Algebra and Sudoku By Ashley MacDonald Math 354: Modern Algebra 1 Saint Francis Xavier University

What is Sudoku? Sudoku is a tremendously popular puzzle that typically consists of a 9x9 grid containing 81 cells (some already occupied) and 9 blocks. To complete the puzzle the participant must fill the remaining cells with the digits 1 through 9. However, the participant may not fill in the cells with any number of their choosing; they must follow the rule that each number may appear at most once in any row, column, or block. This rule is sometimes given the name, “The Rule of One” (Arcos, Brookfield and Krebs 111).

Which one is a real Sudoku puzzle? 1 4 3 1 4 3 3 4 2 4 3 2 1 4 3 3 1 1 4 3 1 1 3 2 ANSWER: NEITHER!

IN GENERAL A Sudoku puzzle can be generalized as a n2 x n2 board with n x n blocks and n2 symbols (where n is an integer). Example: We could have a 4x4 board with 4 blocks and 4 symbols (where the symbols are ☺, ♥, ♪,►) (Newton and DeSalvo 1959).

EXAMPLEs OF A SUDOKU PUZZLEs using SYMBOLS OTHER THEN NUMBERS

Who came up with sudoku anyway? - The game of Sudoku that we know of today is credited to a man by the name of Howard Garns (Newton and DeSalvo 1958). - Howard Garns was an architect. He submitted his game under the name “Number Place” in 1979 to Dell Pencil Puzzles and Word Games (Newton and DeSalvo 1958). - The game of Sudoku however is actually quite old. Sudokus are simply a special case of a Latin square (Gao 3).

What’s a latin square? A Latin square is a n x n table (where n is an integer) consisting of n different symbols such that each symbol can appear at most once in any row or column (Gao 2). How does this differ from Sudoku? Sudoku puzzles differ from Latin squares in that they have a third property to consider: each symbol must appear at most once in any block. Also a Sudoku grid must be a n2 x n2 grid not n x n.

Does anyone recognize this Latin square? + 1 2 3 4 5 ANSWER: Cayley Table of Z6 with addition

CAN WE MAKE A SUDOKU PUZZLE OUT OF A CAYLEY TABLE USING ALGEBRA? Take Z9 (with addition) and < 3 >. What are the left cosets? Let H = < 3 > = {0, 3, 6} 1 + H = {1, 4, 7} 2 + H = { 2, 5, 8}  

(Carmichael, Schloeman and Ward 131)

Other Applications of algebra in sudoku Symmetries

Relabeling 6 4

How many Sudokus exist? According to Felgenhauer & Jarvis there are 6, 670, 903, 752, 021, 072, 936, 960 Sudoku matrices. However it has also been found that there are only 5,472,730,538 different Sudokus (found by Jarvis and Russell) (Arcos, Brookfield and Krebs 1958). Therefore there may be more than one way to solve a Sudoku puzzle depending on how many cells are already filled.

LET’S…. PLAY….. SUDOKU!!!!

Let’s start with an Easy one! 3 1 4 4 3 2

Now a little more difficult! 7 2 8 1 4 9 2 7 3

Proposed Exam Question: a) What rule(s) must a player must follow in order to complete a Sudoku puzzle successfully? ANSWER: Each symbol must appear at most once in any row, column or block (Another acceptable answer would be that the player must follow the rule of one but you must explain what the rule of one is).

b) Complete the below Sudoku puzzle:

c) Are Sudoku puzzles related to Cayley Tables? How? ANSWER: Sudoku puzzles and Cayley tables are the same except that Sudoku puzzles have the added property that each symbol must appear at most once in every block. Cayley tables are nxn grids where as Sudoku tables are n2 x n2 grids. We can make manipulate a Cayley table to have the properties of a Sudoku table.

References Arcos, Carlos, Gary Brookfield and Mike Krebs. "Mini-Sudokus and Groups." Mathematics Magazine 83.2 (2010): 111-122. Web. 23 November 2014. <http://www.jstor.org/stable/pdfplus/10.4169/002557010X482871.pdf?&acceptTC=true&jpdConfirm=true >. Carmichael, Jennifer, Keith Schloeman and Michael B. Ward. "Cosets and Cayley-Sudoku Tables." Mathematics Magazine 83.2 (2010): 130-139. Web. 22 November 2014. <http://www.jstor.org/stable/pdfplus/10.4169/002557010X482899.pdf?acceptTC=true&jpdConfirm=true> . Felgenhauer, Bertram and Frazer Jarvis. "Mathematics of Sudoku I." 2006. Web. 18 November 2014. <http://www.afjarvis.staff.shef.ac.uk/sudoku/felgenhauer_jarvis_spec1.pdf>. Gao, Lei. "Latin Squares in Experimental Design." Michigan State University, 2005. Web. 21 November 2014. <http://www.mth.msu.edu/~jhall/classes/mth880-05/projects/latin.pdf>. Math Pages. Sudoku Symmetries. n.d. http://www.mathpages.com/home/kmath661/kmath661.htm. 20 November 2014. McGuire, Gary, Bastian Tugemann and Gilles Civario. There is no 16-Clue Sudoku: Solving the Sudoku Minimum Number of Clues Problem via Hitting Set Enumeration. Dublin: School of Mathematical Sciences, University College Dublin, Ireland., 2013. Web. Newton, Paul K. and Stephen A. DeSalvo. "The Shannon entropy of Sudoku matrices." Proceedings: Mathematical, Physical and Engineering Sciences 466.2119 (2010): 1957-1975. Web. 22 November 2014. <http://www.jstor.org/stable/pdfplus/25706326.pdf?&acceptTC=true&jpdConfirm=true>. Russell, Ed and Frazer Jarvis. "Mathematics of Sudoku II." 2006. Web.