# KenKen

KenKen and KenDoku are trademarked names for a style of arithmetic and logic puzzle invented in 2004 by Japanese math teacher Tetsuya Miyamoto,[1] who intended the puzzles to be an instruction-free method of training the brain.[2] The names Calcudoku and Mathdoku are sometimes used by those who don't have the rights to use the KenKen or KenDoku trademarks.[3][4]

The name derives from the Japanese word for cleverness ( ken, kashiko(i)?).[1]

As in sudoku, the goal of each puzzle is to fill a grid with digits –– 1 through 4 for a 4×4 grid, 1 through 5 for a 5×5, etc. –– so that no digit appears more than once in any row or any column (a Latin square). Grids range in size from 3×3 to 9×9. Additionally, KenKen grids are divided into heavily outlined groups of cells –– often called “cages” –– and the numbers in the cells of each cage must produce a certain “target” number when combined using a specified mathematical operation (either addition, subtraction, multiplication or division). For example, a linear three-cell cage specifying addition and a target number of 6 in a 4×4 puzzle must be satisfied with the digits 1, 2, and 3. Digits may be repeated within a cage, as long as they are not in the same row or column. No operation is relevant for a single-cell cage: placing the "target" in the cell is the only possibility (thus being a "free space"). The target number and operation appear in the upper left-hand corner of the cage.

In the English-language KenKen books of Will Shortz, the issue of the non-associativity of division and subtraction is addressed by restricting clues based on either of those operations to cages of only two cells in which the numbers may appear in any order. Hence if the target is 1 and the operation is - (subtraction) and the number choices are 2 and 3, possible answers are 2,3 or 3,2. Some puzzle authors have not done this and have published puzzles that use more than two cells for these operations.

## History

In 2007, toy inventor Robert Fuhrer, owner of Nextoy and creator of Gator Golf, Crocodile Dentist, and dozens of other popular toys and games, encountered KenKen books published in Japan by the educational publisher Gakken Co., Ltd. and titled "Kashikoku naru Puzzle" (賢くなるパズル Kashikoku naru pazuru?, lit. "smartness puzzle").[2] Fuhrer's company Nextoy, LLC (now holder of a trademark on "KenKen" and "KenDoku" as a name for brain-training puzzles) and chess International Master Dr. David Levy helped bring the puzzles to the attention of Michael Harvey, an editor of The Times (London).[5] Harvey, impressed with what he calls its "depth and magnitude", arranged for publication of such puzzles, starting in March 2008, in The Times. Other papers, including the New York Times, followed suit. KenKen now appears in more than 100 newspapers in the United States, as well as numerous international publications. KenKen is also being used by over 30,000 teachers throughout the United States to teach math skills, problem solving techniques, logic, and critical thinking.[6] Between its website at www.kenkenpuzzle.com, the New York Times online site and its iOS, Android and Kindle Fire apps, over 150,000 puzzles are played every day by puzzlers of all ages.

## Example

A typical KenKen problem.
Solution to the above problem.

The objective is to fill the grid in with the digits 1 through 6 such that:

• Each row contains exactly one of each digit
• Each column contains exactly one of each digit
• Each bold-outlined group of cells is a cage containing digits which achieve the specified result using the specified mathematical operation: addition (+), subtraction (−), multiplication (×), and division (÷). (Unlike Killer Sudoku, digits may repeat within a cage.)

Some of the techniques from Sudoku and Killer Sudoku can be used here, but much of the process involves the listing of all the possible options and eliminating the options one by one as other information requires.

In the example here:

• "11+" in the leftmost column can only be "5,6"
• "2÷" in the top row must be one of "1,2", "2,4" or "3,6"
• "20×" in the top row must be "4,5".
• "6×" in the top right must be "1,1,2,3". Therefore the two "1"s must be in separate columns, thus row 1 column 5 is a "1".
• "30x" in the fourth row down must contain "5,6"
• "240×" on the left side is one of "6,5,4,2" or "3,5,4,4". Either way the five must be in the upper right cell because we have "5,6" already in column 1, and "5,6" in row 4.
• etc.

## Extensions

More complex KenKen problems are formed using the principles described above but omitting the symbols +, −, × and ÷, thus leaving them as yet another unknown to be determined. Other authors of puzzles include more complex operations, including exponentiation, modulus, and bit-wise operations. Ranges of values can be varied, such as including zero, or having negative values (e.g., -2 to +2 in a 5-by-5 square).

## References

1. ^ a b A New Puzzle Challenges Math Skills, New York Times, February 8, 2009
2. ^ a b Tetsuya Miyamoto creates KenKen. Train your brain, The Times, 21 March 2008
3. ^ KenDoku renamed to CalcuDoku
4. ^ KenKen renamed to Mathdoku
5. ^ Stephey, M. J. "The Next Sudoku?" Time Magazine 23 Mar. 2009: 72.
6. ^ Kulkarni, D. Enjoying Math: Learning Problem Solving With KenKen Puzzles, A textbook for teaching with KenKen Puzzles.