Henry and I say that the number of passes exhibits a removable discontinuity if you change the swimmers' paths. To see why, create a diagram in which the x-axis says how far swimmer #1 has gone and the y-axis says how far swimmer #2 has gone, measured in pool-lengths. The graph of the relation "both swimmers are at the same location" is a union of lines of the form x-y=2n and x+y=2n+1 that form a square grid (obtained from the usual grid by dilating by sqrt(2) and rotating by 45 degrees). Assuming that swimmer #1 always swims faster than swimmer #2, asking "How many times does swimmer #1 pass swimmer #2?" is the same as asking "At how many times t is the point (x(t),y(t)) on that grid?", and as long as the curve {(x(t),y(t)} doesn't pass through a place where grid-lines intersect, the answer is independent of the cuve: it's the number of grid-lines that path must cross. 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. I don't know if Tom is right in saying that the number of passes exhibits a jump discontinuity if one changes the length of the pool, but I'm pretty sure this kind of diagram will help us understand Tom's claim as well. Jim Propp On Sunday, November 12, 2017, Tom Karzes <karzes@sonic.net> wrote:k
It would certainly have been smart of the problem designers to avoid that case.
I would approach the limit from a different perspective, and argue that there's a discontinuity when they bounce simultanrously:
Suppose we shorten the fast swimmer's lane by moving the wall in question closer to the opposite wall. Now if they both bounce simultaneously, there is clearly no passing around the time of the bounces, since at no time are they at the same distance at the same time.
Now slowly increase the length of the fast swimmer's lane by moving the wall back out, almost to the same distance as the slow swimmer's wall. There is still clearly no passing: The fast swimmer is behind the slow swimmer as they approach their walls. They both bounce simultaneously and start to head back. At no point is the fast swimmer as far to the right as slow swimmer (assuming the bounce walls are on the right).
If you continue to move the fast swimmer's wall to the right, you reach the point in question, where the walls are both the same distance from the opposite wall. And if you continue past that, so that the fast swimmer's wall is to the right of the slow swimmer's, then you have a double-pass case: the fast swimmer passes the slow swimmer just before they both bounce, and again immediately after they both bounce (and both passes are "overtaking" passes, as opposed to "swimming past each other" passes.
So there is a discontinuity at the point where both walls are at the same distance, with the passing count jumping from 0 to 2.
Tom
Henry Baker writes:
I agree with Jim's interpretation.
I don't remember the exact speeds from the original problem, but I would imagine that they were chosen in such a way that the *only* time both swimmers hit the wall simultaneously was at the very start. This would eliminate the necessity to explain in great detail exactly what a "pass" entailed.
At 08:03 PM 11/11/2017, Tom Karzes wrote:
Well, if you perturb it by moving the fast swimmer ahead slightly, then you get (a) the fast swimmer overtaking the slow swimmer on the way up, followed by (b) the swimmers swimming past each other. If you perturb it by moving the slow swimmer ahead slightly, then you get (a) the swimmers swimming past each other, followed by (b) the fast swimmer overtaking the slow swimmer on the way back. In other words, the two types of passes occur in the opposite order. I argue that these disappear entirely if they bounce at the exact same time.
Tom
James Propp writes:
I think both swimmers simultaneous bouncing off the wall should count as a "double pass", since that's what it turns into if you perturb the history of the system ever-so-slightly in either direction (having the slow swimmer reach the wall first or having the fast swimmer reach the wall first).
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