AMC8 2025

AMC8 2025 · Q11

AMC8 2025 · Q11. It mainly tests Casework, Geometry misc.

A $\textit{tetromino}$ consists of four squares connected along their edges. There are five possible tetromino shapes, $I$, $O$, $L$, $T$, and $S$, shown below, which can be rotated or flipped over. Three tetrominoes are used to completely cover a $3\times4$ rectangle. At least one of the tiles is an $S$ tile. What are the other two tiles?

一种\textit{四格子}由四个正方形沿边连接而成。有五种可能的四格子形状，$I$、$O$、$L$、$T$和$S$，如下所示，可以旋转或翻转。使用三个四格子完全覆盖一个$3\times4$矩形。至少有一个是$S$格子。另外两个格子是什么？

(A) I$ and $L I$ and $L

(B) I$ and $T I$ and $T

(D) L$ and $S L$ and $S

(E) O$ and $T O$ and $T

Answer

Correct choice: (C)

正确答案：（C）

Solution

The $3\times4$ rectangle allows for $7$ possible places to put the S piece, with each possible placement having an inverted version. One of the cases looks like this: As you can see, there is a hole in the top left corner of the board, which would be impossible to fill using the tetrominos. There are three cases in which a hole isn't created; the S lies flat in the bottom left corner, it lies flat in the top right corner, or it stands upright in the center. All three tilings are shown below. For each of the inverted cases, the L pieces can be inverted along with the S piece. Because the only cases that fill the rectangle after the S is placed are the ones that use two L pieces, the answer must be $\boxed{\textbf{(C)}~I \ and \ L}$.

$3\times4$矩形有$7$个放置S块的位置，每个位置都有反转版本。其中一个情况如下：如你所见，棋盘左上角有一个洞，用四格子无法填充。有三种情况不会产生洞：S平放在左下角，平放在右上角，或竖直立在中心。所有三种铺填如下所示。对于每个反转情况，L块可以与S块一起反转。因为放置S后唯一能填充矩形的情况是使用两个L块，所以答案是$\boxed{\textbf{(C)}~I \ and \ L}$。

Topics