Cryptarithms

Cryptarithms, also referred to as alphametics, present puzzles in which an arithmetic equation is provided, but the digits are substituted with letters, each letter representing a distinct digit. Your task is to decipher the arrangement, so to speak, to uncover the numerical value that each letter represents.

Just for fun, many of the times, the letters actually spell words. One of the most famous alphametics, spelling out SEND + MORE = MONEY was first published by Henry Dudeney, a British puzzlist, in 1924.

The Rules

Five rules govern alphametics:

1. Identical digits are replaced by the same letter. If two letters are the same, they represent the same digit.
2. Different digits are replaced by different letters. Each letter represents a unique digit.
3. After replacing all the letters with digits, the resulting arithmetic expression must be mathematically correct.
4. Numbers cannot start with 0. For example, the number 0900 is illegal.
5. Each problem must have exactly one solution, unless stated otherwise.

The problems will be in base 10 unless otherwise specified. This means that the letters replace some or all of the 10 digits – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Solving Strategy

When solving cryptarithms, look for:

• Carry constraints: If a sum produces a number with more digits than the addends, there must be a carry from the highest column.
• Leading letter constraints: The first letter of any word cannot be 0.
• Column analysis: Work column by column (units, tens, hundreds, etc.), considering possible carries.
• Uniqueness: Each letter must map to a different digit.

Start with columns that have the most constraints or the fewest possibilities. Often the leftmost column or carries give you the first breakthrough.

Famous Example: SEND + MORE = MONEY

The classic cryptarithm SEND + MORE = MONEY has a unique solution. Try working through it step by step:

• Since MONEY has 5 digits and SEND/MORE have 4, there must be a carry, so M = 1.
• This constrains S (which must be 8 or 9 to produce the carry).
• Working through the constraints systematically leads to the solution: 9567 + 1085 = 10652.

Use the interactive walkthrough above to see the full solution process!

Warm-up Problems

Below you'll find 20 mini cryptarithm puzzles to practice your skills. Click each card to reveal the solution. These start simple (3 letters) and gradually increase in complexity.

Practice Problems

Once you've mastered the warm-ups, try the full problems below for a greater challenge!