Question

在一个 $3 \times 3$ 方格中，每个小方格最多放一个 X。若要求竖直、水平或对角线上都不能出现三个 X 连成一排，最多可以放多少个 X？

Placing no more than one X in each small square, what is the greatest number of X's that can be put on a $3 \times 3$ grid without getting three X's in a row vertically, horizontally, or diagonally?

$2$

$3$

$4$

$5$

$6$

答案：E

知识点：极端原理抽屉原理

难度评级：1120

解答：

放 $6$ 个 X 是可以做到的：把一条主对角线上的三个格子留空，填满其余六格。这样每一行和每一列都少一个格子，使用到的那条对角线有空格，另一条对角线经过空的中心，所以没有一条三个连成线。

七个 X 不可能：那时只有两个空格，而两个空格最多只能落在三行中的两行，迫使剩下的一行被三个 X 完全填满。

所以最大数是 $6$ 。

所以正确答案是 E。

A placement of $6$ X's works: leave the three squares along one main diagonal empty and fill the other six. Then each row and each column is missing one square, the used diagonal has an empty square, and the other diagonal passes through the empty center, so no line of three is complete.

Seven X's is impossible: only two squares would be empty, and two empty squares can lie in at most two of the three rows, forcing the remaining row to be completely filled with three X's.

So the greatest number is $6.$

Thus, the correct answer is E .

1988 AMC 8 第 16 题

答案：E

难度评级：1120

解答：

其他年份的第 16 题