I'm puzzled by the following riddle, which seems easy at first, but turns out to be more complex than it looks. I would like to go to the bottom of it, and could not find references online.
You have 9 horses racing on a circular track. they start at the same point, and each of them has a constant speed, but all their speeds are different.
They are only allowed to pass each other at the point of the circle marking their common starting point, otherwise they collide.
Can you choose their speeds so that they can run forever without colliding ?
What I did
After some time I understood that the problem is equivalent to the following: find 9 integers so that the ratio of any two of them can be written $k/(k+1)$ for some integer $k$ (that can depend on the chosen pair).
For instance with 4 horses you can take 6 8 9 12, as 6/8=3/4, 8/12=2/3, and so on.
A solution with 6 horses: 210 216 220 224 225 240.
This constraint can be translated into a condition on the successive differences of the numbers, and we ask that a certain system of modular equations computed from these differences has a solution. The solution is the speed of the slowest horse, and the differences give you the other ones.
Having this, I did a computer bruteforce search on the differences between the integer speeds (with some tricks to fast it up), and managed to get up to 12 horses. Unfortunately I cannot see any structure emerging, which makes me think I missed the point, and another approach could be more explanatory. I'm still not able to explain how to find a solution with 9 horses without the help of a computer.
Is there a solution for any number of horses ?