AMC 10 Daily Practice Round 1

Complete problem set with solutions and individual problem pages

Problem 17 Hard

As shown in the diagram ①, a floor is tiled using a patterned tile. If the tiles are arranged to form a 2 \times 2 square (as shown in Diagram ②), there are a total of 5 complete circles. If the tiles are arranged to form a 3 \times 3 square (as shown in Diagram ③), there are a total of 13 complete circles. If the tiles are arranged to form a 4 \times 4 square (as shown in Diagram ④), there are a total of 25 complete circles.

If the tiles are arranged to form a 15 \times 15 square, how many complete circles are there?

  • A.

    365

  • B.

    366

  • C.

    421

  • D.

    425

  • E.

    440

Answer:C

As shown in the diagram, the size of the square pattern and the number of circles can be summarized in the following table:

\begin{array}{c|c} \text{Square Size} & \text{Number of Circles} \\ \hline 1 \times 1 & 1 \\ 2 \times 2 & 5 = 1 + 4 = 1^2 + 2^2 \\ 3 \times 3 & 13 = 4 + 9 = 2^2 + 3^2 \\ 4 \times 4 & 25 = 9 + 16 = 3^2 + 4^2 \\ \end{array}

From this pattern, we observe that for an n \times n square, the number of circles is given by (n-1)^2 + n^2. Thus, when n = 15, the number of circles is 14^2 + 15^2 = 421. Therefore, the answer is \boxed{421}.

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