Cryptarithmetic Exit
Logical reasoning

Cryptarithmetic

Start from the column that gives the most information: the leftmost carry, a letter that repeats, or a sum that must be 1.

Start with this

SEND + MORE = MONEY: every letter is a different digit. You solve it like a detective: the carry into a new column can only be 1.

Cryptarithm questions reward a few logical starting moves. After that, trial with the remaining digits is quick.

Starting moves

The sum of two numbers can carry at most 1 into a new column. If the answer is one digit longer than both numbers, its first letter is 1.

A letter that starts a number is never 0. In a column where X + X ends in X, X is 0 (or 9 with a carry).

Work from the leftmost column and use carries to limit the choices, then fill the rest by trial.

SENDMORE+MONEY
SEND + MORE = MONEY: M must be 1

Worked example

In SEND + MORE = MONEY, what digit is M?

  1. Two 4-digit numbers add to a 5-digit number.
    Why: The extra digit comes from a carry.
  2. A carry out of the top column can only be 1.
    Why: Even 9999 + 9999 = 19998.
  3. M = 1.

Answer: 1

Worked example

AB + BA = 121, where A and B are digits. Find A + B.

  1. AB + BA = (10A + B) + (10B + A) = 11(A + B).
    Why: Write the two-digit numbers in place value.
  2. 11(A + B) = 121, so A + B = 11.

Answer: 11

Shortcut: Classic: SEND + MORE = MONEY

Use it when: The standard 8-letter puzzle.

  1. M = 1 (leading carry).
  2. S + 1 (+carry) ≥ 10 and gives O → S = 9, O = 0.
  3. Then E, N, D, R, Y follow column by column.

Classic: SEND + MORE = MONEY: try it

SEND + MORE = MONEY. Value of MONEY?

  1. 9567 + 1085 = 10652.

Answer: 10652

Your turn hard

TO + GO = OUT (different letters, different digits). Find OUT.

  1. O = 1 (sum of two 2-digit numbers is at most 198).
  2. Units: O + O = 2 = T, no carry.
  3. Tens: 2 + G must carry. G = 9 would make U = 1, already taken by O.
  4. So G = 8: 2 + 8 = 10 → U = 0, carry 1. 21 + 81 = 102.

Answer: 102

Your turn medium

AB + BA = 121 where A and B are distinct digits and A > B. How many such pairs (A, B)?

  1. 11(A + B) = 121 → A + B = 11.
  2. (9,2), (8,3), (7,4), (6,5).

Answer: 4

Your turn medium

A + A + A = BA (a 2-digit number). Find A and B.

  1. 3A = 10B + A → 2A = 10B → A = 5B.
  2. A = 5, B = 1.

Answer: A = 5, B = 1

Your turn hard

EAT + THAT = APPLE. Find APPLE.

  1. A = 1 (leading carry). THAT < 10000 so T = 9 and P = 0.
  2. Work the columns: E = 8, H = 2, L = 3.
  3. 819 + 9219 = 10038.

Answer: 10038

Recap

Now practise

4 questions with full solutions.

Open practice Back to the lesson