Question

一个长方形板有八列小方格，从左上角开始编号，第一行从左到右编号为一到八，第二行为九到十六，依此类推。一个学生涂色方格一，然后跳过一个方格涂色方格三，跳过两个方格涂色方格六，跳过三个方格涂色方格十，并继续这样做，直到每一列至少有一个涂色方格。

第一次达到这个结果时，被涂色方格的编号是多少？

A rectangular board of 8 columns has squares numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 8, row two is 9 through 16, and so on. A student shades square 1, then skips one square and shades square 3, skips two squares and shades square 6, skips 3 squares and shades square 10, and continues in this way until there is at least one shaded square in each column.

What is the number of the shaded square that first achieves this result?

$36$

$64$

$78$

$91$

$120$

答案：E

知识点：三角形数模运算

难度评级：1630

解答：

被涂色的方格编号是三角形数 $1,3,6,10,15,\ldots$ 。列对应模 $8$ 的余数，其中余数 $0$ 表示第八列。

到 $105$ 为止的三角形数模八的余数为 $1,3,6,2,7,5,4,4,5,7,2,6,3,1$ ，还没有出现第八列的余数零。

下一个三角形数是 $120$ ，且 $120\equiv0\pmod8$ 。这是第一次每一列都有涂色方格。

所以正确答案是 E。

The shaded squares are the triangular numbers $1,3,6,10,15,\ldots$ . Columns correspond to residues modulo $8$ , with residue $0$ representing the eighth column.

The triangular numbers through $105$ have residues $1,3,6,2,7,5,4,4,5,7,2,6,3,1$ , so the eighth-column residue has not appeared yet.

The next triangular number is $120$ , and $120\equiv0\pmod8$ . This is the first time every column has a shaded square.

Thus, the correct answer is E .

1998 AMC 8 第 24 题

答案：E

难度评级：1630

解答：

其他年份的第 24 题