尺子上的蚂蚁：多久能确定全都走下去？
how long must you wait to be sure that all the ants have walked off
题目详情
把 10000 只蚂蚁随机放在一把 1 米（100 厘米）长的尺子上，尺子沿南北方向放置。每只蚂蚁以 1 厘米/秒的速度沿尺子长边方向直线行走，初始方向随机为向北或向南。
蚂蚁只有周边视野，因此可能迎面相撞。若两只蚂蚁迎面相撞，则会立刻同时掉头并保持原速度返回。单只蚂蚁在走下尺子前可能经历多次碰撞。
问：你至少要等多久，才能保证所有蚂蚁都已经走下尺子？
I have dropped 10,000 ants randomly onto a ruler that is one meter (i.e., 100 centimeters) long and oriented to point north- south. The ants are of very small size and mass. Each ant walks at a steady pace of one centimeter per second in a straight line parallel to the long edge of the ruler. Their initial direction is randomly either north or south. The ants are all from the same colony and possess an inherited vision problem: they have peripheral vision only. This means that they can collide with each other if they meet head on (although very small, they are large enough to collide). If two ants do collide head on, however, then they each turn around instantly and head back the way they came at their steady pace. With so many ants in one small space, a single ant may experience multiple collisions before it walks off of the ruler. So,
how long must you wait to be sure that all the ants have walked off of the ruler?
解析
把蚂蚁看作完全相同。
两只蚂蚁迎面相撞并同时掉头，与它们“互相穿过继续前进”在宏观上等价：因为相撞后两只蚂蚁交换了速度方向，而蚂蚁不可区分，所以系统中“某个位置上有一只朝北走的蚂蚁”的演化与穿过模型一致。
因此，只需把每只蚂蚁当作不发生碰撞、以初始方向匀速前进即可。
最晚走下尺子的情况是：某只蚂蚁位于一端点附近但初始方向指向尺子内部，需要走完整条 100 厘米才离开；速度 1 厘米/秒，所以最坏时间为