KenKen
Encyclopedia
KenKen or KenDoku is a style of arithmetic and logic puzzle
invented in 2004 by the Japanese math teacher Tetsuya Miyamoto
, an innovator who says he practices "the art of teaching without teaching". He intends the puzzles as 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 .
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 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 three-cell cage specifying addition and a target number of 6 in a 4×4 puzzle might 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. Some puzzle authors have not done this and have published puzzles that use more than two cells for these operations.
(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, features 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 40 newspapers in the United States
, as well as numerous international publications.
The objective is to fill the grid in with the digits 1 through 6 such that:
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:
The restriction of puzzle size to the range two through nine is not absolute. A KenKen of size two is of little value even as an example, as it can immediately be solved by trying the two possibilities — ones on the "rising" diagonal and twos on the "falling" one, or vice versa. But extension beyond nine presents only difficulties of calculation with larger numbers, and the need, in recording possible values, to avoid confusing multi-digit numbers with items in a list of single digit ones.
Logic puzzle
A logic puzzle is a puzzle deriving from the mathematics field of deduction.-History:The logic puzzle was first produced by Charles Lutwidge Dodgson, who is better known under his pen name Lewis Carroll, the author of Alice's Adventures in Wonderland...
invented in 2004 by the Japanese math teacher Tetsuya Miyamoto
Tetsuya Miyamoto
is a Japanese mathematics teacher who invented the numerical logic puzzle KenKen....
, an innovator who says he practices "the art of teaching without teaching". He intends the puzzles as 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
Japanese language
is a language spoken by over 130 million people in Japan and in Japanese emigrant communities. It is a member of the Japonic language family, which has a number of proposed relationships with other languages, none of which has gained wide acceptance among historical linguists .Japanese is an...
word for .
As in sudoku
Sudoku
is a logic-based, combinatorial number-placement puzzle. The objective is to fill a 9×9 grid with digits so that each column, each row, and each of the nine 3×3 sub-grids that compose the grid contains all of the digits from 1 to 9...
, 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 column (a Latin square
Latin square
In combinatorics and in experimental design, a Latin square is an n × n array filled with n different symbols, each occurring exactly once in each row and exactly once in each column...
). 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 three-cell cage specifying addition and a target number of 6 in a 4×4 puzzle might 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
Will Shortz
Will Shortz is an American puzzle creator and editor, and currently the crossword puzzle editor for The New York Times.-Early life and education:...
, the issue of the non-associativity
Associativity
In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
of division and subtraction is addressed by restricting clues based on either of those operations to cages of only two cells. Some puzzle authors have not done this and have published puzzles that use more than two cells for these operations.
History
In 2005, toy inventor Robert Fuhrer encountered KenKen books published in Japan by the educational publisher Gakken Co., Ltd. and titled . Fuhrer's company Nextoy, LLCLimited liability company
A limited liability company is a flexible form of enterprise that blends elements of partnership and corporate structures. It is a legal form of company that provides limited liability to its owners in the vast majority of United States jurisdictions...
(now holder of a trademark on "KenKen" and "KenDoku" as a name for brain-training puzzles) and chess International Master Dr. David Levy
David Levy (chess player)
David Neil Laurence Levy , is a Scottish International Master of chess, a businessman noted for his involvement with computer chess and artificial intelligence, and the founder of the Computer Olympiads and the Mind Sports Olympiads. He has written more than 40 books on chess and computers.- Life...
helped bring the puzzles to the attention of Michael Harvey, features editor of The Times
The Times
The Times is a British daily national newspaper, first published in London in 1785 under the title The Daily Universal Register . The Times and its sister paper The Sunday Times are published by Times Newspapers Limited, a subsidiary since 1981 of News International...
(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 40 newspapers in the United States
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
, as well as numerous international publications.
Example
- 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 SudokuKiller SudokuKiller sudoku is a puzzle that combines elements of sudoku and kakuro...
, digits may repeat within a group.)
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.The restriction of puzzle size to the range two through nine is not absolute. A KenKen of size two is of little value even as an example, as it can immediately be solved by trying the two possibilities — ones on the "rising" diagonal and twos on the "falling" one, or vice versa. But extension beyond nine presents only difficulties of calculation with larger numbers, and the need, in recording possible values, to avoid confusing multi-digit numbers with items in a list of single digit ones.
External links
- kenken.com (official site)
- Puzzle Guru Will Shortz, Time MagazineTime (magazine)Time is an American news magazine. A European edition is published from London. Time Europe covers the Middle East, Africa and, since 2003, Latin America. An Asian edition is based in Hong Kong...
, March 2, 2009