Calcudoku for Sudoku Players: Rules and First Techniques
If you enjoy Sudoku, try Calcudoku (also known as KenKen-style puzzles). Here are the rules and the first techniques to learn.
Calcudoku is a number puzzle that mixes Sudoku-style logic with simple arithmetic. Puzzles of this type are widely known through the KenKen brand, and many free versions exist under other names. If you already like Sudoku, you will feel at home quickly, but the math adds a new layer of reasoning.
The rules
- The grid is a square of size N×N (common sizes are 4×4 to 9×9).
- Fill each row and each column with the numbers 1 to N, each exactly once. (There are no 3×3 boxes like in Sudoku.)
- The grid is divided into outlined groups of cells called cages. Each cage has a target number and an operation: +, −, × or ÷.
- The numbers in a cage must produce the target using that operation.
- For + and ×, combine all numbers in the cage.
- For − and ÷, cages usually have exactly two cells; subtract or divide the smaller from the larger, in whichever order gives the target.
- A cage with a single cell simply shows the number that goes there.
- Numbers can repeat within a cage as long as they are not in the same row or column.
Technique 1: Start with single cells
Single-cell cages are free answers. Fill them first, then remove those numbers from the rest of their row and column.
Technique 2: List the combinations
For each cage, list the number combinations that could reach the target. In a 6×6 puzzle:
- A two-cell 11+ cage can only be 5 and 6.
- A two-cell 5− cage can only be 1 and 6.
- A two-cell 3÷ cage could be 1 and 3 or 2 and 6.
- A two-cell 30× cage can only be 5 and 6.
Cages with only one possible combination are great places to start, even if you do not yet know the order.
Technique 3: Use the row and column rule
Once you know a cage contains, for example, 5 and 6, those numbers cannot appear elsewhere in that cage's row (if the cage is horizontal) or column (if it is vertical). This is the same "naked pair" logic used in Sudoku.
Technique 4: Row and column totals
Every row and column contains each number from 1 to N once, so every row adds up to the same total:
| Grid size | Row total |
|---|---|
| 4×4 | 10 |
| 5×5 | 15 |
| 6×6 | 21 |
| 7×7 | 28 |
If a row is fully covered by cages whose sums you know except for one cell, you can calculate that cell. This is called the "rule of 45" in Sudoku-like puzzles (because a 9-cell row sums to 45), and it works the same way here with the right total.
Technique 5: Watch for big products
Multiplication cages with large targets usually require large numbers. In a 6×6 grid, a three-cell 120× cage must contain 4, 5 and 6 in some order, because no other three numbers from 1 to 6 multiply to 120.
For other targets, remember that repeated numbers are allowed in a cage when the repeated cells do not share a row or column. Check the cage shape before ruling a combination out.
A solving routine
- Fill single-cell cages.
- Find cages with only one possible combination.
- Use row/column elimination as in Sudoku.
- Apply row and column totals where cages line up.
- Revisit tricky cages with the new information.
Getting started
Begin with 4×4 puzzles. They take a few minutes and teach the arithmetic patterns quickly. Move to 6×6 once you can finish 4×4 without guessing. By then, the combination lists for common targets will start to feel automatic.
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