The standard tool for "can you reach state X?" puzzles - and the answer is usually no, provable by finding a quantity that never changes.
Common invariants: parity (odd or even), a sum modulo some number, or a colouring argument.
The classic: a chessboard with two opposite corners removed cannot be tiled by dominoes. Each domino covers one black and one white square, but the removed corners share a colour, so the counts differ by two. No case analysis required.
Finding the invariant is the entire difficulty; verifying it is trivial.