How to Solve Cryptarithms
Start with the answer’s extra digit and the ones column, then solve the letters one by one.
Play Cryptarithm (free)SEND + MORE = MONEY is 9567 + 1085 = 10652
In the grid, rows are letters and columns are digits. Ruled-out digits are crossed out.
| Letter | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| S | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| E | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| D | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| M | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| O | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| R | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| Y | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
Puzzle
- Think it through, then press “Next”.
- Ten-thousands: two 4-digit numbers make at most 19998, so M is 1.
- Thousands: S + 1 + carry is at most 11, so O is 0 (1 is M).
- Hundreds: with no carry, N would equal E. So 1 carries in: N = E + 1.
- Nothing carries on (that would need N = 0, but 0 is O). E isn’t 1, so N isn’t 2.
- Back to thousands: S + 1 = 10, so S is 9. E isn’t 8 (N would be 9).
- Tens: N = E + 1, so R + carry = 9. 9 is S, so R is 8 (carry 1).
- N can’t be 8 either, so E isn’t 7.
- Ones: D + E = Y + 10. Y is at least 2, so D + E is 12 or more.
- From 2 to 7, only E = 5, N = 6, D = 7, Y = 2 fit: 9567 + 1085 = 10652.
Three rules: one digit per letter, no leading 0
- The same letter always stands for the same digit
- Different letters stand for different digits
- A number with two or more digits never starts with 0
| Pattern | What it tells you |
|---|---|
| The answer is one digit longer | When two numbers are added, its first digit is 1 |
| The ones column | Nothing carries into it, so start here |
| A column like A + B = A | B is 0 (or 9 if 1 is carried in) |
| Carries | Adding two numbers, each carry is 0 or 1. Try both |
| A + A, A × 2 | The ones digit is even |
| Ones digit of a product | Only the ones digits matter; use times tables |
Addition: start with the extra digit and the ones
Puzzle
- Think it through, then press “Next”.
- Thousands: two 3-digit numbers make at most 1998, so D is 1.
- Ones: 8 + B ends in 3 only as 8 + 5 = 13. B is 5, carry 1.
- Tens: 5 + C + 1 ends in 2, so C is 6 and 1 carries.
- Hundreds: A + 6 + 1 = 14, so A is 7. 758 + 665 = 1423.
Subtraction: turn it into an addition
Puzzle
- Think it through, then press “Next”.
- As an addition: 52B + 1C8 = AB5 (answer + amount taken away).
- Ones: B + 8 ends in 5 only as 7 + 8 = 15. B is 7, carry 1.
- Tens: 2 + C + 1 = 7, so C is 4 with no carry.
- Hundreds: A = 5 + 1 = 6, so 675 − 148 = 527.
Multiplication: ones digits, then size
Puzzle
- Think it through, then press “Next”.
- Ones: C × C ends in C only for 0, 1, 5 and 6.
- C = 0 makes the answer 0, and C = 1 makes D = A. So C is 5 or 6.
- Digits: the answer has 3 digits, so ABC is less than 1000 ÷ 5 = 200. A is 1.
- Tens: if C were 5, 25 would carry 2 and B × 5 + 2 would end in 2 or 7.
- That’s never A = 1, so C is 6.
- Tens: 36 carries 3 and B × 6 + 3 ends in 1. So B is 3 or 8.
- With 8, 186 × 6 = 1116 has 4 digits, so B is 3.
- The answer: 136 × 6 = 816, so D is 8.
Tips for solving cryptarithms
- Start with few unknowns: one unknown letter and a known carry is simple arithmetic
- Look for repeated letters: write a solved digit everywhere its letter appears
- Cross out used digits: when only two candidates are left, try one
- Note the carries: write them small above the columns
- Estimate from the number of digits: it gives the size of each number
- Check at the end: do the sum, and make sure no two letters share a digit
Famous alphametics (with answers)
FORTY + TEN + TEN = SIXTY
Show answer
29786 + 850 + 850 = 31486
CROSS + ROADS = DANGER
Show answer
96233 + 62513 = 158746
DONALD + GERALD = ROBERT
Show answer
526485 + 197485 = 723970
BASE + BALL = GAMES
Show answer
7483 + 7455 = 14938
HOCUS + POCUS = PRESTO
Show answer
92836 + 12836 = 105672
EAT + THAT = APPLE
Show answer
819 + 9219 = 10038
Cryptarithm solver: type a puzzle, get every answer
Use the letters A to Z. Nothing you type is sent anywhere.
Practice puzzles
Puzzle 1: AB + B = BA
Show answer
89 + 9 = 98. In the tens column, A plus the carry gives B. With no carry A would equal B, so 1 is carried and B = A + 1. In the ones column, B + B = A + 10, so 2 × (A + 1) = A + 10, which gives A = 8 and B = 9.
Puzzle 2: AB − BA = A
Show answer
98 − 89 = 9. Turn it into the addition A + BA = AB. In the tens column, B plus the carry gives A. With no carry A would equal B, so 1 is carried and A = B + 1. In the ones column, A + A = B + 10, so 2 × (B + 1) = B + 10, which gives B = 8 and A = 9.
Puzzle 3: AB × A = CCC
Show answer
37 × 3 = 111. CCC = 111 × C, and 111 = 3 × 37, so AB × A is a multiple of 37. 37 is a prime number and the single digit A isn’t a multiple of 37, so AB itself is a multiple of 37: 37 or 74. If AB = 37, then A = 3 and 37 × 3 = 111. If AB = 74, then A = 7 and 74 × 7 = 518, which isn’t of the form CCC.
FAQ
- What is a cryptarithm?
- A sum with its digits replaced by letters. With real words, it’s called an alphametic
- What are other names for it?
- Cryptarithmetic, verbal arithmetic, word addition and alphametic
- Who named it?
- Minos (Simon Vatriquant), in 1931. J. A. H. Hunter coined “alphametic” in 1955
- Is a cryptarithm the same as a cryptogram?
- No. A cryptogram swaps letters for other letters; in a cryptarithm, letters stand for digits
- What is the answer to SEND + MORE = MONEY?
- 9567 + 1085 = 10652, its only solution. Henry Dudeney published it in 1924
- Can a cryptarithm have more than one solution?
- Made-up ones often do: TWO + TWO = FOUR has 7 solutions, such as 734 + 734 = 1468
- Can a number start with 0?
- Usually not. The rule doesn’t apply to single digits, such as the D in ABC × D
- Can a letter be a digit already shown in the puzzle?
- It depends on the puzzle. Not in this site’s game; the solver lets you switch it
- How do I type a puzzle into the solver?
- Write A−B=C or A×B=C (- * or x work too). Sums can have three or more numbers
- How long can the numbers be?
- Up to 15 digits each in the solver
- Where can I practice?
- This site’s game has word, addition, subtraction and multiplication puzzles at four levels
