# Tetrahedron car crash prevention

1. Feb 11, 2010

### djuiceholder

1. The problem statement, all variables and given/known data

Imagine a planet in the shape of a regular tetrahedron (its surface consists of 4 equilateral triangles). Suppose that on each face there is a car traveling at a constant speed in clockwise direction along the edges bounding the face. Can they travel without crashing?

2. Relevant equations

I dont know how and where to start working on this problem

3. The attempt at a solution

I really need help :(

2. Feb 12, 2010

### LCKurtz

Have a look at this picture:

Suppose, for example, the red car is on edge 3. Convince yourself that neither the yellow or green cars can be on the "opposite" edge 5 to prevent future collisions. The same idea works for each edge and its opposite. Does that give you any ideas?

Last edited: Feb 13, 2010
3. Feb 14, 2010

### LCKurtz

Well, the OP seems to have disappeared, but I still think it is an interesting question. I think my last post leads to a proof that 4 cars can't do it. Even assuming there are traffic circles at the vertices, they will have a head-on collision along some edge.

But what about three cars? Without traffic circles at the vertices, so they can't arrive at a vertex at the same time, I don't think they can do it either, but I don't have a proof. If you assume they have traffic circles, then three cars can do it precisely by arriving at the vertices at the same time. Killing a bit of time with Maple, I made the following gif to illustrate it:

http://math.asu.edu/~kurtz/pix/cars.gif [Broken]

You have to imagine the traffic circles.

Last edited by a moderator: May 4, 2017
4. Feb 18, 2010

### djuiceholder

wow, thanks a lot. So, that means they can not travel without crashing. Even, 3 cars can't.

5. Feb 18, 2010

### LCKurtz

I haven't proven that three cars can't do it without traffic circles, but I suspect they can't. The attached example shows an effort where they don't arrive at the intersections simultaneously. Unfortunately, there is an accident:

http://math.asu.edu/~kurtz/pix/cars2.gif [Broken]

Last edited by a moderator: May 4, 2017