Bingo! The "constant" speed is actually a red herring: it distracts one from the essence of the problem. The only real constraints are: 1. No backward motion, but continuous nondecreasing forward motion. 2. Constant *lap times*, not constant speed. 3. May need constant speed on the last (partial) lap. How about the following expression for swimmers' positions w.r.t. time: posx1(t)=(L/2)*(1-cos(%pi*t*S1/L)) posx2(t)=(L/2)*(1-cos(%pi*t*S2/L)) where L=25' (Hint: expand the forward & return swimming lanes into a *circle*; swimmers now swim constant speed around the circle, but we only worry about their x-coordinates.) Crossings are the zeros of posx1(t)-posx2(t): L*(cos(pi*t*S2/L)-cos(pi*t*S1/L))/2 which simplifies to the product: L*sin(pi*t*(S1-S2)/(2*L))*sin(pi*t*(S2+S1)/(2*L)) We can now count the zeros of the first factor plus the zeros of the second factor. We may have to "fix up" the last/partial lap. At 04:11 AM 11/12/2017, James Propp wrote:

So in fact one doesn't need to know that the swimmers swim at constant speeds; if one knows how many laps each swimmer completes, and one knows that the faster swimmer is faster throughout, and one knows that they never reach a wall simultaneously (or, alternatively, if one agrees that such occurrences count as double-passes), then the number of times the fast swimmer passes the slow swimmer is a topological invariant.