KenKen Solver

KenKen combines Sudoku's Latin square structure with arithmetic — each row and column must contain each digit exactly once (like Sudoku), but cells are grouped into "cages" outlined in bold, each containing a target number and arithmetic operator. The digits in each cage must combine using that operation to reach the target: a cage showing "12×" means the digits multiply to 12, while "3−" means the larger digit minus the smaller equals 3. Single-cell cages with an equals sign are free placements — they directly tell you what digit goes in that cell. Grid sizes range from 3×3 for beginners to 6×6 for the NYT's daily puzzles. Select your grid size, then add each cage by entering the target number, the operator (+, −, ×, ÷), and the row and column coordinates of each cell in the cage using the format row,column with cells separated by semicolons (for example: 0,0;0,1 for the first two cells in the top row). Hit Solve to get the completed grid.

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About KenKen

KenKen was created by Japanese mathematics teacher Tetsuya Miyamoto in 2004 as an educational tool to develop students' arithmetic and logical reasoning skills without rote memorization. Miyamoto's teaching philosophy — which he calls "the art of teaching without teaching" — uses puzzles to develop problem-solving abilities through self-directed discovery. KenKen (which translates roughly as "wisdom squared" or "clever clever") embodies this philosophy: it teaches arithmetic, logical deduction, and pattern recognition through engaging play rather than instruction.

The New York Times began publishing KenKen in 2008 and the puzzle quickly became one of the most popular features in the paper. The puzzle has since appeared in hundreds of publications worldwide and in educational curricula in multiple countries. KenKen championships are held internationally, with grid sizes ranging from 4×4 for beginners to 9×9 for experts. The puzzle combines two mathematical concepts: the Latin square structure of Sudoku (each digit appears exactly once in each row and column) with arithmetic cage constraints requiring specific operations to reach target numbers.

Each cage — a group of cells outlined in bold — contains a target number and arithmetic operator. The digits in that cage must combine using that operation to produce the target. A cage showing "12×" means the digits multiply to 12. A cage showing "3−" means the larger digit minus the smaller equals 3. Addition and multiplication cages allow digits in any order; subtraction and division cages require specific ordering (though solvers typically need only find a valid arrangement).

KenKen is published under various names due to trademark considerations: Calcudoku, MathDoku, KenDoku, and KENKEN® (the trademarked name owned by KenKen Puzzle LLC). All are functionally identical. The puzzle's educational value has been recognized by mathematics education organizations, and KenKen.com provides free daily puzzles for students and teachers alongside a curriculum resource library.

Single-cell cages with an equals sign (=) give you a free, immediate digit placement. Fill all of these first — they're essentially givens, like the starting digits in Sudoku. Each confirmed digit then constrains multiple rows and columns through the Latin square rule.

A division cage can only be satisfied by specific digit pairs where one divides evenly into the other. In a 4×4 grid with a "2÷" cage, the only options are 1,2'} or 2,4'}. Subtraction cages are similarly constrained: "2−" in two cells in a 4×4 grid allows 1,3'}, 2,4'}, or 3,1'}, 4,2'}. Solve these constrained cages before addition and multiplication cages.

Every row and column must contain each digit exactly once (like Sudoku). As you place digits, immediately eliminate them from other cells in the same row and column. Combined with cage constraints, this elimination often forces single valid options in constrained cells after only a few placements.

For multiplication cages, factorize the target number to find valid digit combinations. A "24×" cage in three cells of a 4×4 grid can only be 1,4,6'}, 2,3,4'}, 2,4,3'} etc. — but limited to digits 1-4 in a 4×4 grid, the only valid set is 1,4,6'} which is impossible (6 not in range), leaving 2,3,4'} as the only option. Factorization combined with grid size range restrictions quickly identifies forced combinations.

Fill single-cell cages first

Division and subtraction cages have very few valid options

Apply the Latin square constraint aggressively

Multiplication cages have unique factorization patterns

Q: What grid sizes does KenKen come in?

Standard KenKen ranges from 3×3 (beginner) to 9×9 (expert). The NYT typically publishes 4×4 through 6×6 daily. Competitive puzzles use up to 9×9. PuzzleUnlock's solver handles 3×3 through 6×6 grids.

Q: Can a cage span non-adjacent cells?

No — all cells in a KenKen cage must be orthogonally connected. L-shapes, zigzags, and other connected shapes are valid as long as every cell touches at least one other cage cell. Disconnected cells cannot form a single cage.

Q: What does the = operator mean in a cage?

A single-cell cage with = and a number means that cell simply contains that number. It's a free placement requiring no calculation — essentially a given digit, similar to Sudoku's starting numbers.

Q: How does KenKen differ from Kakuro?

KenKen uses a Latin square structure (each digit once per row and column) with arithmetic cages using all four operations. Kakuro uses a crossword structure with sum-only constraints on variable-length runs and no Latin square requirement. KenKen involves more operation variety; Kakuro involves more grid complexity.

Q: What is Calcudoku?

Calcudoku is a KenKen variant with slightly different rules regarding cage operations and grid constraints. The core mechanic is identical — arithmetic cages in a Latin square grid. The names KenKen, Calcudoku, MathDoku, and KenDoku all refer to effectively the same puzzle type.