HMMT 二月 2026 · 冲刺赛 · 第 7 题

HMMT February 2026 — Guts Round — Problem 7

[6] A tromino is any connected figure constructed by joining 3 unit squares edge-to-edge. Compute the number of ways to tile a 2 × 6 rectangular grid with 4 nonoverlapping trominoes. (Two tilings that differ by a rotation or reflection are considered distinct.)

解析

[6] A tromino is any connected figure constructed by joining 3 unit squares edge-to-edge. Compute the number of ways to tile a 2 × 6 rectangular grid with 4 nonoverlapping trominoes. (Two tilings that differ by a rotation or reflection are considered distinct.) Proposed by: Sebastian Attlan Answer: 11 Solution: All trominos are either three squares joined in a line or joined in an L shape. Consider the four edges in the middle with no endpoints on the boundary of the grid. We can now perform casework on the intersections of trominos with these edges. • If none of these edges are crossed by a tromino, then the trominos must form a chain tracing the boundary of the grid. There are 3 such tilings depending on where the top-leftmost tromino starts and ends. • Suppose that at least one of these four middle edges are crossed by a tromino. (Assume that the long edges of the grid are horizontal.) If the edge in the 2 nd or 5 th column is crossed, then it leaves an isolated region of area 1 or 2 , which can’t be covered by a tromino. If the edge in the 3 rd or 4 th column is crossed, then we must be able to split the tiling into tilings of two 2 × 3 subgrids, each of which is covered by a chain tracing the boundary of the subgrid. There are 3 possibilities for each, giving us 3 · 3 − 1 = 8 possible tilings since one of those tilings overlaps with the first case above. Then we have 3 + 8 = 11 tilings in total. Here is the diagram that lists all 11 tilings. ©2026 HMMT