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I have this question which I am not sure how to solve:

One hundred indistinguishable ants are dropped on a hoop of diameter 1. Each ant is traveling either clockwise or counterclockwise with a constant speed of 1 meter per minute. When two ants meet, they bounce off each other and reverse directions. Will the ants ever return to their original configuration? After how many minutes?

user121692
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1 Answers1

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There is a well-known trick with these ant problems: instead of them bouncing off each other, just pretend that they pass by each other. So it looks like each ant moves with a constant speed without changing direction. Now it's easy to see that the original configuration will recur every $\pi$ minutes.

TonyK
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  • This is assuming the ants are point particles. It's more complicated if they have nonzero size. – Robert Israel Jan 13 '15 at 19:51
  • @RobertIsrael: A good point! But then what is the answer? Perhaps if the number of ants moving in each direction is equal, the argument can be modified to make it work? Perhaps it's true whatever the numbers? I notice from your edit that you had second thoughts yourself. – TonyK Jan 13 '15 at 19:57
  • But why are the two problems (with bouncing and without boncing) equivalent? I agree if they don't bounce it is trivial, but why are the answers the same for both cases? – gt6989b Jan 13 '15 at 20:18
  • @gt6989b: if you screw up your eyes a bit, and you don't know any of the ants personally, then you won't be able to distinguish the two cases. Give each ant a little hat, if you like, and ask them to exchange hats whenever they bounce; now just watch the hats instead of the ants. – TonyK Jan 13 '15 at 20:47
  • A configuration with $n$ ants, each of size $a$, on a hoop of circumference $L$ can be mapped (by collapsing the ants) to point ants on a hoop of circumference $L-na$, and the argument works there: after time $L-na$ the configuration will recur. – Robert Israel Jan 13 '15 at 20:50
  • @RobertIsrael: I suspect that the resulting configuration might be a rotation of the original. – TonyK Jan 13 '15 at 21:03
  • awesome thanks veryt helpful – gt6989b Jan 13 '15 at 22:55
  • What if we change the assumption that the ants are distinguishable? – user121692 Jan 14 '15 at 03:00
  • @user121692: My assumption was that the ants are _in_distinguishable. But if not, after $\pi$ minutes the configuration will be a permutation of the ants; and a permutation, if repeated enough times, will bring you back to the original. So it might take a bit longer, but the position will still be repeated. – TonyK Jan 14 '15 at 09:16
  • @user121692: In fact, if the permutation is not the identity, it must be cyclic, because the ants never cross each other. So it takes at most $100\pi$ minutes for the original configuration to recur. I thought at first that perhaps all the ants returned to their original positions after $\pi$ minutes, but this is not in general true. – TonyK Jan 14 '15 at 11:03