CRYPTARITHMS

How to Solve Cryptarithms

A cryptarithm replaces digits with letters. Each letter always means the same digit. Your job is to decode the mapping and solve the arithmetic.

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Core rules

1
One letter equals one digit
If A is 7 in one place, A is 7 everywhere in the puzzle.
2
Different letters mean different digits
A and B cannot both be 7. Every letter must map to a unique digit.
3
No leading zeros
A number cannot start with 0. The first letter of any multi-digit number cannot be 0.
4
Use arithmetic consistency
Work column by column. Carries and remainders create strong constraints.
Tip: you do not need to guess. Most puzzles can be solved by logic plus a few pencil notes.
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Using the elimination grid

Many puzzles include an elimination grid to help you track possibilities. Think of it as a clean way to keep your pencil notes organized.

What it represents
  • Each letter has a row (or column) of digits 0–9.
  • Mark digits that a letter cannot be.
  • When only one digit remains for a letter, that letter is solved.
How to use it well
  • Start by eliminating leading zero for any leading letter.
  • When you solve a letter, eliminate that digit from every other letter.
  • Use carries, remainders, and forced columns to eliminate quickly.

A reliable solving strategy

Step 1: Start with forced columns
  • Look for columns where only one or two letters appear.
  • In long division, check the first subtraction step and any remainder column.
  • In addition/subtraction, start at the rightmost column (ones place).
Step 2: Track carries and remainders
  • Write small carry values above columns.
  • In long division, each step must be consistent with the divisor.
  • Even one carry can eliminate many digit options.
Step 3: Use elimination
  • Maintain a small list of possible digits for each letter.
  • When a letter is solved, remove that digit from all other letters.
  • If a contradiction appears, back up one assumption.
Step 4: Confirm by full evaluation
  • When most letters are solved, compute the full arithmetic.
  • Make sure every intermediate step matches.
  • If anything fails, a single letter is wrong. Re-check the earliest forced step.

Logic, algebra, or a mix

There is no single correct approach. Some solvers prefer pure logic, others write equations and solve algebraically, and many use a combination.

Logic-first
  • Use carries, remainders, and column constraints to eliminate options.
  • Great when the puzzle is designed to be solved without guessing.
  • Pairs well with the elimination grid.
Algebraic
  • Translate parts of the puzzle into equations.
  • Use known-digit constraints (0–9, distinct letters, no leading zeros).
  • Best when a few equations quickly narrow the solution.
Combined
  • Start with logic to reduce options.
  • Then use one or two small equations to finish.
  • Verify with full arithmetic at the end.
Tip: if you write equations, always keep the cryptarithm rules in view. Distinct digits and no leading zeros are constraints that eliminate many algebraic possibilities instantly.

FAQ

Usually no. Most puzzles are designed so logic narrows choices quickly. If you do make an assumption, treat it as temporary and verify it as soon as you can.

Track possible digits for letters, note carries and remainders, and circle any columns that become forced. A little pencil bookkeeping makes the puzzle feel much easier.

Return to the earliest steps. Re-check any carries, leading digit rules, and any place where a letter appears as the first digit of a number. Try a different forced column, or test a single carefully chosen assumption and see if it causes a contradiction.

It may take some practice, but anyone who enjoys logic puzzles can learn to solve cryptarithms. Use pencil notes and the elimination grid to help you track possibilities.

Ready to solve?

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Example

This is an example worksheet page showing a partially solved Long Division Cryptarithm.

Example of a partially solved long division cryptarithm worksheet