Cryptarithm

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What Is a Cryptarithm?

A cryptarithm (also called an alphametic) is an arithmetic puzzle written in code: a sum like SEND + MORE = MONEY where every letter hides a digit. Decode all the letters and the sum must be mathematically true. Invented for amusement in the 1800s and named in the 1930s, cryptarithms are the classic meeting point of language and arithmetic — short to read, surprisingly deep to solve.

How Cryptarithms Work: One Letter, One Digit

Each letter stands for exactly one digit from 0 to 9, and keeps it for the whole puzzle.
Different letters are always different digits — no two letters share.
A word's first letter can't be 0 (leading zeros would break the numbers).
The decoded sum must check out exactly, column by column, carries included.

How to Solve a Cryptarithm, Step by Step

Work from the right. The units column sums first, and any overflow becomes a carry of 0 or 1 into the next column — because two three-letter words can't produce a carry bigger than 1, every M in MONEY-type sums is forced: M = 1. From there, hunt for columns with only one unknown, list the candidate digits for the most constrained letters, and test them against the carries. Each confirmed letter instantly limits every remaining one.

The Classic SEND + MORE = MONEY Puzzle

The most famous cryptarithm of all asks what digits make SEND + MORE = MONEY true. It has exactly one solution: O = 0, M = 1, Y = 2, E = 5, N = 6, D = 7, R = 8, S = 9 — giving 9567 + 1085 = 10652. Notice M = 1 falls straight out of the carry rule, and S = 9 plus the final carry is what forces MONEY's leading digit. One elegant chain of logic, no trial and error needed.

Famous Alphametics and Their Answers

I + BB = ILL — a tiny warm-up: B must be 9, forcing a double carry.
CROSS + ROADS = DANGER — a 1960s classic with a unique solution and a satisfying cascade.
DOUBLE + DOUBLE + TOIL + TROUBLE = 444,438 — Shakespeare's witches as arithmetic.
EARTH + AIR + FIRE + WATER = NATURE — the four elements balance into one word.

Play Free Cryptarithm Puzzles Online

The board above is our daily cryptarithm game: a hidden sum appears, you assign digits letter by letter, and the puzzle checks itself the moment every letter is decoded. There's a free practice mode with unlimited codes, a fresh daily cryptarithm at midnight UTC that's the same for everyone, and ★★★ for solving without hints.