AMC 8 2009 Problem 18

AMC8 2009 年第 18 题

数列与规律★★★☆☆

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?

该图表示一块 7 英尺乘 7 英尺的地板，该地板由 1 平方英尺的黑色瓷砖和白色瓷砖铺成。请注意，四个角都是白色瓷砖。如果某个 15 英尺乘 15 英尺的地板要以同样的方式铺成，需要多少块白色瓷砖？

A.49
B.57
C.64
D.96
E.126

正确答案 · Correct Answer

解析 · Solution

Solution 1

In a 1 -foot-by- 1 -foot floor, there is 1 white tile. In a 3 -by- 3, there are 4. Continuing on, you can deduce the

n^{th}

positive odd integer floor has

n^2

white tiles. 15 is the

8^{th}

odd integer, so there are 64 white tiles.

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