KenKen Rules: How to Play, Strategies and Variations

What is KenKen?

KenKen is an arithmetic logic puzzle played on a square grid, usually from 3×3 up to 9×9. You fill every cell with a digit so that no digit repeats in any row or column, just as in Sudoku. On top of that, the grid is divided into outlined groups of cells called cages, and each cage shows a target number and an operation.

A cage labeled 12× means its digits multiply to 12. A cage labeled 7+ means they add to 7. Subtraction and division cages work on pairs of cells: 2− means the two digits differ by 2, and 3÷ means one digit is three times the other.

You will also see the puzzle called Calcudoku or Mathdoku. KenKen is a trademarked name, so other publishers use those generic labels for the same rules. Whatever it is called, the puzzle rewards quick mental math and careful elimination in equal measure.

A short history

KenKen was invented in 2004 by Tetsuya Miyamoto, a Japanese mathematics teacher, as a way to help his students build arithmetic skills and the habit of logical thinking. The name comes from the Japanese word ken, meaning cleverness or wisdom.

The puzzle reached English-speaking readers when The Times in London began printing it in 2008, and The New York Times followed in 2009. It has since become a regular in newspapers, puzzle books and classrooms, where teachers use small grids to practice times tables and number facts. You can read more in Wikipedia's KenKen article.

The core rules

KenKen's rules fit on an index card. For a grid of size N (4 for a 4×4, 6 for a 6×6 and so on):

1. 1Use the digits 1 to NA 4×4 grid uses 1, 2, 3 and 4. A 6×6 grid uses 1 to 6. There are no zeros.
2. 2No repeats in any row or columnEach row and each column contains every digit from 1 to N exactly once, which makes the finished grid a Latin square.
3. 3Each cage must hit its targetCombine the digits in the cage using the operation shown and you must get the target. For subtraction and division, take the larger digit first, so order within the cage never matters.
4. 4Single-cell cages are givensA cage of one cell shows just a number, with no operation. Write that number straight in.
5. 5Digits may repeat inside a cageA cage can contain the same digit twice, as long as the two cells are not in the same row or column. There are no Sudoku-style boxes.

In standard KenKen, subtraction and division cages always have exactly two cells, while addition and multiplication cages can be any size. A well-made puzzle has exactly one solution and never requires a guess.

The four operations

Every cage uses one of four operations. Here is how to read each one:

• Addition (+). The digits add up to the target. In a 4×4 grid, a three-cell cage clued 10+ could be 2, 4 and 4, provided the two 4s are in different rows and columns.
• Subtraction (−). Two cells whose difference is the target. In a 4×4 grid, 3− can only be 1 and 4, while 1− has many possibilities and is usually a weak clue.
• Multiplication (×). The digits multiply to the target. Large products are very restrictive: in a 6×6 grid, a two-cell 30× cage can only be 5 and 6.
• Division (÷). Two cells where the larger digit divided by the smaller gives the target. In a 6×6 grid, 3÷ is either 1 and 3 or 2 and 6.

A worked example

This 4×4 KenKen is solved. Each colored area is a cage, with its target and operation in the top-left corner.

Start with the free cell: the top-left cage is a single 1. Next, look at the 36× cage. In a 4×4 grid the only way to multiply three digits to 36 is 3 × 3 × 4, and the two 3s cannot share a row or column, so they must sit at the two ends of the L with the 4 in its corner. The 3÷ cage can only be 1 and 3, and the 10+ cage shows the repeat rule in action: 2 + 4 + 4, with the two 4s safely in different rows and columns.

How to play: a step-by-step method

This routine works on any size of grid:

1. Fill the single-cell cages. They are free answers. Write them in and remove those digits from the rest of their row and column.
2. List the options for tight cages. Large products, extreme sums and big differences have few combinations. Work out every possibility and pencil them in.
3. Apply the row and column rule. If a row already has a 3, no other cell in it can be 3. Cross off candidates that clash.
4. Look for forced positions. If a digit can only go in one cell of a row or column, place it there, even if the cage still has other options.
5. Revisit cages as neighbors fill. Every placed digit narrows the cages it touches. Recalculate what each cage still needs.
6. Check the finished grid. Confirm each row and column holds every digit once, then recheck each cage's arithmetic.

Strategies for tougher grids

Beyond the basics, these techniques do most of the work on 6×6 and larger puzzles.

Popular variations

Once you know the standard rules, these variants are easy to pick up:

• Calcudoku and Mathdoku. Generic names for the same puzzle, used by publishers who do not license the KenKen name.
• No-op puzzles. The cages show only a target, with the operation hidden. You have to work out both the digits and which operation produces the target.
• Addition-only or multiplication-only grids. Beginner puzzles often use a single operation so new solvers can focus on the row and column logic.
• Killer Sudoku. A close cousin that uses sum cages on a full Sudoku grid, and does not allow repeats inside a cage. See the Killer Sudoku rules.
• Futoshiki. Another Latin-square puzzle that swaps arithmetic for greater-than and less-than signs between cells. Read the Futoshiki guide.

Common mistakes

The mistake almost everyone makes early on is assuming cage digits must be different. That rule belongs to Killer Sudoku. In KenKen, digits may repeat inside a cage whenever the cells are in different rows and columns, and missing this often hides the only valid combination.

The second is forgetting the grid size. A 12× cage in a 4×4 grid cannot use 6, because 6 does not exist there. Third, solvers sometimes do subtraction or division in the wrong order and reject a correct pair. Always take the larger digit first. Finally, do not ignore the row and column totals: they are the easiest way to finish a nearly complete line.

KenKen vs Sudoku and Kakuro

KenKen shares its row and column rule with Sudoku, but it has no 3×3 boxes and no printed digits other than single-cell cages. The clues are all arithmetic, and the grid size changes from puzzle to puzzle.

Kakuro also uses sums, but its layout is a crossword-shaped grid of runs, and it uses addition only. KenKen offers all four operations, which makes it the most mathematical of the three and a favorite for classroom practice.

Frequently asked questions