[bonanova]

Given that ...

1. Twenty-five Robots, all named after BrainDenizens, are placed at random on a rail 1 mile long.
   
2. The robot named Prime begins as the thirteenth robot from the North end of the rail at its midpoint.
   
3. Each robot faces North or South with equal probability, and travels at 1 mile/hour in the direction it faces.
   
4. When two robots meet or reach the end of the rail, they reverse direction.
   

On average, what is the net distance Prime has traveled after one hour?

[bonanova]

In that case, it will always be exactly zero.

A distant observer will not be able to tell whether the robots turn about upon meeting each other, or simply continue on their original course. It will therefore appear as though the only thing that causes them to change direction is reaching an end of the rail. Thus, after one hour, it will appear as though the robot that started at a position x from the north end of the rail will end up at a position x from the south end, regardless of which direction it was initially traveling. In other words, the starting position will be reflected about x=0.5 (but with different robots), and because a robot began at x=0.5, there will be a robot at that same position after 1 hour. Because no positions were actually exchanged, that robot must be Prime.

Note that after two hours, all robots will have returned to their starting positions, regardless of the initial distributions of position and direction.

d3k3 has it.

Nice job.

asking "on average" how far from the center Prime might end up.

Octopuppy's analysis of the previous robot puzzle [using flags] works well here: It's basically the same reasoning d3k3 uses.

Suppose the robots carry flags and exchange them when they meet.

All the flags travel one mile, bouncing off an end, to finish the same distance from the North end as they were initially from the South.

In other words, the final pattern of flags is the mirror image of the initial pattern.

Prime's flag is the only flag that ends up where it started.

It's still the 13th flag, [mirror image doesn't change that] and Prime remains the 13th robot [order of robots never changes.]

So Prime ends up with his initial flag, exactly where he started.

The maximum distance he can be from his starting position is therefore zero.

[Prime]

d3k3 has it.

Nice job.

asking "on average" how far from the center Prime might end up.

Octopuppy's analysis of the previous robot puzzle [using flags] works well here: It's basically the same reasoning d3k3 uses.

Suppose the robots carry flags and exchange them when they meet.

All the flags travel one mile, bouncing off an end, to finish the same distance from the North end as they were initially from the South.

In other words, the final pattern of flags is the mirror image of the initial pattern.

Prime's flag is the only flag that ends up where it started.

It's still the 13th flag, [mirror image doesn't change that] and Prime remains the 13th robot [order of robots never changes.]

So Prime ends up with his initial flag, exactly where he started.

The maximum distance he can be from his starting position is therefore zero.

So zero it is. On to openning the ends of the rail and calculating average distance (time) before the robot falls off?

[bonanova]

So zero it is. On to openning the ends of the rail and calculating average distance (time) before the robot falls off?

Sure!

Have at it ...

[Prime]

Sure!

Have at it ...

Do you want to make it a new topic? I mean, you are the original designer of those robots. And you can formulate a problem so much better.

On a related note. We do need an "unsolved problems" category on this forum.

[Guest]

But you said they reverse direction, and so wouldn't fall off. So they would travel until their battery died?

Given that ...

1. Twenty-five Robots, all named after BrainDenizens, are placed at random on a rail 1 mile long.
   
2. The robot named Prime begins as the thirteenth robot from the North end of the rail at its midpoint.
   
3. Each robot faces North or South with equal probability, and travels at 1 mile/hour in the direction it faces.
   
4. When two robots meet or reach the end of the rail, they reverse direction.
   

On average, what is the net distance Prime has traveled after one hour?