← All problem sets

Invariants

Fall 2026 · 10 problems

Find the quantity that never changes, then use it to show something is forced or outright impossible: a chocolate bar whose game is decided before anyone moves, chameleons that can never agree on a colour, a knight that cannot cross the board, and the fifteen puzzle.

Download the problem set (PDF)

What is in it

Problems only — the club works these out together at meetings. Topic and attribution are listed as a starting point.

1. Breaking the Bar InvariantsEasy
2. The Chameleons InvariantsEasy
3. A Bean in the Can InvariantsEasy
4. The Blackboard InvariantsMedium
5. A Knight from Corner to Corner ColoringMedium
6. Coins in a Line Invariants / StrategyMedium
7. He Loves Me, He Loves Me Not Symmetry / StrategyMedium
8. The Fifteen Puzzle Parity of PermutationsHard
9. Red Squares, Black Squares Coloring / InvariantsMediumPutnam 2001 B1
10. Pegs in a Row Invariants / StrategyHardPutnam 2020 B2