Back to the robots problem

Here is the problem from my previous post.
Four robots are initially at the vertices of a unit square.  They start moving simultaneously with the same speed, each one keeping its direction to the right closest neighbor.  Where do they meet and what distance will each one cover to the meeting point?
The answer to the first question is obvious due to the symmetry of the problem - they meet at the center of the square.  In order to answer the second question we can introduce a rotating frame of reference. 

The origin is where the robot A is at the moment.  The axis x is always directed to the robot D, and the axis y is always directed to the robot B.  Due to the symmetry of the problem the robots will always be at the vertices of the shrinking square up to the moment they meet at the origin.  If we look at the motion of the robot B, it moves along the axis y from the point (0, 1) to the origin.  So it moves exactly one unit to the meeting point.  The same is true about distances for other robots due to the symmetry as mentioned before.

I have known this problem since, probably, age 14, but today I did a Google search and found a lot of sites talking about it.  So I was planning to discuss the same situation for different regular polygons but decided just to put a link to the site with the short and meaningful discussion.
http://www.cut-the-knot.org/Curriculum/Geometry/FourTurtles.shtml

On a different note, I asked my algebra students to solve the following problem.  

Two cars start moving towards each other from A and B respectively with speeds 60 mph and 40 mph. The
distance AB is 750 mi. What will be the distance between these cars 1 hour before they meet?

It took them a few minutes to realize how simple it is, and at the end the strongest students were a bit upset at themselves.

I continue my list of favorite problems

I remember this problem amazed me.  I was surprised by its beautiful solution that I will show you later not to spoil some fun to those of you who wants to try to solve it.

Four robots are initially at the vertices of a unit square.  They start moving simultaneously with the same speed, each one keeping its direction to the right closest neighbor.  Where do they meet and what distance will each one cover to the meeting point?

Another favorite problem

One more from the list of my favorites.

Prove that there are 100 consecutive composite (nonprime) natural numbers.

I like it because everyone can understand and solve it, but the idea of the proof is relatively deep.

My favorite problem

I should have started with my favorite problem.  I gave it to different audiences, ages 10 to whatever.  Usual results are always the same: two or three people do it correct.  The problem is many times older than I am.  Here it is (more modern variation).

The distance between A and B is 100 km.  Two bicyclists start moving simultaneously towards each other from A and B respectively with speeds 10 km/h and 15 km/h.  A dog starts running from A at the same moment as bicyclists do with the speed 20 km/h.  When it reaches the second bicyclist, it turns back, runs until meeting the first one, turns again etc.  The dog runs back and forth until bicyclists meet.  What distance does it run? 

As I mentioned above, the number of fifth graders who solved this problem is approximately the same as the number of high school seniors.  BTW the results were the same in USA, Russia, and Ukraine :-)