Invariants, Monovariants, and Coloring

Identify invariant properties to prove the impossibility of reaching specific states in grid and board problems.

Construct monovariants to prove that certain iterative combinatorial processes must eventually terminate.

Analyze grid-based puzzles by applying checkerboard (2-coloring) patterns.

Apply non-standard coloring patterns (e.g., using multiple colors or modular arithmetic) to solve Olympiad-style problems.