BALKAN 2026 Problem 2

Let $n$ be a positive integer. A $2n \times 2n$ board is tiled with $2 \times 1$ and $1 \times 2$ dominoes. To pivot a domino is to select one of its two unit squares and rotate the entire domino by $90^\circ$ clockwise, $90^\circ$ anticlockwise, or $180^\circ$ about the centre of that unit square. Prove that it is always possible to simultaneously pivot every domino such that, after all pivots have been performed, the dominoes still tile the board.