1. Amazons Puzzles are NP- Complete

2. G∞ is the infinite grid. Cubic Subgrid Graphs are subgraphs of G∞ where nodes have degree at most three. HC3G = {G | G is a cubic subgrid graph with a Hamilton circuit} HCB3P = {G | G is a bipartite cubic planar graph with a Hamilton circuit} HCB3P is known to be NP-complete, so we prove that HC3G is NP-complete by reducing from HCB3P to HC3G.

5. Definition: A collision path is an edge disjoint path with at most one node repetition which ends right after the repetition Corollary: The set of all cubic subgrid graphs G with a collision path of length |V G |-1 with a specified starting point is NP- complete

6. Theorem: The set AP = {(p, b) | Amazons puzzle p has a solution length at least b} is NP-complete. Proof by reduction from cubic subgrid graphs with a collision path. Prove the equivalent statement: G has a collision path of length n-l (n=|V G |) starting in a specified node s if and only if the amazon can make at least b moves in position p

9. Let m be the maximum number of moves the amazon can make in position p and L be the maximum length of collision paths in G. Let k = 6n (i.e., the corridors have length 12n). Set b to 12 (n 2 –n) to get: Claim: L >= n-1  m >= 12(n 2 –n) m >= L(2k + 1) m <= L(C + 2k + 1) +C, where C is the maximum number of empty squares in the 7x7 regions.

10. Let C=11, k = 6n Then m >= L(12n+1) and m <= (12n+12)+11 Therefore: L >= n-1 => m >= (n-1)(12n+1) = 12n 2 -11n-1 L m <= (n-2)(12n+12)+11 = 12n 2 -12n-13 L >= n-1  m >= 12(n 2 -n) follows from this, and so AP is NP-complete

11. Corollary: SAE = {p | Black wins simple Amazons endgame p} is NP-equivalent (?)