Transfinite Chomp Scott Huddleston and Jerry Shurman Presented by Ehren Winterhof

Chomp Invented by David Gale, 1974 Invented by David Gale, 1974 Non-partisan combinatorial Non-partisan combinatorial Played on ℕ 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 Transfinite chomp uses ordinals for notation

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

Chomp Notation Each ordinal a is the set of all ordinals less than a. ie. 5 = { } Each ordinal a is the set of all ordinals less than a. ie. 5 = { } A rectangular game is written as a x b A rectangular game is written as a x b 5 x 3 = { } x { } 5 x 3 = { } x { } A bite from a two dimensional game is 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) Every Chomp position X has ordinal size, size(X) Decompose position into finite, overlapping sum of boxes S Decompose position into finite, overlapping sum of boxes S Each component box has each side length ω e, for non-negative integer e Each component box has each side length ω e, for non-negative integer e Discard any box contained within another to form Discard any box contained within another to form S’ If Y is reachable from X, size(Y) < size(X) If Y is reachable from X, size(Y) < size(X) Chomp terminates after finitely many moves 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 } G(X) = mex{G(Y) : Y is reachable from X } Poison Cookie has Grundy value 1 Poison Cookie has Grundy value 1 P-Positions have Grundy value 1 because they are reversible P-Positions have Grundy value 1 because they are reversible P-positions typically have value 0, but unrestricted misere Chomp is “tame” 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 < ) Two Chomp Positions A and B of dimension d and d-1, (with 1 < d < ω) Ordinal h Ordinal h E(A, B, h) = A + (B x Ω) - ⌐(0,…,0,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 A plus an infinite “column” of B, truncated to height h in the last direction “Extension of A by B to height h “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 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 Uniqueness is easy given existence Existence requires complicated double- induction 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)) 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: Assuming we can find h, such that E(A, B, h) is a P-position, we can: Find the Grundy Value of a position Find the Grundy Value of a position Construct positions of arbitrary Grundy value Construct positions of arbitrary Grundy value For finite A and ordinal h, G(A + (1 d-1 x h)) has the same highest term as h. For finite A and ordinal h, G(A + (1 d-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 A Chomp Position is P-Ordered if its P subpositions are totally ordered by inclusion 2 x 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)

Side-Top Positions A 2 dimensional Position of the form A 2 dimensional Position of the form U = 2 x U = 2 x ω + (2,0) + (0, ω) Def. ☐S = (2 x s) + (2, 0) Side-Top Theorem- In a S.T. Position, if H(S,2) is finite, U is an N-Position. If it is infinite: U is P position iff G

Two-Wide Chomp Two Columns h, k of ordinal height Two Columns h, k of ordinal height h = ω i * u + a k = ω j * 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 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? In the sum of these three 2-wide Chomp positions, what is the winning move that reduces the game size the most? A. ( A. ( ω * 2 + 3) x 2 B. (ω 4 * ), (ω 4 * ) C. (ω 3 * ), (ω 3 * )

Other Topics Covered but omitted here Side – Top Theorem Side – Top Theorem N and P analysis of 3 wide chomp N and P analysis of 3 wide chomp ω ω x 3 is a P Position Open Questions