AMC10 2012 A

AMC10 2012 A · Q14

AMC10 2012 A · Q14. It mainly tests Basic counting (rules of product/sum), Area & perimeter.

Chubby makes nonstandard checkerboards that have 31 squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard?

Chubby 制作非标准棋盘，每边有 31 个方格。棋盘四个角都是黑方格，每行每列红黑方格交替。这样的棋盘上有多少黑方格？

(A) 480 480

(B) 481 481

(D) 483 483

(E) 484 484

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): Separate the modified checkerboard into two parts: the first 30 columns and the last column. The larger section consists of 31 rows, each containing 15 black squares. The last column contains 16 black squares. Thus the total number of black squares is \(31\cdot 15 + 16 = 481\). OR There are 16 rows that have 16 black squares and 15 rows that have 15 black squares, so the total number of black squares is \(16^2 + 15^2 = 481\).

答案（B）：将修改后的棋盘分成两部分：前 30 列和最后一列。较大的部分由 31 行组成，每行有 15 个黑格。最后一列有 16 个黑格。因此黑格总数为 \(31\cdot 15 + 16 = 481\)。或者有 16 行每行有 16 个黑格，另有 15 行每行有 15 个黑格，所以黑格总数为 \(16^2 + 15^2 = 481\)。

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