SUDOKU

1. SUDOKU Patrick March, Lori Burns

2. History • Islamic 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?

3. Latin Squares • 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.

4. Where Have WE Seen Latin Squares? • All the Z mod addition and multiplication tables!!!!! Z mod 4- addition table Z mod 4- multiplication table

5. How to complete a Sudoku? • The object of sudoku: given an m2× m2 grid divided into m × m distinctsquares 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

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

7. 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: Xg= known elements |X/G|=1/|G|Σg in G|Xg|,

8. Solutions:

9. 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.

10. 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

11. 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.

12. 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.

13. New Versions of Sudoku!

14. References • http://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 .

15. 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

16. Find 2 different solutions!