Patrick March, Lori Burns. History Islamic thinks and the discovery of the latin square. Leonhard...

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SUDOKU Patrick March, Lori Burns

Transcript of Patrick March, Lori Burns. History Islamic thinks and the discovery of the latin square. Leonhard...

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!