Question

一名无聊的学生沿着一条走廊行走，走廊里有一排编号为 $1$ 到 $1024$ 的关闭储物柜。他打开 $1$ 号柜，此后在仍关闭的储物柜中交替跳过一个、打开一个。走到走廊尽头后，他转身往回走。他打开遇到的第一个关闭储物柜，此后又在仍关闭的储物柜中交替跳过一个、打开一个。学生按这种方式来回行走，直到所有储物柜都打开。最后打开的储物柜编号是多少？

A bored student walks down a hall that contains a row of closed lockers, numbered $1$ to $1024.$ He opens locker $1,$ and then alternates between skipping and opening each closed locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens the first closed locker he encounters, and then alternates between skipping and opening each closed locker thereafter. The student continues wandering back and forth in this manner until every locker is open. What is the number of the last locker he opens?

答案

答案：342

知识点：过程模拟找规律等差数列

难度评级：2270

小提示：

每次走完后，仍关闭的储物柜编号构成一个等差数列

After each trip, the still-closed lockers form an arithmetic sequence

大提示：

每次走完后，只记录首项、公差和项数

Record only the first term, common difference, and number of terms after each trip

解答：

每次行走时，学生都会按照当前行走方向，在剩余储物柜中打开第一个、第三个、第五个，依此类推。追踪每次走完后仍关闭的等差数列，得到：

次数	首项	公差	项数
$1$	$2$	$2$	$512$
$2$	$2$	$4$	$256$
$3$	$6$	$8$	$128$
$4$	$6$	$16$	$64$
$5$	$22$	$32$	$32$
$6$	$22$	$64$	$16$
$7$	$86$	$128$	$8$
$8$	$86$	$256$	$4$
$9$	$342$	$512$	$2$

因此第九次走完后，只剩下 $342$ 号柜和 $854$ 号柜仍关闭。第十次从右侧出发时，先打开 $854$ 号柜，而 $342$ 号柜保留下来。所以最后打开的是 $342$ 号柜。

On every trip the student opens the first, third, fifth, and so on among the remaining lockers in his direction of travel. Tracking the closed arithmetic sequence after each trip gives:

trip	first	difference	count
$1$	$2$	$2$	$512$
$2$	$2$	$4$	$256$
$3$	$6$	$8$	$128$
$4$	$6$	$16$	$64$
$5$	$22$	$32$	$32$
$6$	$22$	$64$	$16$
$7$	$86$	$128$	$8$
$8$	$86$	$256$	$4$
$9$	$342$	$512$	$2$

Thus only lockers $342$ and $854$ remain after the ninth trip. On the tenth trip, starting from the right, locker $854$ is opened and $342$ remains. Therefore the last locker opened is $342.$

1996 AIME 第 9 题

答案：342

难度评级：2270

小提示：

大提示：

解答：

其他年份的第 9 题