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Pudding Game

Source: 2019 Taiwan TST Round 1

March 31, 2020
analytic geometrycombinatorics

Problem Statement

Alice and Bob play a game on a Cartesian Coordinate Plane. At the beginning, Alice chooses a lattice point (x0,y0) \left(x_{0}, y_{0}\right) and places a pudding. Then they plays by turns (B goes first) according to the rules
a. If A A places a pudding on (x,y) \left(x,y\right) in the last round, then B B can only place a pudding on one of (x+2,y+1),(x+2,y−1),(x−2,y+1),(x−2,y−1) \left(x+2, y+1\right), \left(x+2, y-1\right), \left(x-2, y+1\right), \left(x-2, y-1\right)
b. If B B places a pudding on (x,y) \left(x,y\right) in the last round, then A A can only place a pudding on one of (x+1,y+2),(x+1,y−2),(x−1,y+2),(x−1,y−2) \left(x+1, y+2\right), \left(x+1, y-2\right), \left(x-1, y+2\right), \left(x-1, y-2\right)
Furthermore, if there is already a pudding on (a,b) \left(a,b\right) , then no one can place a pudding on (c,d) \left(c,d\right) where c≡a(modn),d≡b(modn) c \equiv a \pmod{n}, d \equiv b \pmod{n} .
1. Who has a winning strategy when n=2018 n = 2018
1. Who has a winning strategy when n=2019 n = 2019
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