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The New Coffee Room

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  3. Puzzle time - more bugs on a meter stick

Puzzle time - more bugs on a meter stick

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  • jon-nycJ Online
    jon-nycJ Online
    jon-nyc
    wrote on last edited by
    #4

    Try experimenting with, say, 4 bugs.

    "You never know what worse luck your bad luck has saved you from."
    -Cormac McCarthy

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    • jon-nycJ Online
      jon-nycJ Online
      jon-nyc
      wrote on last edited by jon-nyc
      #5

      As a hint here's a suggested approach:

      Imagine Bugsy has a baton and passes it to whatever bug he comes into contact with, who then passes it along similarly, etc.

      It's clear from our last problem that the baton will make a revolution of the circle in 100s, ending up at Bugsy's starting point.

      How many 'contacts' or 'handoffs' must occur for the baton to end up back in Bugsy's hands in the end?

      "You never know what worse luck your bad luck has saved you from."
      -Cormac McCarthy

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      • KlausK Offline
        KlausK Offline
        Klaus
        wrote on last edited by
        #6

        Hm. Thinking about it a little, I'd say the probability is 50%, depending on whether there's an even number of direction changes or an odd one.

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        • jon-nycJ Online
          jon-nycJ Online
          jon-nyc
          wrote on last edited by jon-nyc
          #7

          It seems clear to me that, for the baton to return to Bugsy, the number of 'handoffs' needs to be a multiple of 24, since the relative position of the bugs doesn't change.

          What does that imply about the starting position of the bugs?

          "You never know what worse luck your bad luck has saved you from."
          -Cormac McCarthy

          KlausK 1 Reply Last reply
          • jon-nycJ Online
            jon-nycJ Online
            jon-nyc
            wrote on last edited by
            #8

            I should add I won’t see the solution until Saturday morning so I haven’t gotten confirmation of my approach, but it seems sound.

            "You never know what worse luck your bad luck has saved you from."
            -Cormac McCarthy

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            • jon-nycJ jon-nyc

              It seems clear to me that, for the baton to return to Bugsy, the number of 'handoffs' needs to be a multiple of 24, since the relative position of the bugs doesn't change.

              What does that imply about the starting position of the bugs?

              KlausK Offline
              KlausK Offline
              Klaus
              wrote on last edited by
              #9

              @jon-nyc said in Puzzle time - more bugs on a meter stick:

              It seems clear to me that, for the baton to return to Bugsy, the number of 'handoffs' needs to be a multiple of 24, since the relative position of the bugs doesn't change.

              What does that imply about the starting position of the bugs?

              So what's your answer?

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              • jon-nycJ Online
                jon-nycJ Online
                jon-nyc
                wrote on last edited by jon-nyc
                #10

                There are two handoff events for every bug traveling in the opposite direction of the baton. So we need 12 bugs heading one way and 12 the other. So you get C(24,12) * (1/2) ^ 24 ~= 0.16118

                Well, that's the engineer's answer. Bugsy also gets to the same spot if the bugs are all heading in the same direction, so for the mathematician's answer you need to add (1/2)^23 to the above.

                "You never know what worse luck your bad luck has saved you from."
                -Cormac McCarthy

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                • jon-nycJ Online
                  jon-nycJ Online
                  jon-nyc
                  wrote on last edited by
                  #11

                  Interestingly the originator of the puzzle sent out the following hint today:

                  "Consider the angular momentum of the mass of bugs"

                  That still seems to be pointing you to 12 bugs going one way and 12 the other leading to an angular momentum of zero, but I'm not sure exactly how.

                  "You never know what worse luck your bad luck has saved you from."
                  -Cormac McCarthy

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                  • jon-nycJ Online
                    jon-nycJ Online
                    jon-nyc
                    wrote on last edited by
                    #12

                    Thinking about their hint, the bugs must shift positions as a function of #clockwise - #counterclockwise, since they can't shift relative position and must end in a state with a bug in each of the original starting positions.

                    I'll get their exact reasoning on Saturday

                    "You never know what worse luck your bad luck has saved you from."
                    -Cormac McCarthy

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                    • jon-nycJ Online
                      jon-nycJ Online
                      jon-nyc
                      wrote on last edited by
                      #13

                      Here’s their solution. Same answer as me but arrived via the angular momentum concept. Frankly I thought my approach was simpler.

                      SOLUTION: We know from last week's puzzle, "Bugs on a Stick," that the final positions for the bugs collectively are the same as if they passed through one another instead of bouncing. If they pass one another, every bug simply travels once around the hoop and ends up where it began.

                      With bouncing, some bug will end up where Bugsy began, but will it be Bugsy? It certainly will be if it happens that all the bugs begin facing the same way, since then there will be no collisions. But that happens with probability only 2/(224) = 0.0000001192, approximately.

                      Wait, maybe there's another way. Note that the whole process preserves the angular momentum of the mass of bugs. Suppose, say, that 16 of the 24 bugs face clockwise; the remaining 8 are counterclockwise. Then the bugs will, as a whole, rotate clockwise at 1/3 cm/sec, and at the end, each bug will be, on average, 1/3 of the way around from where it began.

                      But since they never pass through one another, the bugs remain in the same order; thus every bug will end up (in this case) in the position of the bug that began 8 bugs in front of it, clockwise.

                      So all the bugs (including Bugsy) change position in this case, and indeed in every case unless the mass clockwise speed is an integer number of cm/sec. That integer can only be 1 (all bugs moving clockwise), -1 (all counterclockwise), or 0 (12 bugs in each direction).

                      The latter is quite a lively possibility, occurring with probability (24 choose 12)/(224) ≈ 0.1611802578. Adding the 0.0000001192 from before gives us a total probability of 0.1611803770 for the event that Bugsy ends up where he started.

                      "You never know what worse luck your bad luck has saved you from."
                      -Cormac McCarthy

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