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). Jim On Saturday, November 11, 2017, Tom Karzes <karzes@sonic.net> wrote:
The notion of "physical passing" can be formalized as follows: Assume the pool lane runs from left to right. Denote a swimmer's position at any moment in time as the swimmer's distance from the left end of the lane. So the positions increase from 0 to L, then decrease from L down to 0, and the pattern repeats until the race ends (assuming the swimmers start at the left end).
Let d1(t) and d2(t) denote the positions of swimmers 1 and 2 at time t.
Let d(t) = d1(t) - d2(2). Then d(t) can range from -L to L.
If during some interval of time the sign of d(t) changes, either one pass occurred, or some other odd number of passes occurred. If it does not, then either no pass occurred, or else an even number of passes occurred.
If you narrow your intervals to those in which there was a single sign change throughout the entire interval, then those intervals correspond to the passes.
Under this definition, my example holds: A simultaneous bounce is *not* a pass.
On the other hand, suppose at the instant the swimmers touch the wall, they swap swimming speeds. In this case, the fast swimmer catches up to the slow swimmer, they bounce, then the slow swimmer speeds up and the fast swimmer slows down, so the formerly slow swimmer starts moving back more quickly than the formerly fast swimmer. They have changed positions, and have physically passed each other, even though they have not "logically" passed each other. It's the exact reverse of the normal case, in which they have not physically passed each other, but have logically passed each other (ignoring the difference in lap counts).
Tom
Tom Karzes writes:
Understood, but it seems to me that that's a different problem. Under that model, there is no passing at all, since the fast swimmer immediately overtakes the slow one and remains in the lead until the end. The question is asking about one swimmer physically moving past the other swimmer, which can occur in two circumstances: If they are moving in the same direction, or if they are moving in the opposite direction. I don't see bouncing at the same time as either of those cases.
Tom
Henry Baker writes:
Consider a plot of distance v. time.
This is a long strip 25' wide and 30 minutes long.
But you can *unfold* it -- e.g., make it 50' wide and 30 minutes long, which means that the wall doesn't exist and the swimmer keeps going.
The elastic bounce has become a straight line.
At 01:41 PM 11/11/2017, Tom Karzes wrote:
Considering a short window in which both swimmers hit the same wall at nearly the same time:
If the slow swimmer reaches a pool end an instant before the fast swimmer, there are two passes: the slow swimmer hits the wall, bounces, passes the fast swimmer in the opposite direction, then very shortly after the fast swimmer bounces and overtakes the slow swimmer, passing the slow swimmer once again.
If the fast swimmer reaches a pool end an instant before the slow swimmer, there is one pass: The fast swimmer hits the wall, bounces, passes the slow swimmer, and keeps going, with the slow swimmer falling further and further behind.
But what if they reach a pool end at the exact same moment? Does that still count as the fast swimmer passing the slow swimmer? Certainly the fast swimmer overtakes the slow swimmer, but it's not clear that the fast swimmer "passes" the slow swimmer, since the fast swimmer remains furthest from the wall at all times except at the moment of contact, so in a sense it's not much different than if the fast swimmer turned around just before reaching the wall, in which case there would clearly be no passing. I think this case could be argued either way.
Tom
Henry Baker writes:
I recall a question from my high school AP math or math SAT exam circa 1964:
Two swimmers are swimming laps of a 25' pool, but they swim at different (constant) speeds, say S1 & S2. (Assume that they bounce elastically off the pool ends!)
How many times do they pass one another in 30 minutes?
This problem is pretty complicated, as one swimmer could pass the other in either the same or opposite direction. The time is also long enough, that you can't simply simulate the first 1,2,3 laps, but must come up with a more general answer.
I don't recall how I solved it back then, but I must have gotten the right answer, because I did very well on that test.
At 05:06 AM 11/11/2017, James Propp wrote:
Anyone have a favorite puzzle in which miscounting plays a role?
My favorite is the classic bookworm puzzle (see https://math.stackexchange.com/questions/1271651/how-is- this-true-bookworm-puzzle ).
Another example: A man was born in 50 BC and died on the same day in 50 AD. How old was he when he died?
I'm especially interested in puzzles that lend themselves to solvers committing fencepost errors, and off-by-one errors more generally.
Jim Propp
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