### 9.08.2005

## Procrastination

I just spent the last hour working on this brainteaser:

My solution is below; highlight over it to read it.

Number the guards 1 to 12. Let's march with guard 1 around the castle, but everytime there's a collision, let's switch to following the other guard (so that we maintain the same direction). After one hour, we've walked around the castle once, but we may be following a different guard. So after one hour, there's a guard at each tower, and that guard is walking the same direction as the guard who was stationed there one hour ago.

Now let's consider the actual path that each guard walks. Since each guard walks at the same pace, and since guards reverse direction every time there's a collision, no guard passes another. Hence the ordering of the guards remains constant. So after 1 hour, the guards need not be at their original tower, but they will all have advanced the same number of towers. Furthermore, the number of towers advanced is the same each hour. Therefore, after 12 hours, each guard will return to his original tower.

Note that this puzzle works for all integer values of 12.

There is an irregularly shaped castle wall with 12 irregularly spaced guard towers around the perimeter. The towers may be evenly spaced; they may all be clustered together; they may be somewhere in between those two options. There is a guard at each tower. Each guard patrols the castle perimeter, walking at a pace that allows him to make a complete loop around the perimeter in exactly one hour. (Thus, all 12 guards walk at the same pace as each other.) At noon, each guard starts at his own station and begins to walk either clockwise or counterclockwise (to be determined randomly). Whenever two guards meet each other, they immediately each turn around and start walking back in the direction from which they came. Their turnaround is immediate and they lose no time in switching directions.

Prove that at midnight each guard is back at his original tower.

My solution is below; highlight over it to read it.

Number the guards 1 to 12. Let's march with guard 1 around the castle, but everytime there's a collision, let's switch to following the other guard (so that we maintain the same direction). After one hour, we've walked around the castle once, but we may be following a different guard. So after one hour, there's a guard at each tower, and that guard is walking the same direction as the guard who was stationed there one hour ago.

Now let's consider the actual path that each guard walks. Since each guard walks at the same pace, and since guards reverse direction every time there's a collision, no guard passes another. Hence the ordering of the guards remains constant. So after 1 hour, the guards need not be at their original tower, but they will all have advanced the same number of towers. Furthermore, the number of towers advanced is the same each hour. Therefore, after 12 hours, each guard will return to his original tower.

Note that this puzzle works for all integer values of 12.