The Rat game and the Mouse game

We define three new take-away games, the Rat game, the Mouse game and the Fat Rat game. Three winning strategies are given for the Rat game and outlined for the Mouse and Fat Rat games. The efficiencies of the strategies are determined. Whereas the winning strategies of nontrivial take-away games are based on irrational numbers, our games are based on rational numbers. Another motivation stems from a problem in combinatorial number theory. 1 Description of the Game The Rat game is played on 3 piles of tokens by 2 players who play alternately. Positions in the game are denoted throughout in the form (x, y, z), with 0 ≤ x ≤ y ≤ z, and moves in the form (x, y, z) → (u, v, w), where of course also 0 ≤ u ≤ v ≤ w (see below). The player first unable to move — because the position is (0, 0, 0) — loses; the opponent wins. There are 3 types of moves: (I) Take any positive number of tokens from up to 2 piles. (II) Take ` > 0 from the x pile, k > 0 from the y pile, and an arbitrary positive number from the z pile, subject to the constraint |k − `| < a, where a = { 1 if y − x 6≡ 0 (mod 7) 2 if y − x ≡ 0 (mod 7). (III) Take ` > 0 from the x pile, k > 0 from the z pile, and an arbitrary positive number from the y pile, subject to the constraint |k − `| < b, where b = 3 if w = u; otherwise, b = { 5 if w − u 6≡ 4 (mod 7) 6 if w − u ≡ 4 (mod 7).