The lonely runner conjecture is a good example of an open mathematical problem that can be explained easily to a nonspecialist.  Suppose that k runners are running in the same direction around a circular track.  A runner is said to be lonely if none of the other runners are within 1/k of a lap of that runner.  For example, if there are three runners on the track, and one of the runners is at least 1/3 of a lap from all the others, then that runner is said to be "lonely".

Now suppose in addition that the k runners are running at distinct, fixed, integer speeds.  The lonely runner conjecture says that if the runners run for long enough, then at some point, each runner will experience loneliness.  For example, if three runners are running around a track at speeds of exactly 4, 5 and 6 miles per hour, then if they run for long enough, each runner will eventually have the experience of being at least 1/3 of a lap away from each of the others.

As Clayton L. Barnes, II explains in the recent paper, the lonely runner conjecture is over 50 years old.  It is known to be true for small values of k, but is still an open question in general.  The case of 2 runners is trivial, and the case of 3 runners is not difficult.  However, as the number of runners increases, the problem becomes extremely challenging.  The first proof of the conjecture for 6 runners is a 50 page paper by three authors from MIT which appeared in 2001, although a short proof was found in 2004.  The case of 7 runners was solved in 2008.

There is a known reformulation of the lonely runner conjecture that superficially seems unrelated.  This is in terms of view obstruction.  Figure 3 shows a quadrant of the plane filled with squares centred at half-integer points (meaning coordinates such as (0.5, 1.5), (6.5, 3.5), etc.)  The squares are exactly 1/3 of a unit wide, and this is just wide enough so that, if viewed from the origin, this quadrant of squares would appear to have no gaps in it.

Figures 4, 5 and 6 show a similar situation in three dimensions, again with cubes 1/3 of a unit wide centred at half integer points.  The number of cubes increases from left to right.  In this case, it seems that if viewed from the origin, there will still appear to be gaps in the cubes even if the number of cubes is large.  This turns out to be true: in order not to leave any apparent gaps, one needs wider cubes that are 1/2 of a unit across;  this is shown in Figures 7, 8 and 9.

So what happens in n dimensions?  It is conjectured that one would need cubes of side at least (n-1)/(n+1) in order not to leave any apparent gaps. This number works out to be 1/3 if n = 2, and 1/2 if n = 3.  The remarkable thing is that this conjecture is equivalent to the lonely runner conjecture!  So a positive solution to one of these two conjectures would lead to a positive solution of the other conjecture, and vice versa.

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