Problem

Source: Indonesia National Math Olympiad 2021 Problem 8 (INAMO 2021/8)

Tags: combinatorics, grid, board, trimino



On a $100 \times 100$ chessboard, the plan is to place several $1 \times 3$ boards and $3 \times 1$ board, so that Each tile of the initial chessboard is covered by at most one small board. The boards cover the entire chessboard tile, except for one tile. The sides of the board are placed parallel to the chessboard. Suppose that to carry out the instructions above, it takes $H$ number of $1 \times 3$ boards and $V$ number of $3 \times 1$ boards. Determine all possible pairs of $(H,V)$. Proposed by Muhammad Afifurrahman, Indonesia