How to Solve Cryptarithm Puzzles, With Three Originals

A cryptarithm is an addition sum written in letters instead of digits. NRICH, the mathematics enrichment project at the University of Cambridge, defines it as "a mathematical puzzle where the digits in a sum have been replaced by letters," with each letter standing for a different digit and no number starting with 0.

The fun is that a handful of letters can hide a surprising amount of structure. Below are the rules, the column tricks that do most of the work, and three original puzzles built from ordinary words. Each was checked against every possible assignment of digits to letters, and each has exactly one solution.

1. The rules

• Each letter stands for one digit from 0 to 9.
• Different letters stand for different digits, and the same letter always means the same digit.
• No number may start with 0.
• The sum has to be correct when you swap the letters back to digits.

2. Why "exactly one answer" is not guaranteed

Many word sums have several solutions or none at all. NRICH's Two and Two problem asks how many solutions TWO + TWO = FOUR has, and its guidance on working systematically walks through the cases: when T is 7 there are two solutions, and when T is 8 there are three. That is why it pays to check uniqueness before calling a puzzle finished. The three below have only one solution each.

3. The column tricks

Trick 1: a new leading digit is 1. When two 3-digit numbers add up to a 4-digit number, the total is less than 2,000, so the first letter of the answer must be 1. NRICH's walkthrough of TWO + TWO = FOUR starts exactly this way, with F = 1.

Trick 2: track the carries. Each column can pass a carry of 0 or 1 to the next. Write the possible carries above the columns and treat them as extra unknowns. A column like "K + K ends in E" really means "2K = E, or 2K = E + 10."

Trick 3: the same letter in the same column. If a letter appears in both numbers and the answer of one column, it often forces a carry or a 0. For example, if A + B ends in A with no carry in, B must be 0.

Trick 4: work from the most constrained column. Start where the fewest options remain, usually the leftmost column, then the rightmost.

4. Puzzle 1

ELK + YAK = LAKE

5. Puzzle 2

RED + SEA = STAR

6. Puzzle 3

TEA + TOE = BOAT

Each puzzle uses five or six different letters. Try them before reading the solutions.

7. Solutions

Puzzle 1: 814 + 534 = 1348. By Trick 1, L = 1. In the ones column, K + K ends in E. In the tens column, L + A plus any carry ends in K, so 1 + A + carry ends in K. In the hundreds column, E + Y plus any carry must produce A with a carry of 1 into the thousands. Testing small values of K quickly settles it: K = 4 gives E = 8 with no carry, then 1 + A = 4 gives A = 3 with no carry, and finally 8 + Y = 13 gives Y = 5. So E = 8, L = 1, K = 4, Y = 5, A = 3, and 814 + 534 = 1348 spells LAKE.

Puzzle 2: 925 + 124 = 1049. By Trick 1, S = 1. Now look at the hundreds column: R + S plus a carry must reach at least 10, because it feeds the thousands. Since R is at most 9 and S is 1, the total is at most 11, so T, the hundreds digit of the answer, is 0 or 1. S already uses 1, so T = 0. That leaves R plus the carry equal to 9. With R = 9 and no carry from the tens, the tens column becomes E + E = A, and the ones column becomes D + A = 9. The digits E = 2, A = 4, D = 5 fit, giving RED = 925, SEA = 124, and STAR = 1049.

Puzzle 3: 578 + 507 = 1085. By Trick 1, B = 1. The hundreds column T + T must carry, so T is at least 5. Now compare the tens and ones columns. In the tens, E + O plus a carry gives A. In the ones, A + E gives T, possibly with a carry. Trying T = 5 makes O = 0 in the hundreds column. Then the tens column says A equals E plus the carry from the ones. If that carry were 0, A and E would be the same digit, which is not allowed, so the carry is 1 and A = E + 1. The ones column then needs A + E = 15, so E = 7 and A = 8. The result is TEA = 578, TOE = 507, and BOAT = 1085.

For every puzzle, all other assignments were ruled out by checking each possibility, so these are the only answers.

8. Make your own

NRICH invites readers to create their own cryptarithms, and the Cryptarithms problem adds a challenge: can all of the digits 1 to 9 appear exactly once in a 3-digit plus 3-digit sum? If you write your own word sums, test them before sharing. A sum with two answers is frustrating, and a sum with none is worse.

9. Key takeaways

• In a cryptarithm, each letter is a different digit, the same letter is always the same digit, and no number starts with 0.
• When two 3-digit numbers make a 4-digit total, the leading letter of the total is 1.
• Treat carries as small unknowns (0 or 1) and write them above each column.
• Start from the most constrained column, usually the leftmost, and work outward.
• Not every word sum has exactly one answer, so a good puzzle is checked for uniqueness before it is shared.