# KenKen

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

The name derives from the Japanese word for cleverness (賢 ken, kashiko(i)).

As in sudoku, the goal of each puzzle is to fill a grid with digits -- 1 through 4 for a 4x4 grid, 1 through 5 for a 5x5, etc. -- so that no digit appears more than once in any row or any column (a Latin square). Grids range in size from 3x3 to 9x9. 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 4x4 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"). 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). 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 200 newspapers in the United States and worldwide. In 2014, KenKen signed an agreement with DTI, the software division of inflight entertainment expert Advanced Flight Alliance AG, and its parent company Global Eagle Entertainment, to provide KenKen on international flights. In 2015, KenKen partnered with German news organization Der Spiegel. The magazine, which is one of Europe's largest news publications, offers KenKen puzzles on their website and on their mobile app. 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. Today KenKen can be played online at www.kenkenpuzzle.com, the New York Times and Yahoo! Games online sites, as well as on its iOS, Android and Kindle Fire apps.

## 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 (x), 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"
• "20x" in the top row must be "4,5".
• "6x" 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"
• "240x" 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 +, −, x 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).

## Education

One of KenKen's main uses is in the classroom as a means of teaching basic arithmetic and logic skills. In 2009, KenKen LLC began the KenKen Classroom (KKCR) Program, which allowed teachers to subscribe to a weekly newsletter providing KenKen puzzle sets, as well as other brain teasers, and KenKen news. Today, over 25,000 educators are a part of the KKCR Program. In addition to KKCR, KenKen also helps educators organize their own KenKen clubs and tournaments in their schools. KenKen continues to partner with educators to help students learn while having fun. KenKen has also collaborated with the National Council of Teachers of Mathematics (NCTM), the world's largest educational organization, to create a mobile app in order to help students develop their math and logic skills. KenKen is also featured on Scholastic Corporation's online sites Scholastic Math and Scholastic Dynamath, as well as on The Math Forum @ Drexel.

## Tournaments

Beginning in 2010, KenKen CEO Robert Fuhrer has organized yearly KenKen tournaments held at the Chappaqua Library in Chappaqua, New York. Tournaments are open to all ages and skill levels. In addition to the Chappaqua tournament in December, Fuhrer partnered with IBM to host an international tournament in May 2015. Five Indian students beat out 35,000 competitors in their home country to earn a spot in the tournament. In addition to the visitors, many local students ages 8-18 competed in the tournament. The tournament was won by 8-year-old Gaurav Pandey, from India, who beat out competitors up to twice his age.

Year 1st place 2nd place 3rd place
2010 Molly Olonoff Rebecca Shapiro Martin Eiger
2011 Molly Olonoff Martin Eiger Adam Marcus
2012 Martin Eiger Adam Marcus Molly Olonoff
2013 Martin Eiger Matthew Zander Daniel Gritz
2014 Mack Meller Martin Eiger Hobart Chin
2015 (International) Gaurav Pandey Devika Pillai Eleanor Grueskin