1. Combinatorial Designs Dr. David R. Berman

2. Sudoku puzzle 134 1 243 341 2

3. Sudoku puzzle solution 1234 4321 2143 3412 3

4. Sudoku is Latin square with additional property Latin square of order n: Each number {1, 2, 3, …, n} appears exactly once in each row and column. Order 4 Latin square, not a Sudoku: 4 1234 4123 3412 2341

5. The Fano plane Seven points Three points on each line Every two points define a line Seven lines Three lines through each point Every two lines meet at a point 5

6. The Fano plane as a set system {0,1,4}, {0,2,5}, {0,3,6}, {1,2,6}, {4,2,3}, {4,5,6}, {1,3,5} 0 546 3 2 1 6

7. Round robin tournament 7 Directed edge between every pair of vertices X  Y means X beats Y {(1,2),(1,4),(2,4),(3,1),(3,2),(4,3)}

8. Doubles tournament Each game: a, b v c, d Tournament has many games Tournament usually has structure (e.g. everyone plays in the same number of games) 8

9. Whist tournament every pair of players partner once and oppose twice. Tournament is played in rounds. Example: Whist with 8 players 9 Table 1Table 2 Round 1∞0v4513v26 Round 2∞1v5624v30 Round 3∞2v6035v41 Round 4∞3v0146v52 Round 5∞4v1250v63 Round 6∞5v2361v04 Round 7∞6v3402v15

10. Research Strategies Use theoretical techniques to prove that a given design exists (or doesn’t exist) for certain sizes. Use experimental techniques to prove that a given design exists (or doesn’t exist) for certain sizes. 10

11. Field Operations + and * with properties: commutative, associative, identity, inverses, distributive Examples: real numbers, complex numbers Finite field: integers modulo a prime (Z p ) Primitive element ω of Z p generates all non- zero elements, i.e., Z p – {0} = {ω i : 0 ≤ i ≤ p-2} 11

12. Whist with 13 players outTable 1Table 2Table 3 R10112v85211v31049v67 R2120v96312v411510v78... R1312011v74110v2938v56 12

13. Theorem If p is a prime of the form 4K+1, then there exists a whist tournament with p players. 13

14. Examples of experimental work http://people.uncw.edu/bermand/Java.txt http://people.uncw.edu/bermand/C.txt http://people.uncw.edu/bermand/Mathemati ca.pdf http://people.uncw.edu/bermand/Mathemati ca.pdf 14

15. Applications of combinatorial designs Experimental designs (statistics) Coding, cryptography Software and hardware testing Network design and reliability 15

16. Resources C.J. Colbourn, J.H. Dinitz, Handbook of Combinatorial Designs, second edition, 2007, http://www.emba.uvm.edu/~dinitz/hcd.html http://www.emba.uvm.edu/~dinitz/hcd.html C.J. Colbourn, P.C. van Oorschot, Applications of combinatorial designs in computer science, ACM Computing Surveys, 1989. (Available in ACM Digital Library at Randall Library web site.) D.R. Berman, M. Greig, D.D. Smith, Brother Avoiding Round Robin Doubles Tournaments II, submitted to J. Comb. Des, http://people.uncw.edu/bermand/BARRDT.pdf http://people.uncw.edu/bermand/BARRDT.pdf 16

17. Thank you Are there questions?