
x x x
-------
x x x x
x x x x
=======
```
Ref: American Mathematical Monthly 49 (1932) 489

• The lonesome 8
```x x x ) x x x x x x x x ( x x 8 x x
x x x
-------
x x x x
x x x
-------
x x x x
x x x x
=======
```
Author: P. L. Chessin
Ref: American Mathematical Monthly (Apr. 1932)
Ref: W. B. Carver, American Mathematical Monthly (Dec. 1954) 712
Ref: W. R. Ransom, One hundred Mathematical Curiosities, 1955, p134
Ref: Martin Gardner; The Second Scientific American Book of Mathematical Puzzles and Diversions, 1961, Chap. 14
Ref: L. A. Graham; The surprise Attack in Mathematical Problems, 1968, Prob. 31
Ref: Ch. W. Trigg, Mathematical Quickies, 1985, Problem 31
Ref: P. v Delft & J. Botermans, Creative Puzzles of the World, p161
Ref: Zweisteins Zahlenlogeleien, Insel it 1510, Problem 36

• The lonesome 8 again
```x x x ) x x x x x x x x ( x x x x x
x x x
-------
x x x x
x x x
-------
x x x x
8 x x x
=======
```
Ref: W. Engel (Ed.), Mathematische Olympiade-Aufgaben, 1979, A.2.42
- 2. Math. Olympiade DDR, 1962/1963, Klasse 9, Stufe 1

• A fairly easy lonesome 8
```x x ) x x x x x x x x x x ( x x x x x 8 x x
x x x
-----
x x x
x x x
-----
x x x
x x x
-----
x x
x x
---
x x x
x x x
=====
```
Ref: B. A. Kordemsky, The Moscow Puzzles, 1972, Problem 272 f

• The eight 8
```8 x x x ) 8 8 x x x x x x ( x x x x
x x x x x
---------
x x x x x
x x x x x
---------
x 8 x x x
8 x x x 8
---------
x x x x x
x x x 8 8
=========
```
Ref: Zeitschrift Archimedes, vor 1950
Ref: Walter Lietzmann, Lustiges und Merkwürdiges von Zahlen und Formen, 11.Ed, p129

• completely hidden
```x x ) x x x x ( x x.x x x
x x
---
x x x
x x
-----
x x
x x
---
x x x
x x x
-----
x x
x x
===
```
Ref: Joseph S. Madachy, Mathematics on Vacation, 1966, p192

• completely hidden II
```x x ) x x   ( x.x x x x
x x
---
x x x
x x x
-----
x x x
x x x
-----
x x
x x
===
```
Ref: Zweistein, Logeleien für Kenner, 1974, Problem 7

• nothing
```x x x ) x x x x x x ( x x x x.x x x x
x x x
-----
x x x
x x x
-----
x x x
x x x
-----
x x x
x x x
-----
x x x x
x x x x
=======
```
Author: A. Corrigan in Strand Magazine
Ref: H. E. Dudeney, The Strand Magazine 65 (1923) 311-312, 405, 538
Ref: P. v Delft & J. Botermans, Creative Puzzles of the World, p161
Ref: Heinrich Hemme, Der Wettlauf mit der Schildkröte, Problem 44

• nothing else
```x x x ) x x x x x x x x ( x x x x x x
x x x
-----
x x x x
x x x
-------
x x x x
x x x
-------
x x x x
x x x x
=======
```
Determine the unique divisor.
Ref: H. Hemme & M. Schwoerer, Mathematischer Denkspass, 1998, p139

• the complete ghost
```x x x ) x x x x x x x x x ( x x x x x x
x x x
-------
x x x x
x x x
-------
x x x
x x x
-----
x x x x
x x x x
=======

x x ) x x x x x x ( x x x x x
x x
---
x x x
x x
-----
x x x
x x x
-----
x x x
x x x
=====
```
The six-digit quotient of the first must be equal the dividend of the second.
Ref: H. E. Dudeney, Puzzles and Curious Problems, 1931, 36-37, 141.
Ref: Geoffrey Mott-Smith, Mathematical Puzzles, 1954, Problem 122

• Sense of humore
```x x ) x x x x 0 x ( x x x x
x x
-----
x x x
x x 1
-----
x x
2 x
===
```
Ref: L. A. Graham; The surprise Attack in Mathematical Problems, 1968, Problem 9
Ref: L. A. Graham; Mathematik aus dem Hinterhalt, 1981, Problem 9

```x x ) x x x x 0 x ( x x x x
x x
-----
x x x
x x 1
-----
x x
3 x
===
```
Ref: Ch. W. Trigg, Mathematical Quickies, 1985, Problem 173

• It sounds greek to me
```x x  +  x  =  x
```
Author: Jaime Poniachik
Ref: www.g4g4.com

## All the digits

• all digit sums
```X X X  +  X X X  =  X X X
```
Put 1-9 once each to make the equation honest. Ref: Recreational Mathematics Magazine, No. 7 (Feb. 1962) 35-36

• fractions with digits 1-9
``` X     X     X
--- + --- + ---  =  1
X X   X X   X X
```
Put 1-9 once each to make the equation honest. Ref: REC 8:5 (1993) 5-6 (Nob)
Ref: L. Mittenzwey, Mathematische Kurzweil, Leipzig und Wien, 1880, Problem 141

```X X
X
---  *
X X
X X
---  +
X X
```
Put 1-9 once each to make the equation honest.
Author: H. E. Dudeney
Ref: M. Gardner's Sixth Book of Mathematical Games from Scientific American, 1971, Chap. 6.7 Three Cryptarithms
Ref: P. v Delft & J. Botermans, Creative Puzzles of the World, p161

• All the digits
```X X X X X  =  k  *  X X X X X
```
Put 0-9 once each to make the equation honest for k=7 and for k=9.
Let s(k) be the number of solutions without leading zero. David Daykin computed the table
```   k  :  2  3  4  5  6  7  8  9
s(k) : 48  6  8 12  0  1 16  3
```
Ref: Angela Fox Dunn, Mathematical Bafflers, 1980, p99 (k=9)
Ref: Angela Fox Dunn, Second Book of Mathematical Bafflers, 1980, p21 (k=7)
Ref: David Daykin, Mathematics Magazine, 42 (1969) 102-103 (k=2..14, base=3..15)
Ref: M. Gardner, The Magic Numbers of Dr. Matrix, 1985
Chap. 13: The Moon (k=7 and the table of s(k))
```X X X X X  =  k  *  X X X X
```
Put 1-9 once each to make the equation honest for k=2..9.
Ref: H. E. Dudeney, Amusements in Mathematics, 1958, problem 88
Ref: P. v Delft & J. Botermans, Creative Puzzles of the World, p160

• Years
```X X X X X X X  /  X X X  =  1984
```
Put 0-9 once each to make the equation honest.
Ref: Stewart Metchette, Years Expressed as Distinct-digit Fractions, JoRM 13:1 (1980-81) 26-28

• A Curious Multiplication
```    x x x
x x x
-----  *
x x x
x x x
x x x
---------
x x x x x
```
No digit occurs more than twice.
Ref: Fred. Schuh, The Master Book of Mathematical Recreations, Dover Publ. 1968, p287-291
Ref: M. Gardner's Sixth Book of Mathematical Games from Scientific American, 1971, Chap. 6.7 Three Cryptarithms

• Multiplication: fill in the digits
```X O X O X  *  O X O X O  =  X O X O X O X O X O
```
All of the X's are different, as are all of the O's, as are all digits in the product (whose final O is 0).
Author: Jim Ferry
Ref: rec.puzzles 1999-06

## German Alphametics

• Kinderleicht?
```  B A C K E
B A C K E
-----------
K U C H E N
```
Ref: Zweisteins Zahlenlogeleien, Insel it 1510, Problem 47

• A family problem
```  V A T E R
M U T T E R
-----------
E L T E R N
```
Ref: J. Lehmann, Mathe mit Herz, Urania (1991), p82, problem 22
Ref: J. Lehmann, Mathe mit Pfiff, Manz Verlag 1977, p72, problem 22
Ref: W. Engel (Ed.), Mathematische Olympiade-Aufgaben, 1979, A.2.43
There are 6 solutions

```    L A G E
L E M G O
-----------
S C H O E N
```
Author: Ingo Althöfer
soluble in base 12.

• Patriot of Lage-Lippe
```    L A G E
L I P P E
-----------
S C H O E N
```
Author: Torsten Sillke (but the Patriot is Udo Sprute)
unique soluble in base 11.

• Suche nach Atlantis
```A T L A N T I S  =  S C H L A M M  +  S C H L I C K
```
Stern Nr. 13 (1996)

• Auf den ersten Blick
```Z W E I  +  B L I T Z  =  L I E B E
```
Stern Nr. 17 (1996)

• Auf den ersten Blick II
```Z W E I  +  B L I T Z  =  E R O T I K
```
Stern Nr. 18 (1996)

• Steithammel, Streithähne
```H A M M E L  +  H A E H N E  =  S T R E I T
```
Stern Nr. 18 (1996)

• Kulturerlebnisse
```M U S I K  +  K U N S T  =  G E N U S S
```
Stern Nr. 35 (1996) (several solutions)

• Olympia'96
```A T L A N T A  =  R U M M E L  +  P L E I T E N
```
Stern Nr. 37 (1996)

• suchen und finden
```S U C H E N  -  M A C H T  =  S P A S S
```
```N A M E N  =  S C H A L L  -  R A U C H
```
Two solutions but "schall" is unique.
```F I N D E  +  D I E S E  =  K I N D E R
```
```M A C H  *  M A L  =  L A U T E R
```
```K E I N E  +  A N G S T  =  S I G R I D
```
Ref: Zeit Magazin, Nr 47, 15. Nov. 1996, p51, Logelei

• Kai und Ina
```I N A  *  I S T  =  S U E S S
```
```V I E L E N  :  D A N K  =  K A I
V E E V
---------
A V U E
A L K O
---------
I N U N
I N U D
-------
K
```
```D E I N E  +  B E I N E  =  S U P E R
```
This alphametic has two solutions but the sum remains equal. What number is SUPER?
```N I E M A N D  :  Z U  =  H A U S E
N I H
-------
I M A
I N S
-------
Z I N
U D M
-------
N H D
N H D
=====
```
```K U S S  +  L U S T  =  O L A L A
```
Ref: Zeit Magazin, Nr 23, 30. Mai 1997, p42, Logelei

• no magazin any more
```D I E  *  Z E I T  =  R A E T S E L
```
```D E N K E R  +  G E R N E  =  L O G E L N
```
Ref: Zeit Nr 20, 12. Mai 1999, Logelei

• Trinken
```      S E K T * B I E R
-----------------
T B T S R
B E R A S
S E K T I
-------------
S A E U F E R
```
Ref: Zeit Magazin, Nr 23, 30. Mai 1997, p42, Logelei

• tierisch
```S C H W I M M T  :  E I N  =  F I S C H
S N E I
---------
I C I I
E I N
---------
C E T M
C H F T
---------
C M W M
H C S C
---------
H C C T
N S C W
-------
E M S
```
Ref: Zeit Magazin, Nr 35, 22. Aug. 1997, p42, Logelei

• schlau und klug?
```U N S E R  +  O S C A R  =  S C H L A U
```
```K L U G E  +  L E U T E  =  L O G E L N
```
Ref: Zeit Magazin, Nr 35, 22. Aug. 1997, p42, Logelei
Ref: Zweisteins Zahlenlogeleien, Insel it 1510, Problem 77

• jeder liebt es
```J E D E R  +  L I E B T  =  B E R L I N
```
This alphametic has two solutions, but EIN BIER is unique.
Ref: Zeit Magazin, Nr 35, 22. Aug. 1997, p42, Logelei

• sparen
```W A I G E L  +  S P A R E  =  E I F R I G
```
Ref: Zeit Magazin, Nr 3, 10. Jan. 1997, Logelei

• Wer ist Zweistein?
```W E R  +  I S T  +  Z W E I  =  S T E I N
```
This alphametic has two solutions,
Ref: Zweisteins Zahlenlogeleien, Insel it 1510, Problem 75

## Musicmetic

```+ C E G E F e c
+ C c B A D B c
+ C E G A F D c
---------------
c c d d B f f C
```
One frequency is false and must be corrected!
Ref: Zeit Magazin, Nr 38, 30. Sep. 1994, p44, Logelei

## Cross Number Puzzles (Arithmogriphs)

• ``` ABC *   DA = BEFA
+      *      -
GHHA +  GJE = GFDC
------------------
GJEJ + HJJJ = FFDC
```
Ref: M Sändig, Hobbymatik, 1984, Titelpage
The Solution of Matthew Daly.

• ```ABCD :  EFG =   HC
-      *      +
FHA -   IG =  FJD
------------------
AHBE - ACFE =  FAJ
```
Ref: Hör Zu, 17/1972
The Solution of H. Müller-Merbach.

• ``` ABC *   DE = CFGH
+      *      -
JDHJ +  DGC = JGKK
------------------
JEDK + EBAH = FAGH
```
Ref: Hör Zu, 38/2001
The Solution

• ```  AB *  CDC = BABE
*      +      -
AFD + GHJE = GGDC
------------------
HFKH + GJEF = KDAH
```
Ref: Hör Zu, 10/1995
The Solution

• ``` EGA *   AH = DGFG
*      +      -
LL + EFFC = EFDL
------------------
ABCD + EFGH = KLBK
```
Ref: Hör Zu, 00/1995
The Solution

• ```ABAC :  DEF =   GB
-      +      *
HEGF - HBID =  HED
------------------
EKBI - HKGI = ICDC
```
Ref: Hör Zu, 45/2002
The Solution

• ```ABBC :  DDE =   DF
+      +      *
AGFG - AHIB =  KKG
------------------
KGEH + ACKE = FCII
```
Ref: Hör Zu, 29/2003
The Solution

• ```ABCD + DCEF = GHFI
-      :      +
AHGE +  AIC = AKGF
------------------
KEG *   KI = EBKB
```
Ref: Hör Zu, 31/2003
The Solution

• ```ABCD + EBFG = FDBH
:      -      +
AE + EBII = EBAE
------------------
AHC *   FG = KFGK
```
Ref: Hör Zu, 37/2003
The Solution

• ```ABCD - EABF = GDGA
:      -      -
HII + EGKD = EFCK
------------------
IH *  EDH = HABF
```
Ref: Hör Zu, 51/2003
The Solution

• ```ABCD + EFCF = BGHE
-      :      +
AAIF +  GDE = AEDC
------------------
FCH *   GK = KHHI
```
Ref: Hör Zu, 01/2004
The Solution

• ```ABBC :   DE =  FGD
-      +      *
FBHI - FGKD =   FD
------------------
EHGI - FBCG = CCEK
```
Ref: Hör Zu, 05/2004
The Solution

• ```ABCD + EFGG = HEBG
:      -      +
CI + EHKI = EFCH
------------------
CKB *   GH = IKFD
```
Ref: Hör Zu, 07/2004
The Solution

• ```ABCD :   EE =  EFD
+      +      *
GHEH - GIKC =   IF
------------------
HHDI + GICF = DBCA
```
Ref: Hör Zu, 53/2004
The Solution

• ```ABCD + EFGH = KLBK
:      -      +
LL + EFFC = EFDL
------------------
EGA *   AH = DGFG
```
Robert Israel solved it with a Maple Program.
The Solution

• ```ABCDE - FGEE = AHBDE
:      -       -
DI *  HDB =  BKDB
--------------------
CDE + FKCD =  KDGD
```
Ref: Lehmann
The Solution

• ```ATU + IAS  = IITE
-     -       :
NEG : IOG  =    E
-----------------
PAU -  NS  =  PPA
```
Ref: Boris A. Kordemsky, Köpfchen muss man haben, 1975, P241
The Solution If you order the letters according there values you get a mathematical term in russian.

• ```HEB + EUS  =  LFS
-     -       :
WUB :  EH  =   US
-----------------
EHB - ESU  =   RA
```
Ref: Boris A. Kordemsky, Köpfchen muss man haben, 1975, P241
The Solution If you order the letters according there values you get a german word.

• ``` ABC *   DC = EAEF
*      +      -
BF + AEGG = AEBF
------------------
FFEE + AEDC = DBHG
```
Ref: Torsten Sillke
The Solution

• three solutions
```  AB *   CD = EFCB
*      +      -
EC +  GBB =  GEC
------------------
EBDB +  GCD = EDBD
```
Ref: Torsten Sillke
The Solution

• four solutions
``` ABC *   BB = ADBC
*      +      -
E +  ACC =  ACE
------------------
FGEC +  ABB = ABCB
```
Ref: Torsten Sillke
The Solution

• ``` ABC :    C =   DE
-      *      +
AE +   AF =   DD
------------------
BGA -  FHC =   IJ
```
Ref: J. Petigk, Mathematik in der Freizeit, 1998, p151
The Solution

``` A  +  B  =  CD
+     +      +
CE  +  F  =  CF
---------------
CA  + CC  =  DE
```
Ref: J. Petigk, Mathematik in der Freizeit, 1998, p159
Note there was a printer error in the last line in the book where the wrong equation CG + CC = HE is given. Show that this modified problem has no solution. The Solution

``` ABC - DBB  = EFG
+     +      +
ABB - ECC  = BBB
-----------------
DBEB - HBB  = IBC
```
Ref: Jürgen Köller, Symbolrätsel 2
The Solution

• multiplication and division
```  AB * BC  =  DAD
*    :       *
EC :  F  =    F
-----------------
BCD *  G  = BBEA
```
Ref: Jürgen Köller, Symbolrätsel 4
The Solution

```HJ  +  AD  =  DF
+      +      +
BG  +  EC  =  DF
----------------
AC  +  GE  = HEK
```
Ref: Trend-Nüsse 1981, p52
The Solution

```ISE + EID  = ALK
-      +     -
RAS -  KL  = RUM
----------------
ERR + MDR  = LLM
```
Ref: Zeit Magazin 19/1972
The Solution

• ```ABC - DEF  = EG
+      -     :
HIE : DAH  =  H
---------------
FBD :  HB  = DF
```
Ref: Eckstein, Symbolrätsel, www.eckstein.de
The Solution

• ```AB  +   C  =  AC
+      +      +
----------------
FD  +   G  =  EC
```
Ref: W. N. Bolchowitinow, B. I. Koltowoi, I. K. Lagowski; Spass für freie Stunden, 1980, problem 194 A
Ref: J. Lehmann, Mathe mit Pfiff, Manz Verlag 1977, p72, problem 23
The Solution

• ```AB  *   A  = CCD
-      *      +
AE  :   F  =   G
----------------
B  *  CF  = CHE
```
Ref: J. Lehmann, Mathe mit Pfiff, Manz Verlag 1977, p72, problem 25
The Solution

• ``` E  *   K  =  BE
+      *      :
B  :   N  =   N
----------------
DN  +  DN  =  NB
```
Ref: Walter Lietzmann, Lustiges und Merkwürdiges von Zahlen und Formen, 11.Ed, p192
The Solution

• ```AB  *   C  =  CD
-      +      :
E  :   B  =   F
----------------
F  +   G  =   H
```
Ref: Walter Lietzmann, Lustiges und Merkwürdiges von Zahlen und Formen, 11.Ed, p192
The Solution

• ```AB  *   C  = DEB
+      +      -
AC  *   B  = FBB
----------------
GE  -  FE  =  CE
```
Ref: Walter Lietzmann, Lustiges und Merkwürdiges von Zahlen und Formen, 11.Ed, p192
The Solution

• ```AB  *   B  = CDD
:      *      :
CB  :   B  =   E
----------------
E  *   D  =  BD
```
Ref: Walter Lietzmann, Lustiges und Merkwürdiges von Zahlen und Formen, 11.Ed, p192
The Solution

• ```AB  *   C  =  DE
*      +      -
F  *   F  =  AC
----------------
GH  +  AI  =  CH
```
Ref: Walter Lietzmann, Lustiges und Merkwürdiges von Zahlen und Formen, 11.Ed, p192
The Solution

• ```ABB  :   C  =  DB
*      +      -
E  *   A  =   C
-----------------
FC  :  GA  =   B
```
Ref: Walter Lietzmann, Lustiges und Merkwürdiges von Zahlen und Formen, 11.Ed, p192
The Solution

• ```ABC :  CD  =   EF
-     *       +
FDF +  EG  =  AHD
-----------------
BB + FGA  =  AIJ
```
Ref: Bernhard Berchtold, www.mathematik.ch/puzzle/, Puzzle 9
The Solution

• ```A  -  B  =  C
*
D  :  E  =  F
=
G  +  H  =  I
```
The Solution

• magic cryptarithm
``` VV  RE  HR
AH  VE  RL
RV  HE  VR
```
This is a magic square with magic constant `HEH`. If you order the letters according there values you get the hidden word.
Ref: Praxis der Mathematik 39:4 (1997) 153
The Solution

• a bigger magic cryptarithm
``` HG GH UF LI  U
IF  I EU GG UH
NU UG IH FF EI
FH HF NI RU IG
RI LU FG HH GF
```
This is a magic square with magic constant `RHH`. The 25 numbers form an arithmetic progression. If you order the letters according there values you get the hidden word.
Ref: Zweisteins Zahlenlogeleien, Insel it 1510, Problem 22
The Solution

## Alphacipher

The numbers 1 - 26 have been randomly assigned to the letters of the alphabet. The numbers beside each word are the total of the values assigned to the letters in the word. e.g for LYRE L,Y,R,E might equal 5,9,20 and 13 respectively or any other combination that add up to 47.

The problem - What is the value of D ?

```  BALLET   45      POLKA     59
CELLO    43      QUARTET   50
CONCERT  74      SAXOPHONE 134
FLUTE    30      SCALE     51
FUGUE    50      SOLO      37
GLEE     66      SONG      61
JAZZ     58      SOPRANO   82
LYRE     47      THEME     72
OBOE     53      VIOLIN    100
OPERA    65      WALTZ     34
```
A problem form 'Tough Puzzles'. Solution

## Mysterious Computation

• explain
```161 + 134 + 145 = 503
```
Ref: Geoffrey Mott-Smith, Mathematical Puzzles, 1954, Problem 32

• find the clue
```R O M E  -  S U M  =  R U S E
```
Ref: Stephen Barr, Mathematical Brain Benders 2nd Miscellany of Puzzles, 1969, Problem 22

• Sense
```What SENSE does it make, if nine HENS give seven EGGS?
```
Author: J.A.H. Hunter

• Cut
```W E ) G E T ( I T
W E
---
C U T
x x x
=====
```
Author: J.A.H. Hunter

## Printers Errors

• 2^5*9^2 = 2592
Ref: Dudeney, Amusements in Math, 1917, Problem 115, A printer's error, pp.20 & 162.
Ref: Hubert Phillips, Question Time, 1937, Problem 133, pp.87 & 219.
Ref: Donald L. Vanderpool, Printer's `errors', Rec Math Mag 10 (Aug 1962) p.38. (Some extensions)
Ref: Printers Errors - Eric W. Weisstein

## Operator Puzzles

• ascending order
```1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9  =  10
```
Replace the 'x' by '+', '-', '*', '/'. No operator occurrs more than twice. Evaluate left to right.
Ref: Zeit Magazin, Nr 13, 22. Mar. 96, Logelei

## References

• W. W. Rouse Ball
Mathematical Recreations and Essays, Dover publications, New York, 12. Edition 1974 (edited by H. S. M. Coxeter) Chapter I.8: Arithmetical Restoration (p20-26, 12 problems)
• W. N. Bolchowitinow, B. I. Koltowoi, I. K. Lagowski
Spass für freie Stunden, Verlag für die Frau, Leipzig, 1980, 2. Auflage
(18 Cross Number Puzzles, 10 alphametic problems)
• Maxey Brooke
150 Puzzles in Crypt-Arithmetic, Dover Publ. 1963, 72pp
(156 problems with great varity in type and level)
• Pieter van Delft, Jack Botermans
Creative Puzzles of the World, (german: Denkspiele der Welt, Hugendubel, 1977)
(8 alphametic problems)
• Angela Dunn
Mathematical Bafflers, Dover Publ. 1980, ISBN 0-486-23961-6
(3 alphametic problems)
• Angela Fox Dunn
Second Book of Mathematical Bafflers, Dover Publ. 1983, ISBN 0-486-24352-4
(7 alphametic problems)
• Jack van der Elsen
Alphametics, (Dutch), Shaker Publishing B.V. ISBN: 90-423-0084-1
60 pages
• William A. Ewbank
Cryptarithms: Math Made Me Daft, Mama, Mathematics Teacher (Jan 1988) 54-60
• Dick Hess
Puzzles from around the world, Apr. 1997, p37, distributed by the author,
Add: 4100 Palos Verdes Drive East, Rancho Palos Verdes, CA 90275-6462
Contains 4 alphametics and 3 cross number puzzles
• P. Holzamer
Kreuzzahlenrätsel und Zahlenknobeleien, Harri Deutsch Verlag, 1994, Tb. Band 80, ISBN 3-8171-1321-8, 150p
Contains 178 alphametics and cross number puzzles.
• J. A. H. Hunter
Mathematical Brain-Teasers, Dover Publ. 1976, ISBN 0-486-23347-2
Part: Alphametics (40 problems)
• J. A. H. Hunter
Challenging Mathematical Teasers, Dover Publ. 1980, ISBN 0-486-23852-0
Part: Alphametics (p87-96, 40 problems)
• J. A. H. Hunter
Entertaining Mathematical Teasers - and How to Solve Them, Dover Publ. 1983, ISBN 0-486-24500-4
Part: Alphametics (p87-100, 40 problems)
• Steven Kahan
HAVE SOME SUMS TO SOLVE, the completest alphametics book, Baywood Publ. Company, Inc., Amityville, New York, ISBN 0-89503-007-1, 128p
(51 problems)
• Steven Kahan
AT LAST!! ENCODED TOTALS SECOND ADDITION, Baywood Publ. Company, Inc., Amityville, New York, ISBN 0-89503-171-X, 138p
(42 problems)
• Steven Kahan
TAKE A LOOK AT A GOOD BOOK, Baywood Publ. Company, Inc., Amityville, New York, ISBN 0-89503-142-6, 130p
(41 problems)
• Donald E. Knuth
The Art of Computer Programming
Section 7.2.1.2: Generating all permutations. Zeroth printing (2002)
(20 problems)
• Harry Langman
Play Mathematics, Hafner Publ. 1962
Chap. IV: Letter Division (49 problems)
Chap. V: Skeleton Division (24 problems)
• Johannes Lehmann
Mathe mit Pfiff, Manz Verlag 1977, ISBN 3-7863-0383-5
Chap: Kryptarithmetik (p71-72, 29 problems)
• Johannes Lehmann
Mathe mit Herz, Urania Verlag 1991, ISBN 3-332-00479-4, new edition of "Mathe mit Pfiff"
Chap: Kryptarithmetik (p81-83, 29 problems)
• Walter Lietzmann
Lustiges und Merkwürdiges von Zahlen und Formen, Verlag Vandenhoeck & Ruprecht, Göttingen 1955, 8. Auflage, ISBN 3-525-39112-9
Chap II.5: Vergilbte Manuskripte (p128-131, 21 problems)
Chap II.13: Anagramme, Kryptogramme, Geheimschriften und dergleichen (p191-192, 10 problems)
Madachy's Mathematical Recreations, Dover Publ. 1979, ISBN 0-486-23762-1
Chap. 7: Alphametics (p178-200, 37 problems)
• Geoffrey Mott-Smith
Mathematical Puzzles - for beginners and enthusiasts, Dover Publ. 1954, 2nd Ed., ISBN 0-486-20198-8
Chap. 7: Properties of Digits (p67-78, 10 problems)
• Randall K. Nichols (LANAKI)
Classical Cryptography Course (Vol I and II), Aegean Park Press, 1996-1997, 301 and 450pp., ISBN 0-89412-263-0 and 0-89412-264-9 respectively.
How to solve them. Lessons by LEDGE (Dr. Gerhard. D. Linz) in Vol I, chap. 8 and Vol II, chapt. 14 and 18.
• Jürgen Petigk
Mathematik in der Freizeit, Aulis Verlag Deubner & Co, Köln, 1998, ISBN 3-7614-1997-X
Wir bauen ein Zahlenrätsel (p150-159)
• Dr. Martin Sändig
Hobbymatik, Über 200 Symbolrätsel mit Lösungs- und Bastel-Tips,
Verlagsbuchhandlung Elmar Sändig, Walluf, 1984, ISBN 3-924395-01-2, 246p
over 200 problems, 100 with extended solution.
• Zweistein
Zweisteins Zahlenlogeleien, Insel Tb. 1510, 1993, ISBN 3-458-33210-3, 100p
Contains alphametics (40 problems) and cross number puzzles only
• N. V. Findler, P. Bunting
Some Ideas about the Solution of Cryptarithms, Journal of Recreational Puzzles 7:4 (1974) 309-14
An alphametics solver
• Journal of Recreational Puzzles
Each issue contains several Alphametics.
• Zeit Magazin
Section 'Logelei' contains now and than an alphametic.
• Usenet
alphametic articles found in rec.puzzles.