Patrick March, Lori Burns

History Islamic thinks and the discovery of the latin square. Leonhard Euler, a Swiss mathematician from the 18 th 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?

Latin Squares ABC CAB BCA 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 tableZ mod 4- multiplication table

How to complete a Sudoku? The object of sudoku: given an m 2 × m 2 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 m 2 exactly once. 2. Each column of cells contains the integers 1 to m 2 exactly once. 3. Each m×m square contains the integers 1 to m 2 only once

Sudoku Tactics If 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 solution Burnside Lemma: X g = known elements |X/G|=1/|G|Σ g in G |X g |,

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’s Let x= known numbers in the sudoku grid Each 3x3 sub grid is called a band Each of these sub grids has a (m-x)! permutations

Group Properties The 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 Life Sudoku 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!

References 2EE4FE8PQ/1?accountid= B3C4F4D5FPQ/2?accountid= B3C4F4D5FPQ/7?accountid= df *******

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 6 Band 8 with Band 9

Find 2 different solutions!