Solution: Beetles

We publish the solution to the Beetles divertimento.

This problem was proposed by Jean-Paul Delahaye, based on an earlier problem popularised by Peter Winkler.

The fun

Nine beetles are placed in a circular loop whose length is 100 m, so that the distances in metres between the beetles are distinct prime numbers. At the initial moment, each beetle starts walking clockwise or counterclockwise, at random, at a speed of 1 metre per minute. When two beetles collide, they change direction.

What are the distances between the beetles when 50 minutes have passed?

The solution

Solution submitted by Alberto Castaño.

The answer is that the distances will be the same (which are the nine primes less than 25), but the beetles could be cyclically rearranged.

Firstly, since the beetles bounce off each other perfectly when they meet, we could imagine that they pass each other, and that each one goes its own way. Therefore, after 50 metres, each one would arrive at the opposite point from where it started. Now, beetles cannot pass each other, but the above reasoning shows us that there will always be a beetle that arrives at the point opposite to the starting point of any other beetle. And since they do not pass each other, each one will always find itself between the same two as at the beginning of its “long march”, hence the greatest possible disorder between them is nothing more than a cycle of maximum length.

As for concrete distances, the only way to add up to no more than 100 with nine different primes is to take precisely the first nine primes, 2, 3, 5, 7, 11, 13, 17, 19 and 23.