Lessons / Logical reasoning
Logical reasoning · Day 12

Cryptarithmetic

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

Practise (4 questions) Present as slides

Why this topic matters

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
Example 1In SEND + MORE = MONEY, what digit is M?
  1. Two 4-digit numbers add to a 5-digit number.The extra digit comes from a carry.
  2. A carry out of the top column can only be 1.Even 9999 + 9999 = 19998.
  3. M = 1.
Example 2AB + BA = 121, where A and B are digits. Find A + B.
  1. AB + BA = (10A + B) + (10B + A) = 11(A + B).Write the two-digit numbers in place value.
  2. 11(A + B) = 121, so A + B = 11.

Formula summary

Leading carry
When two n-digit numbers give an (n + 1)-digit sum, the new front digit is 1.
Same letter, same digit
Different letters, different digits. Leading letters are not 0.
Column rule
Digit + digit + carry-in = result + 10 × carry-out.
X + X = X
Then X = 0 (or X = 9 with carry-in 1 and a carry out).

Shortcuts and tricks

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.
ExampleSEND + MORE = MONEY. Value of MONEY?
  1. 9567 + 1085 = 10652.

Before you move on, you should be able to...

  • set the leading carry to 1
  • never put 0 at the front
  • use place value for two-digit letter numbers

Practice questions (4)

Try each one before opening the solution. Or practise them one by one so your score is saved.

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

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

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

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

Class plan (2 hours, for trainers)

A tested order for teaching this topic in one 2-hour session. Present mode follows the same order.

10 minWarm-up.
30 minStarting moves: leading 1, no leading zero, carries.
30 minSEND + MORE worked on the board.
15 minPlace-value tricks (AB + BA).
30 minPractice set.
5 minRecap.
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CryptarithmeticPractise now