Question

一个每边有 $31$ 个方格的棋盘按黑白相间染色，并且四个角都是黑色。黑色方格有多少个？

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?

$480$

$481$

$482$

$483$

$484$

答案：B

知识点：基本计数奇偶性找规律

难度评级：1020

解答：

共有 $15$ 行各含 $15$ 个黑格，另有 $16$ 行各含 $16$ 个黑格。

第一行有 $16$ 个黑格，其余各行按 $15$ 个与 $16$ 个交替。

因此共有 $16$ 行各含 $16$ 个黑格，其余各行含 $15$ 个黑格。

因此黑格总数为 $15^2 + 16^2 = 481.$

所以正确答案是 B。

Note that there are $15$ rows with $15$ black tiles and $16$ rows with $16$ black tiles.

This can be seen by observing that the first row has $16$ black tiles, and all the other rows alternate with $15$ and $16$ tiles.

Then, due to the alternating pattern, there will be a total of $16$ rows with $16$ tiles, and the other rows have $15$ tiles.

The total number of black squares is then $15^2 + 16^2 = 481.$

Thus, B is the correct answer.

2012 AMC 10A 第 14 题

答案：B

难度评级：1020

解答：

其他年份的第 14 题