2000 AMC 12 Problem 8

Below is the professionally curated solution for Problem 8 of the 2000 AMC 12, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2000 AMC 12 solutions, or check the answer key.

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Concepts:pattern recognition perfect square

Difficulty rating: 1370

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 $100?$

$10401$

$19801$

$20201$

$39801$

$40801$

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 $100^2 + 101^2 = 10000 + 10201 = 20201$ unit squares.

Thus, the correct answer is C.

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