What Is Cryptarithmetic? A Complete Guide to Letter-for-Digit Puzzles

What Is Cryptarithmetic?

Cryptarithmetic (also called alphametics or verbal arithmetic) is a type of mathematical puzzle where letters replace digits in an arithmetic equation. Each letter represents a unique digit from 0 to 9, and no multi-digit number can start with zero. The challenge is to figure out which digit each letter stands for so that the equation becomes mathematically correct.

The most famous example is SEND + MORE = MONEY, created by Henry Dudeney in 1924. The solution — S=9, E=5, N=6, D=7, M=1, O=0, R=8, Y=2 — means that 9567 + 1085 = 10652, which is correct.

The Rules of Cryptarithmetic

• Each letter represents exactly one digit from 0 to 9.
• Different letters must represent different digits.
• The same letter always represents the same digit throughout the puzzle.
• No number can start with zero. So the first letter of any word cannot be 0.
• The equation must be mathematically correct when all letters are replaced by their digits.

Types of Cryptarithmetic Puzzles

Addition Alphametics

The most common form. Two or more words are added together to produce a result word. Each column follows normal addition rules, including carries. Examples include SEND + MORE = MONEY and DONALD + GERALD = ROBERT.

Multiplication Alphametics

Less common but often harder. Two words are multiplied to produce a result. The constraint propagation works differently because partial products create additional sub-constraints. An example is ABC × DE = FGHIJ.

Doubly-True Puzzles

A special variant where the words themselves describe the arithmetic. For example, THREE + THREE + FIVE = ELEVEN, where the equation is literally true both in words and in digits.

Why Cryptarithmetic Matters

Beyond entertainment, cryptarithmetic puzzles train several important cognitive skills: logical deduction, working memory, constraint satisfaction, and systematic problem-solving. These are the same skills used in programming, mathematics, and scientific reasoning.

In computer science, cryptarithmetic is a classic example of a constraint satisfaction problem (CSP), and solving them efficiently is a standard topic in artificial intelligence courses.

How to Get Started

The best way to learn cryptarithmetic is to start with easy puzzles that have fewer unique letters, then work your way up to harder ones. You can practice with our free collection of over 200 puzzles, use the step-by-step solving guide, or try the automatic solver to check your work.