The 5 × 5 grid shown contains a collection of squares with sizes from 1 × 1 to 5 × 5. How many of these squares contain the shaded center square?

2004 AMC 10A Problem 16 — Solution

2004 AMC 10A Problem 16

Below is the professionally curated solution for Problem 16 of the 2004 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2004 AMC 10A solutions, or check the answer key.

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Concepts:counting shapes in figures casework

Difficulty rating: 1480

16.

The $5 \times 5$ grid shown contains a collection of squares with sizes from $1 \times 1$ to $5 \times 5.$ How many of these squares contain the shaded center square?

$12$

$15$

$17$

$19$

$20$

Solution:

Every $5\times5,$ $4\times4,$ and $3\times3$ square contains the center cell, and there are $1^2 + 2^2 + 3^2 = 14$ of them.

Among the smaller squares, $4$ of the $2\times2$ squares and $1$ of the $1\times1$ squares cover the center, giving $14 + 4 + 1 = 19.$

Thus, the correct answer is D.

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