By Kardi Teknomo, PhD.


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CryptArithmetic or verbal arithmetic is a class of mathematical puzzles in which the digits are replaced by letters of the alphabet or other symbols. Usually it is required that each letter would be replaced by a unique digit. Each letter having different value from the other letters.
Alphametic problem is a subset of CryptArithmetic where the arithmetic operation is summation. In artificial intelligence, this problem is under category of Constraint Satisfaction Problem (CSP), or Constraint programming.

Example

Note that if you set X=2, Y =9 the meaning of XY is not 2*9 but a number 29.

To answer the above problem, we know that 2Y = X, thus X must be even number. We also know that 2X = Z, thus Z is also even number. Since the digits are limited from 2 to 9, then we can simply try Y = 2, this will give us X = 4 and Z = 8. Thus, {X=4, Y=2 and Z=8} is one of the solutions to the puzzle above.

When we try Y = 3, it will give us X = 6 and Z = 12 as invalid answer (because Z has two digits). You may think there is no other solutions. However, when you keep trying on higher number, say Y=6 then X = 12 and this is allowed if we consider the 1 can be carried the next digit and the actual value of X is 2. Thus, the value of Z will be 2*2+1 = 5. Thus, {X=2, Y=6 and Z=5} is also one of the solutions to the puzzle above.

If the digit integers are limited from 1 to 9, the above puzzle has another solutions of {X=2, Y=1 and Z = 4}. If the domain of digit integers are 0 to 9, the above puzzle has yet another solution of {X=0, Y=5 and Z=1}.

Thus, for domain of digits from 0 to 9, here are the possible solutions:
{X=2, Y=1 and Z=4}
{X=0, Y=5 and Z=1}
{X=2, Y=6 and Z=5}
{X=4, Y=2 and Z=8}



          21     05     26     42
21 05 26 42
--- + ---- + ---- + --- +
42 10 52 84


The following are Alphametic puzzles. Can you find the solutions?
SEND + MORE = MONEY
WIRE + MORE = MONEY
WRONG + WRONG = RIGHT
LETTERS + ALPHABETH = SCRABBLE
BASE + BALL = GAMES
GERALD + DONALD = ROBERT
ATTRACTIONS + INTENTIONS = REGENERATION
LIONNE + TIGRE = TIGRON (French)
WEIN + WEIB = LIEBE (German)
SEVEN - NINE = EIGHT
ABCDE x 4 = EDCBA










The solutions can be searched using backtracking algorithm or by searching from the list of all possible permutations.

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