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Transfinite Chomp

Transfinite Chomp. Scott Huddleston and Jerry Shurman Presented by Ehren Winterhof. Chomp . Invented by David Gale, 1974 Non-partisan combinatorial Played on ℕ d for d in ℤ + A move consists of choosing a lattice point in the position and removing it along with all points outward

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Transfinite Chomp

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  1. Transfinite Chomp Scott Huddleston and Jerry Shurman Presented by Ehren Winterhof

  2. Chomp • Invented by David Gale, 1974 • Non-partisan combinatorial • Played on ℕd for d in ℤ+ • A move consists of choosing a lattice point in the position and removing it along with all points outward • Transfinite chomp uses ordinals for notation

  3. Ordinals • Ordinals Ωextend Natural Numbers ℕ to include infinite numbers • Totally ordered (mex, sup) • ⊎, ⋆ (not commutative) • Smallest infinite number is ω (little omega) • In ascending order: ω, ω⊎1, ω⊎2, …, ω⋆2, ω⋆2⊎1, …, ω⋆3, …, ω2, ω2 ⊎1, …, ω2⋆2, ω3, …,ωω …

  4. Chomp Notation • Each ordinal a is the set of all ordinals less than a. ie. 5 = { 0 1 2 3 4 } • A rectangular game is written as a x b • 5 x 3 = { 0 1 2 3 4 } x { 0 1 2 } • A bite from a two dimensional game is ⌐(a b) = ⌐a x ⌐b = { y | y ≥ a } x { z | z ≥ b } • Notation extends to any number of dimensions

  5. Chomp Size • Every Chomp position X has ordinal size, size(X) • Decompose position into finite, overlapping sum of boxes S • Each component box has each side length ωe, for non-negative integer e • Discard any box contained within another to form S’ • If Y is reachable from X, size(Y) < size(X) • Chomp terminates after finitely many moves

  6. Size Example Size (X) = Size (S’) = ω *3 + 1

  7. Grundy Values • G(X) = mex{G(Y) : Y is reachable from X } • Poison Cookie has Grundy value 1 • P-Positions have Grundy value 1 because they are reversible • P-positions typically have value 0, but unrestricted misere Chomp is “tame”

  8. Extension • Two Chomp Positions A and B of dimension d and d-1, (with 1 < d < ω) • Ordinal h • E(A, B, h) = A + (B x Ω) - ⌐(0,…,0,h) • A plus an infinite “column” of B, truncated to height h in the last direction • “Extension of A by B to height h

  9. Fundamental Theorem • For any A and B, there is a unique ordinal h such that E(A, B, h) is a P position • Uniqueness is easy given existence • Existence requires complicated double-induction • h is tricky to calculate, but if you choose B to be the d-1 dimension poison square, h is bounded by size(A – (B x Ω))

  10. Consequences • Assuming we can find h, such that E(A, B, h) is a P-position, we can: • Find the Grundy Value of a position • Construct positions of arbitrary Grundy value • For finite A and ordinal h, G(A + (1d-1 x h)) has the same highest term as h. (General Beanstalk Lemma)

  11. P-Ordered Positions • A Chomp Position is P-Ordered if its P subpositions are totally ordered by inclusion • 2 x ω • { (1 x (i+1) + (2 x i) : 0 ≤ I < ω } • { (1 x a) + (a x 1) : 0 < a } • If P is a P ordered Chomp Position, then G(X x P) = G(X)

  12. Two-Wide Chomp • Two Columns h, k of ordinal height • h = ωi * u + a • k = ωj * v + b • If h and k differ by a factor of ω, by an extension of the beanstalk lemma, the Grundy value is infinite • Limiting examination to i=j and u=v we get the following

  13. Finite Two Wide Grundy Values If columns are of finite heights u, v If i = j = 1, and u = v

  14. More Two Wide Grundies When 2 < i = j < ω, and u = v When i = j < ω and u > v When ω ≤ i = j

  15. Question • In the sum of these three 2-wide Chomp positions, what is the winning move that reduces the game size the most? • A. (ω * 2 + 3) x 2 • B. (ω4 * 6 + 26) , (ω4 * 6 + 10) • C. (ω3 * 10 + 36), (ω3 * 4 + 15)

  16. Other Topics Coveredbut omitted here • Side – Top Theorem • N and P analysis of 3 wide chomp • ωωx 3 is a P Position • Open Questions

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