SMO Open · 12. Combinatorics: games, invariants and the pigeonhole principle
Pigeonhole boxes and averaging, colourings, invariants and games, and extremal proofs made of a bound plus a construction.
Choose pigeonhole boxes and use averaging over a table of successes.
Use colourings to prove that a tiling or a configuration is impossible.
Find invariants for repeated moves and solve games with strategies for both players.
Write extremal answers as a bound plus an example, using the equality case.
Write complete Round 2 proofs for these ideas.