2000 AMC 12 Problem 8

Problem 8 of 25EasierNumber Theory

Figures

0,

1,

2,

and

3

consist of

1,

5,

13,

and

25

nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure

100?

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Solution

Figure

n

is a diamond whose row lengths increase through the odd numbers and back down, giving a total of

n^2 + (n + 1)^2

unit squares. This matches

1, 5, 13, 25

for

n = 0, 1, 2, 3.

Therefore figure

100

has

\begin{aligned} 100^2 + 101^2 &= 10000 + 10201 \\ &= 20201 \end{aligned}

unit squares. Thus, the correct answer is C.