AMC8 2009

AMC8 2009 · Q18

AMC8 2009 · Q18. It mainly tests Area & perimeter, Patterns & sequences (misc).

The diagram represents a 7-foot-by-7-foot floor that is tiled with 1-square-foot black tiles and white tiles. Notice that the corners have white tiles. If a 15-foot-by-15-foot floor is to be tiled in the same manner, how many white tiles will be needed?

图示是一个7英尺×7英尺的地板，用1平方英尺的黑白瓷砖铺成。注意角落处是白瓷砖。如果要以相同方式铺一个15英尺×15英尺的地板，需要多少白瓷砖？

(A) 49 49

(B) 57 57

(D) 96 96

(E) 126 126

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): To maintain the pattern, white squares will always occupy the corners, and every edge of the square pattern will have an odd number of tiles. Create a table, starting with a white square in the corner of the pattern, and increase the sides by 2 tiles. Following the pattern, an $11 \times 11$ area has 36 squares, a $13 \times 13$ area has 49, and a $15 \times 15$ has 64.

答案（C）：为了保持该图案，白色方块总是占据四个角，并且该正方形图案的每条边都会有奇数块瓷砖。建立一个表格，从图案角上的一个白色方块开始，并且每次将边长增加 2 块瓷砖。按照该规律，$11 \times 11$ 的区域有 36 个方块，$13 \times 13$ 的区域有 49 个，$15 \times 15$ 的区域有 64 个。

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