Cryptarithm Solver

Alphametics (also called cryptarithm) solving involves deduction and use of calculation tricks.

Method 1 (automatic): use the above solver, it tries all possible digits for all letters.

Some Alphametics comes under the form:
AB
×C
--------
DE
+FG
---------
HI
Please rewrite it in two parts AB*C=DE+FG && DE+FG=HI for the solver

Method 2 (manual): deduction, logic and principles of mathematical calculations according to a few rules:

- Each letter can be associated to only one digit (appropriate numeral from 0 to 9). and reciprocally each digit can be associated to only one letter

If your calculation do not follow this rule, then use the missing numbers solver.

- Numbers usually do not start with a zero 0

The resolution then goes through the search for particular cases:

- addition/subtraction with 0 or 9

Example: ??A+??A=??A then A is 0

Example: ?A?+?A?=?A? then A is 0 or 9, same for ?A?+?B?=?B? or ??A+??B=??B

- first digits and last digits

Example: ???+???=A??? then A is 1 because it is impossible that the sum of 2 numbers less than 1000 is superior to 1999.

- multiplications by 0, 1 or 5 and squaring

Example: A*A=?B then A is not 0, 1 or 5

- multiplications of numbers with n and m digits that create numbers with n + m digits

- divisions by 1 ou 5

Do not hesitate to make attempts when there are few possibilities

dCode cryptarithm solver handles classical mathematical operators like additions + (plus), subtractions - (minus), multiplications * (times) and divisions /. It also handles logical conditions like && = ET, || = OU and also comparators superior and inferior > and <.

The solver takes for unknown between 1 and 10 distinct letters.

Example: A+A=B && B*C=AB is solved with 2+2=4 && 4*6=24

Example: DCODE+CODING=SOLVED => 92095+209764=301859.

This alphametic solver uses brute-force, ie. it tries all combinations of digits and displays the possible ones.

A cryptarithm is a number puzzle representing an arithmetic equation (with an equal = sign) in which some or all of its digits has been replaced by a substitution of letters or symbols. The goal is to find the digits represented by the letters. The term alphametics is also used when the letters form a real word.

Example: DONALD + GERALD = ROBERT
BASE + BALL = GAMES
LLP + LINEAR + LOGIC = PROLOG
LOGIC + LOGIC = PROLOG
CROSS + ROADS = DANGER
SATURN + URANUS = PLANETS
ABC + ABC + ABC = BBB
AB + BC + CA = ABC
XX + YY + ZZ = XYZ
XXXX + YYYY + ZZZZ = YXXXZ
XXXX + YYYY + ZZZZ = XYYYZ

It is the most known example of cryptarithm (or alphametic), published in 1924 in Strand Magazine, by Henry Dudeney: SEND+MORE=MONEY. Solution is O=0, M=1, Y=2, E=5, N=6, D=7, R=8, and S=9.