Tool/solver to resolve cryptarithms, a numeric puzzle which consists in a mathematical calculation in which letters have been replaced by digits to find.
Cryptarithm Solver - dCode
Tag(s) : Number Games, Arithmetics
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Cryptarithm Solver
Cryptarithm/Alphametic Solver
Other Calculation with Letters?
To get more results associating several numbers with the same letter or several letters with the same number, replace the letters by '?' (question mark) and use the fill-the-blank solver:
dCode has an equation solver to solve calculations with unknowns:
Answers to Questions (FAQ)
What is a cryptarithm? (Definition)
A cryptarithm (or alphametic, or cryptarithmetics) a mathematical game representing an arithmetic equation (with an equals sign =) in which one or more numbers are replaced by a substitution of letters or symbols. The objective of the game is to find which numbers correspond to which letter so that the equation is correct.
Example: DONALD + GERALD = ROBERT
BASE + BALL = GAMES
LLP + LINEAR + LOGIC = PROLOG
LOGIC + LOGIC = PROLOG
CROSS + ROADS = DANGER
SATURN + URANUS = PLANETS
TWO + TWO = FOUR
ABC + ABC + ABC = BBB
AB + BC + CA = ABC
XX + YY + ZZ = XYZ
XXXX + YYYY + ZZZZ = YXXXZ
XXXX + YYYY + ZZZZ = XYYYZ
How to solve cryptarithms?
Cryptarithm solving involves deduction and use of calculation tricks.
Method 1 (automatic): use the above solver, it tries all possible digits for all letters (brute-force method).
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
How does the cryptarithm solver work?
The cryptarithm solver handles classic mathematical operators like addition + (plus), subtraction - (minus), multiplication * (times) and division /.
The solver takes as unknowns between 1 and 10 distinct letters A-Z (which will represent the numbers 0 to 9).
Example: DCODE+CODING=SOLVED => 92095+209764=301859.
The solver also handles the logical conditions && for AND, || for OR and the upper and lower operators > and <. Additional criteria/conditions can be written with &&.
Example: To solve ABC+BCD=DEF knowing that B is less than C and F has the value 6. Write: ABC + BCD = DEF && B < C && F = 6 which resolves to 537+379 = 916
The cryptarithm solver uses a brute-force method: it tries all combinations of numbers and displays the possible ones. He does not provide a detailed explanation of his reasoning.
Some cryptarithms arrive in the form:
AB
Ă—C
--------
DE
+FG
---------
HI
Please rewrite it in 2 parts AB*C=DE+FG && DE+FG=HI for the solver
Can a cryptarithm have several solutions?
Yes, some cryptarithms can have multiple valid solutions, but many are designed to have only one.
dCode displays all solutions if multiple answers are possible for a given cryptarithm.
Can a cryptarithm have several equations?
Yes, as long as the rules of cryptarithm apply, it is possible to have multiple equations. The main thing is that the letters of the first equation are compatible with the letters of the following equations.
dCode handles multiple equations by separating them with the logical operator && (AND).
Why SEND+MORE=MONEY?
It is the most known example of cryptarithm, 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.
Source code
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