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Combinatorial Designs
Dr. David R. Berman
Sudoku puzzle
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Sudoku puzzle solution
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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:
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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
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The Fano plane as a set system
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{0,1,4}, {0,2,5}, {0,3,6}, {1,2,6},
{4,2,3}, {4,5,6}, {1,3,5}
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Round robin tournament
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)}
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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)
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Whist tournament
every pair of players partner once and oppose twice.
Tournament is played in rounds.
Example: Whist with 8 players
Table 1
Table 2
Round 1
∞
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Round 2
∞
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v
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Round 3
∞
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Round 4
∞
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Round 5
∞
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Round 6
∞
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Round 7
∞
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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.
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Field
• Operations + and * with properties:
commutative, associative, identity, inverses,
distributive
• Examples: real numbers, complex numbers
• Finite field: integers modulo a prime (Zp)
• Primitive element ω of Zp generates all nonzero elements, i.e., Zp – {0} = {ωi: 0 ≤ i ≤ p-2}
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Whist with 13 players
out
Table 1
Table 2
Table 3
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R2
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...
R1
R13
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Theorem
If p is a prime of the form 4K+1, then there
exists a whist tournament with p players.
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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
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Applications of combinatorial designs
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Experimental designs (statistics)
Coding, cryptography
Software and hardware testing
Network design and reliability
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Resources
• C.J. Colbourn, J.H. Dinitz, Handbook of Combinatorial Designs,
second edition, 2007,
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
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Thank you
Are there questions?