KenKen rules
Cage arithmetic meets Latin squares. Here is how it all works.
What is KenKen?
KenKen is a number-placement puzzle invented by Japanese math teacher Tetsuya Miyamoto in 2004. The name means "cleverness squared" in Japanese. You fill an N×N grid so that every row and column contains each digit from 1 to N exactly once, and the digits inside each outlined cage must combine using the given arithmetic operation to produce the cage's target number.
You might also see it called Calcudoku or Mathdoku. Same puzzle, different names. KenKen is the trademarked version that appears in The New York Times and many other outlets.
The rules
- Latin square: Every row and every column must contain each digit from 1 to N exactly once (where N is the grid size).
- Cage arithmetic: Each cage shows a target number and an operation (+, −, ×, ÷). The digits in the cage must combine using that operation to hit the target.
- Single-cell cages have no operation — the target is simply the digit that goes in that cell.
- Order does not matter for addition and multiplication cages. For subtraction and division, the operation applies to the larger minus/divided by the smaller, regardless of the cell positions.
Walking through a 4×4 example
Take a 4×4 grid (digits 1-4). Suppose you have these cages:
- Top-left 2 cells (horizontal): 3+
- Top-right 2 cells (horizontal): 8×
- A single cell in row 2: 2
The single-cell cage is straightforward — write 2 in that cell. For the 3+ cage, the two digits must add to 3 using digits 1-4 with no repeats in the row. Possibilities: 1+2. Check which goes where by looking at column constraints.
The 8× cage needs two digits from 1-4 that multiply to 8: that is 2×4. Now check which arrangement satisfies the row and column rules, and you have three cells placed. The rest of the grid follows from elimination.
Operations explained
| Symbol | Operation | Example |
|---|---|---|
| + | Addition | 7+ with three cells could be 1+2+4 |
| − | Subtraction | 1− with two cells: |4−3| = 1 or |2−1| = 1 |
| × | Multiplication | 12× with two cells: 3×4 or 2×6 |
| ÷ | Division | 2÷ with two cells: 4÷2 or 6÷3 |
Subtraction and division cages always have exactly two cells. The result is always the larger value minus (or divided by) the smaller, so cell order does not matter.
Grid sizes by difficulty
| Level | Grid | Notes |
|---|---|---|
| Easy | 3×3 – 4×4 | Addition only, small cages |
| Medium | 4×4 – 5×5 | Addition and subtraction |
| Hard | 5×5 – 6×6 | All four operations |
| Expert | 6×6 – 7×7 | Larger cages, trickier combinations |
| Einstein | 7×7 | Complex cage layouts, all operations |
Tips for beginners
- Fill single-cell cages first. They are free digits and immediately constrain their row and column.
- List all possible combinations for each cage. On a 4×4 grid, a 6+ two-cell cage can only be 2+4 — there is only one option.
- Division and subtraction cages on small grids often have just one or two possible digit pairs. Start with those.
- Use row/column counting. If a row is missing just two digits, check which cage constraints are compatible with each arrangement.
- Think about what a cage cannot contain. If a cage sits entirely in one row, no digit can repeat inside it.
Frequently asked questions
What is the difference between KenKen and sudoku?
Both require no-repeat rows and columns. Sudoku also has 3×3 box constraints and is always 9×9. KenKen replaces boxes with arithmetic cages that require actual math. KenKen grids range from 3×3 to 9×9.
What does the number in a cage mean?
It is the target. The digits inside must combine using the shown operation to produce that number. "12×" means the digits multiply to 12. "3−" means the difference between the two digits is 3.
Can digits repeat inside a cage?
Yes, if the cage cells are in different rows and columns. The no-repeat rule is per row and per column, not per cage. An L-shaped cage spanning two rows could have the same digit twice.
What is Calcudoku?
Calcudoku is another name for the same puzzle type. KenKen is trademarked by Nextoy LLC. Calcudoku and Mathdoku are the unbranded versions. The rules are identical.
What grid sizes does KenKen come in?
From 3×3 (digits 1-3) up to 9×9 (digits 1-9). On this site, difficulty levels use progressively larger grids with more complex cage layouts.
Related puzzle rules
- Sudoku rules — The classic 9×9 digit-placement puzzle
- Killer Sudoku rules — Sudoku with cage sums, no given digits
- Futoshiki rules — Latin square with inequality constraints
Ready to play? Start with an easy puzzle or pick your difficulty.