2024 AMC 8 第 7 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
7.
一个 的矩形用下方 种瓷砖无重叠覆盖：、、。最少可能使用多少块 瓷砖？
A rectangle is covered without overlap by shapes of tiles: and shown below. What is the minimum possible number of tiles used?
小提示：
非单位瓷砖覆盖的面积必须是 的倍数
The area left for non-unit tiles must be a multiple of
大提示：
若只用一块单位瓷砖，则有两行需要由大瓷砖覆盖奇数个格子
If only one unit tile were used, two rows would need an odd number of cells covered by larger tiles
视频讲解：
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文字解答：
和 瓷砖面积都是 。矩形面积为 ，所以 瓷砖数必须同余于 。选项中只可能是 或 。
不能只用一块 瓷砖。否则大瓷砖覆盖其余 个格子，会有两行的 个格子全由大瓷砖覆盖。但每块 瓷砖在某行覆盖 格，每块 瓷砖在某行覆盖 格，所以一行由大瓷砖覆盖的格数应为偶数，不可能是 。
下图说明 块单位瓷砖可以做到。
正确答案是 E。
The and tiles each have area . Since the rectangle has area , the number of tiles must be congruent to . Among the choices, only and are possible by area.
It is impossible to use just one tile. If the larger tiles covered the other cells, then two rows would have all cells covered by larger tiles. But each tile covers cells in any row it meets, and each tile covers cells in one row, so each row would have an even number of cells covered by larger tiles. A row cannot have such cells.
The following tiling shows that unit tiles are possible.
Thus, E is the correct answer.
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