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Transfinite Chomp Scott Huddleston and Jerry Shurman Presented by Ehren Winterhof 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 Ordinals Ordinals Ω extend 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 … 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 5x3={01234}x{012} 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 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 Size Example Size (X) = Size (S’) = ω *3 + 1 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” 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 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 doubleinduction 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)) 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) P-Ordered Positions A Chomp Position is P-Ordered if its P subpositions are totally ordered by inclusion 2xω { (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) 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 Finite Two Wide Grundy Values If columns are of finite heights u, v If i = j = 1, and u = v More Two Wide Grundies When 2 < i = j < ω, and u = v When i = j < ω and u > v When ω ≤ i = j 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) Other Topics Covered but omitted here Side – Top Theorem N and P analysis of 3 wide chomp ωω x 3 is a P Position Open Questions