Written: Dec. 31, 2017

The great French mathematical columnist, Jean-Paul Delahaye, recently (Pour La Science (Sept. 2017), 80-85) posed the following brain-teaser, adapting a beautiful puzzle, of unknown origin, popularized by Peter Winkler in his wonderful book "Mathematical Mind-Benders". Here is a free translation from the French.

One places nine beetles on a circular track, where the nine distances,
measured in meters, between two consecutive beetles are the first nine prime numbers,
2,3,5,7,11,13,17,19 and 23. The order is arbitrary, and each number appears exactly
once as a distance.
At starting time, each beetle decides *randomly* whether she would go, traveling at a speed of 1 meter
per minute, clockwise or counter-clockwise. When two beetles bump into each other, they immediately
do "U-turn", i.e. reverse direction (like billiard balls). We assume that the size of the beetles
is negligible. At the end of 50 minutes, after many collisions, one notices the
distances between the new positions of the beetles. The nine distances are exactly as before,
the first nine primes numbers! How to explain this miracle?

Before looking at the answer, try to do it yourself.

This variation, due to Delahaye, is **much** harder than the original version
posed in Peter Winkler's book, where also the initial distances were arbitrary.
In Delahaye's rendition, the solver is bluffed into trying to use the fact that the distances
are primes. Something analogous happened to the great Paul Erdős, the patron saint
of combinatorics and number theory, who introduced **covering systems** and posed
a problem that was open since 1950 until two years ago, when Bob Hough found
a great refutation.
A close reading of his proof reveals that essentially all you need is the asymptotic property of
the primes, namely that p_{n} is asymptotic to n times log(n), and in hindsight,
Erdős' guess (that there are distinct covering systems with arbitrarily large minimal moduli)
was unlikely to be true by a healthy heuristic argument.

Also the recent breakthroughs in the twin prime conjectures (that are still very far from the true state of affairs), use very clever sieving arguments, but the same arguments should work if one replaces the sequence of primes by more general sequences, with similar asymptotic properties.

Hence, who knows? perhaps even the Riemann Hypothesis, that is equivalent to a statement about primes, would be made tractable if one can find a more general statement, valid for a wide class of sequences, that since it is more general, would be (as often is the case) much easier to prove. That far more general statement should reduce to RH (and even GRH) when one specializes to the primes. Good luck!

Opinions of Doron Zeilberger