KenKen 賢くなるパズル

KenKen is an arithmetic logic puzzle played on a square grid. Like Sudoku, every row and every column must contain each number from 1 to N exactly once, but there are no boxes and no starting digits. Instead the grid is divided into heavily outlined groups of cells called cages, and each cage carries a small target number with an operation: 7+, 12×, 1−, 2÷. The numbers you write in a cage must produce that target with that operation. A cage of one cell simply shows the number that belongs there.

The puzzle was created in 2004 by Tetsuya Miyamoto, a mathematics teacher in Japan, as a classroom exercise that makes students reason for themselves. Ken (賢) is Japanese for wisdom or cleverness, and the doubled name is usually glossed as cleverness squared or wisdom squared. The Times of London began printing KenKen in 2008 and The New York Times followed the next year.

On this page you can play a new 5×5 KenKen every day and unlimited puzzles in 4×4, 5×5 and 6×6 sizes. Each grid is generated and then run through an exact solver that proves it has one solution and a logic grader that finishes it with deductions alone, so every puzzle can be solved without guessing. A keypad, pencil marks, a timer and a Check button are built in, and nothing needs an account.

Start with the cages that leave no choice. A single-cell cage is a given: write its number in and strike it from that row and column. Then look for two-cell cages with few combinations. A 1− cage holds two consecutive numbers. A 2÷ cage holds a number and its double, so in a 4×4 grid it is 1 and 2 or 2 and 4. Large multiplication targets usually factor one way: in a 5×5, a 20× pair must be 4 and 5, and a 60× triple must be 3, 4 and 5. A 3+ pair must be 1 and 2.

Write the candidates you find as pencil marks rather than committing early. Once a few cages are narrowed down, the row and column rule does the rest. If a number can go in only one cell of a row, it must go there (a hidden single); if a cell has one candidate left, place it (a naked single). Two cells in a line that share the same two candidates lock those numbers out of the rest of that line, exactly as a naked pair does in Sudoku.

Line sums separate experienced solvers from beginners. Every row and column of an N×N grid adds up to 1+2+…+N: 10 for 4×4, 15 for 5×5 and 21 for 6×6. When addition cages fill a row apart from one cell, that cell equals the row total minus the cage targets. The same trick works on columns and on pairs of rows.

Common mistakes: assuming the larger number in a − or ÷ cage must come first (order does not matter), forgetting that an L-shaped cage may hold the same number twice in different rows and columns, and guessing when the candidate lists get long.

KenKen was invented in 2004 by Tetsuya Miyamoto, a Japanese mathematics teacher who wanted a puzzle that would train his students in logic and arithmetic without a lecture. The name comes from ken (賢), the Japanese word for wisdom, and in Japan his puzzle books carry the title 賢くなるパズル, puzzles that make you smarter. The Times of London began printing KenKen in 2008 and The New York Times followed in 2009; its crossword editor Will Shortz later lent his name to a series of KenKen collections. KenKen is a trademarked name, which is why other publishers print the same mechanic under the generic names Calcudoku and Mathdoku. Unlike Kakuro, Slitherlink or Nurikabe, it is not a Nikoli puzzle.