More Computational Curiosities of Sudoku ( 数独 ) Bryson R. Payne, Ph.D. North Georgia College & State University

What is Sudoku? Usually 9x9 grid with 9 3x3 subgrids Usually 17 to 32 “givens” Must fill in the grid so that each digit 1-9 appears only once in each column, row, and 3x3 subgrid

What’s in a name? Japanese name ( 数字は独身に限る ) = "Sūji wa dokushin ni kagiru" = the digits must occur only once 数独 = “su doku” = number single = Chinese “shu du” In each row, column, and area (usually 3x3 box), each digit must occur only once

Computational Curiosities Sudokus are special cases of magic squares (in which all rows and columns add up to 45) Sudokus are also special cases of Latin Squares (in which digits appear only once in each row & column) with the added requirement that digits appear only once in each 3x3 grid

More Computational Curiosities There are approximately 6x10 21 valid sudoku grids Taking into account rotation & reflection, there are only around 4 trillion relatively unique sudoku grids Considering relabeling leaves only around 5 billion unique sudoku puzzle grids

History of Sudoku Magic squares were printed in China at least back to the 1 st & 2 nd centuries A.D. First believed to be introduced to the West in the 9 th century in Thabit ibn Qurrah Europe’s famous appearance of a magic square is in Albrecht Dürer's 'Melancholia' in 1514 where a 4x4 magic square is shown with an arrangement of the numbers 1-16 and a sum of 34 in each row, column and diagonal

History continued The Swiss mathematician Leonhard Euler (of i and π fame) developed the closest historical relative to the modern sudoku, which he called Latin Squares, while teaching at the St. Petersburg Academy in Russia Believed to have presented the puzzle on October 17, 1776 – published 6 years later

Into the 20 th Century Euler’s puzzle remained largely unused for over 200 years, until Howard Garnes published the first known 9x9 Sudoku puzzle in Dell Puzzle Magazines in 1979 Title of the puzzle was “Number Place” Garnes added the rule that each 3x3 block had to satisfy the same requirements as each row and column

Into Japan Garnes’ puzzle quickly caught on in Japan One possible explanation for Sudoku’s popularity in Japan is a property of the Japanese language: Japanese characters are more symbolic than phonetic, and do not lend themselves easily to crosswords Japanese characters are more symbolic than phonetic, and do not lend themselves easily to crosswords Sudoku was first printed in the Monthly Nikolist magazine in April 1984

Back to the USA… almost In 1989, Loadstar/Softdisk introduced DigitHunt for the Commodore 64, the first known computerized version of Sudoku In 1997, a retired judge from New Zealand saw a partially solved Japanese sudoku puzzle and spent years developing & promoting a program to quickly produce the puzzles In November 2004, the London Times launched the puzzle, which quickly became a craze in Great Britain

Back to the USA In the spring of 2005, sudokus finally returned to the US in strong form

Hints & Heuristics Hints and Heuristics from American Scientist: The most elementary strategy for solving the puzzle is to examine each cell and list all its possible occupants—that is, all the numbers not ruled out by a conflict with another cell. If you find a cell that has only one allowed value, then obviously you can write that value in. The complementary approach is to note all the cells within a row, a column or a block where some particular number can appear; again, if there is a number that can be put in only one position, then you should put it there.

More Hints & Heuristics These are known as Level 1 rules: They restrict a cell to a single value or restrict a value to a single cell. At level 2 are rules that apply to pairs of cells within a row, column or block; when two such cells have only two possible values, those values are excluded elsewhere in the neighborhood. Level-3 rules work with triples of cells and values in the same way. In principle, the tower of rules might rise all the way to level 9.

It’s Just Not That Easy Part of the fun lies in the fact that not all Sudokus can be solved by these rules alone; some of the puzzles seem to demand analytic methods that don't have a clear place in the hierarchy. A few of these tactics have even acquired names, such as "swordfish" and "x-wing." The subtlest of them are nonlocal rules that bring together information from across a wide swath of the matrix.

Variations on Sudoku

Samurai Sudoku

Killer Sudoku

Killer Samurai Solution

Great Resources Free, online Sudoku sites: (my favorite ) (my favorite ) (hints, solves) (hints, solves)

Sudokus to print out: cube & code-doku - cube & code-doku Pay-per-view Sudokus:

Sudoku History: /assetid/48550g /assetid/48550g /assetid/ /assetid/

The Holy Grail: A Sudoku Solver

Another Great Site My web site: Has this presentation, as well as the “Computational Curiosities of Sudoku” handout Has this presentation, as well as the “Computational Curiosities of Sudoku” handout

Conclusion Finally – numbers are part of a positive, fun “craze” in the US! Sudokus have a rich, multicultural history Solving sudokus is primarily non- mathematical, but might help reduce “number anxiety” prevalent in our society There’s a sudoku level and variation for every taste – catch the fever today!

Thank You!