Play KenKen Online
Fill the grid so every row and column has unique numbers and every cage hits its arithmetic target. A maths puzzle you can solve with pure logic.
Created by Brian Hamilton
Step 1 — Grid Size
Step 2 — Operations
Step 3 — Difficulty
How to play KenKen
Fill the grid so every row and every column contains each digit from 1 to the grid size exactly once.
- The dashed outlines are cages. The small number and symbol in a cage say what its digits must produce with that operation.
- Subtraction and division are unordered: 2− means the two digits differ by two, either way round.
- A cage of one square simply shows its digit.
- A cage may repeat a digit, as long as no row or column does. This is the opposite of Killer Sudoku.
- Every puzzle has exactly one solution.
Controls: tap a square and type or tap a digit. Notes lets you pencil in candidates, and Check highlights mistakes.
A multiplication cage tells you far more than an addition cage
Every KenKen cage looks about equally informative — a number, an operation, a couple of squares. They are nothing like equal. On a 6×6 grid, taking every two-square cage and asking how many digit pairs satisfy it gives a clear ranking.
| Operation | Possible targets | Targets that fix the digits | Share |
|---|---|---|---|
| Multiply | 18 | 15 | 83% |
| Divide | 6 | 3 | 50% |
| Add | 11 | 4 | 36% |
| Subtract | 6 | 1 | 17% |
Five out of six multiplication targets name their digits outright. That is because products are unforgiving: 15 on a 6×6 can only be 3×5, while 8 can only be 2×4 — the pair 1×8 does not exist when the largest digit is 6.
Addition is the opposite. A cage reading 7+ could be 1+6, 2+5 or 3+4, and the
middling sums are the vaguest clues on the board. Subtraction is worse again: only the
largest possible difference pins anything down.
So when you are looking for a way in, read the operations before you read the numbers. A × cage in a corner is usually worth more than a + cage anywhere.
Every row adds to the same total
Because each row and column holds the digits 1 to n exactly once, every line on the board sums to the same fixed amount — and you can use that as an arithmetic check across cages, not just within them.
| Grid | Each row and column totals |
|---|---|
| 4×4 | 10 |
| 5×5 | 15 |
| 6×6 | 21 |
| 7×7 | 28 |
| 9×9 | 45 |
The practical move: if the cages covering a row are all addition cages and you know all but one of their totals, the missing one is forced. On a 5×5, cages adding to 6 and 4 in a row leave exactly 5 for whatever is left.
A worked board
Here is a 5×5 from the generator on this page. Dashed lines mark the cages.
Start with the single-square cages, then the multiplication cages, then use the row total of 15 to squeeze the rest. That order is not a stylistic preference — it is the ranking in the first table, applied.
Difficulty is cage size, and the free squares vanish
The setting does two things at once, and the second is the one you feel.
| Difficulty | Cages | Single-square (free) | Largest cages |
|---|---|---|---|
| Easy | 22.3 | 8.6 | 2 squares |
| Medium | 18.4 | 6.3 | 3 squares |
| Hard | 14.3 | 4.0 | up to 5 squares |
On easy, more than a third of the cages are single squares — digits handed to you. By hard that drops to about four on the whole board, and cages stretch to five squares, where a target says much less about any individual digit.
Where solvers get stuck
Assuming a cage cannot repeat a digit. It can. KenKen only forbids repeats within a row or column, so a cage bending across two rows may hold the same digit twice. This is the biggest difference from Killer Sudoku and it catches people arriving from there. It is not a rare technicality either: across 48 generated boards, 75 of 724 cages contained a repeated digit, and 35 of the 48 boards had at least one.
Treating subtraction and division as ordered. A cage reading
2− means the two digits differ by 2, in either order. Likewise
3÷ means one is three times the other, either way round.
Ignoring the line totals. Cage arithmetic is what people focus on, but the fixed row and column total is often what breaks a deadlock — especially on larger grids where cages sprawl.
Two things you can check about these puzzles
Every puzzle has exactly one solution, and it is verified. A cage layout is only accepted once a solver has confirmed a single filling satisfies it; otherwise the layout is thrown away and another is built. Checked across every grid size and difficulty, all the puzzles tested had a single answer.
The digits are a Latin square, so a size change is a real change. Every row
and column holds each digit once, which means a 4×4 uses only the digits 1–4 and
a 9×9 uses 1–9. Larger grids are not just longer — they widen the range
each cage has to work with, which is why a 12× cage means something quite
different on a 4×4 than on a 9×9.
KenKen, Calcudoku and Mathdoku
KenKen is a trademarked name; the same puzzle appears as Calcudoku, Mathdoku, KenDoku and Square Wisdom. It is generally credited to the Japanese teacher Tetsuya Miyamoto, who is said to have devised it in 2004 under a name usually translated as “the puzzle that makes you smarter”.
It sits between Sudoku and Killer Sudoku: like both it is built on a no-repeats grid, but its cages use four operations rather than sums alone, and — unlike Killer Sudoku — a cage may repeat a digit as long as no row or column does.
More number puzzles
If you like arithmetic doing the deducing, try these: