SUDOKUPatrick March, Lori Burns

HistoryIslamic thinks and the

discovery of the latin square. Leonhard Euler, a Swiss

mathematician from the 18th century, used this idea to attempt to solve the following problem:

Is it possible to arrange 36 officers, each having one of six different ranks and belonging to one of six different regiments, in a 6-by-6 square, so that each row and each file contains just one officer of each rank and just one from each regiment?

2404001250

Latin Squares

A B C

C A B

B C A

•an m x m grid with m different elements, each element only appearing once in each row and column.

•Row permutation= ρ•Column permutation= β•Element permutation ={α}• All elements in a latin square follow:(ρ, β, α) •All permutations to rows, columns and elements are a bijection to the previous latin square.

Where Have WE Seen Latin Squares?All the Z mod addition and

multiplication tables!!!!!

Z mod 4- addition table Z mod 4- multiplication table

How to complete a Sudoku?The object of sudoku: given an m2 × m2 grid

divided into m × m distinct squares with the goal of filling each cell. The following 3 aspects must be met:1. Each row of cells contains the integers 1 to

m2 exactly once.2. Each column of cells contains the integers 1

to m2 exactly once.3. Each m×m square contains the integers 1 to

m2 only once

Sudoku TacticsIf ρ=2 β=1 α= x.Solve for X, and write it as a permutation.

Try it Out! What is the minimal number of starting

numbers given that will yield one unique solution?

|Knowns ≥ 17| = 1 unique solutionBurnside Lemma:Xg = known elements|X/G|=1/|G|Σg in G|Xg|,

Solutions:

Nowadays:The Sudoku is just a 9X9 Latin Square with 3x3

boxes as restrictions.The cardinality of a 9x9 Sudoku is

5,472,730,538 different Sudoku's without including reflections or rotations of the board.

The Math Behind Sudoku’sLet x= known

numbers in the sudoku grid

Each 3x3 sub grid is called a bandEach of these sub

grids has a (m-x)! permutations

Group PropertiesThe symmetries of a grid form a group G by

the following properties:1) Closure : l,mЄG, then so is (l·m)ЄG. 2) Associatively: l,m,k Є G, then l·(m·k)=(l·m)·k. 3) Identity: There is an element e ЄG such that

l·e=e·l=l for all l Є G. 4) Inverse: For all l Є G, there exists and inverse

m such that mЄ G, l·m=m·l=e where e is the identity element.

Sudoku in Real LifeSudoku algorithms have

inspired new algorithms that help with the automatic detection/ correction of errors during transmission over the internet

DNA Sudoku: a new genetic sequencing technique that helps with genotype analysis by sequences small portions of a persons genome to assist in identifying diseases.

New Versions of Sudoku!

Referenceshttp://search.proquest.com/docview/

918302381/803E07CB2EE4FE8PQ/1?accountid=13803 http://search.proquest.com/docview/

1113279814/977DD97B3C4F4D5FPQ/2?accountid=13803

http://search.proquest.com/docview/1450261661/977DD97B3C4F4D5FPQ/7?accountid=13803

 http://www2.lifl.fr/~delahaye/dnalor/

SudokuSciam2006.pdf *******http://theory.tifr.res.in/~sgupta/sudoku/expert.html http://www.geometer.org/mathcircles/sudoku.pdf

.

If you swapped band 2 and band 3 what else would you need to swap to keep the following grid all valid?

Band 5 with Band 6Band 8 with Band 9

Find 2 different solutions!