2003 AMC 12A 第 13 题

Question

图中实线围成的多边形由

4

个全等正方形边对边连接而成。再把一个全等正方形接到图中标出的九个位置之一的某条边上。九种所得多边形中，有多少种可以折成一个缺少一个面的立方体？

The polygon enclosed by the solid lines in the figure consists of

4

congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing?

2

3

4

5

6

答案：E

知识点：展开图（立体几何）正方体

难度评级：1530

小提示：

缺少一个面的立方体需要

5

个正方形。

A cube with one face missing needs

5

squares

大提示：

先把四个正方形的部分围着立方体折起，再检查附加正方形是否落在仍未覆盖的面上。

Fold the four-square piece partway around a cube, then check which attached square lands on a still-open face

解答：

缺少一个面的立方体有

5

个面，所以第五个正方形必须折到四正方形拼块尚未占据的面上。追踪每个连接位置折到的面可知，位置

1

，

2

，和

3

会折到已占据的面并与之重叠。位置

4

，

5

，

6

，

7

，

8

，和

9

都会折到未占据的面。因此九个位置中有

6

个可行。

所以正确答案是 E。

A cube missing one face has

5

faces, so the fifth square must fold onto a face not already occupied by the four-square piece. Tracing the face reached by each attachment, positions

1,

2,

and

3

fold onto a face already occupied and overlap it. Each of positions

4,

5,

6,

7,

8,

and

9

folds onto an unoccupied face. Therefore

6

of the nine positions work.

Thus, the correct answer is E.

2003 AMC 12A 第 13 题

答案：E

难度评级：1530

小提示：

大提示：

解答：

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