BMO Round 1, 2021

Combinatorics Difficulty 6.0 National Olympiad Find the answer United Kingdom

Evie and Odette are playing a game. Three pebbles are placed on the number line; one at $-2020$ , one at $2020$ , and one at $n$ , where $n$ is an integer between $-2020$ and $2020$ . They take it in turns moving either the leftmost or the rightmost pebble to an integer between the other two pebbles. The game ends when the pebbles occupy three consecutive integers.
Odette wins if their sum is odd; Evie wins if their sum is even. For how many values of $n$ can Evie guarantee victory if:
(a) Odette goes first;
(b) Evie goes first?

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