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The New Coffee Room

  1. TNCR
  2. General Discussion
  3. Puzzle time - Beetles edition

Puzzle time - Beetles edition

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  • KlausK Offline
    KlausK Offline
    Klaus
    wrote on last edited by
    #2

    I don't understand. If they walk to the beetle to the left, how would they ever meet in the middle?

    1 Reply Last reply
    • jon-nycJ Online
      jon-nycJ Online
      jon-nyc
      wrote on last edited by jon-nyc
      #3

      They donโ€™t stay on the square. They continuously head toward the next beetle in a straight line, who is moving toward the next in a straight line, etc

      Only non-witches get due process.

      • Cotton Mather, Salem Massachusetts, 1692
      1 Reply Last reply
      • KlausK Offline
        KlausK Offline
        Klaus
        wrote on last edited by
        #4

        Oh, now I see. You mean they constantly change direction to take the shortest path to the next beetle, whereever that beetle is at that time, right? But they do not "foresee" the future movement of that other beetle to adapt their "shortest path" calculation, right?

        That sounds like a hard puzzle.

        A 1 Reply Last reply
        • jon-nycJ Online
          jon-nycJ Online
          jon-nyc
          wrote on last edited by
          #5

          Exactly.

          There are several ways to solve it, my friend had two, I came up with a third.

          Only non-witches get due process.

          • Cotton Mather, Salem Massachusetts, 1692
          1 Reply Last reply
          • KlausK Klaus

            Oh, now I see. You mean they constantly change direction to take the shortest path to the next beetle, whereever that beetle is at that time, right? But they do not "foresee" the future movement of that other beetle to adapt their "shortest path" calculation, right?

            That sounds like a hard puzzle.

            A Offline
            A Offline
            AndyD
            wrote on last edited by
            #6

            @Klaus said in Puzzle time - Beetles edition:

            That sounds like a hard puzzle.

            Good enough answer for me. Advanced maths or what ! Trying to imagine an increasingly spiral path?

            1 Reply Last reply
            • KlausK Offline
              KlausK Offline
              Klaus
              wrote on last edited by
              #7

              I also wonder about the limit case: infinite number of beetles on a circle. I'd say the solution is that no traveling occurs and they don't meet in the middle. But for finite n it seems to be the case that they would meet in the middle. That's a strange discontinuity.

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

                Itโ€™s a limit, not a discontinuity. The distance approaches infinity as n does.

                Only non-witches get due process.

                • Cotton Mather, Salem Massachusetts, 1692
                1 Reply Last reply
                • KlausK Offline
                  KlausK Offline
                  Klaus
                  wrote on last edited by
                  #9

                  Oh I see. So in the limit case the beetles just all walk simultaneously on the perimeter of the circle.

                  1 Reply Last reply
                  • jon-nycJ Online
                    jon-nycJ Online
                    jon-nyc
                    wrote on last edited by
                    #10

                    I would start with the square case then generalize, not start with the general case.

                    Only non-witches get due process.

                    • Cotton Mather, Salem Massachusetts, 1692
                    1 Reply Last reply
                    • KlausK Offline
                      KlausK Offline
                      Klaus
                      wrote on last edited by Klaus
                      #11

                      I'll start with the 2-sided polygon.

                      If the length of the line is s, then every beetle travels s/2.

                      And as a bonus, for a 1-sided polygon, the distance is 0.

                      Do I get the prize?

                      1 Reply Last reply
                      • jon-nycJ Online
                        jon-nycJ Online
                        jon-nyc
                        wrote on last edited by
                        #12

                        What do you think this is, everybody gets a trophy?

                        Only non-witches get due process.

                        • Cotton Mather, Salem Massachusetts, 1692
                        1 Reply Last reply
                        • KlausK Offline
                          KlausK Offline
                          Klaus
                          wrote on last edited by Klaus
                          #13

                          :::

                          OK, slightly more seriously:

                          I'd say the beetles will always form a square at each point in time, but that square rotates and shrinks.

                          At every time, the movement vectors will hence be orthogonal to each other. Which means that every beetle has to move distance s before the size of the square becomes 0.

                          :::

                          1 Reply Last reply
                          • jon-nycJ Online
                            jon-nycJ Online
                            jon-nyc
                            wrote on last edited by
                            #14

                            Thatโ€™s right!

                            Now generalize to N sides.

                            Only non-witches get due process.

                            • Cotton Mather, Salem Massachusetts, 1692
                            1 Reply Last reply
                            • KlausK Offline
                              KlausK Offline
                              Klaus
                              wrote on last edited by Klaus
                              #15

                              But then the "orthogonality" thing doesn't hold anymore and things get complicated. Please don't tell me you want me to write down some complicated trigonometric functions and differential equations! ๐Ÿ˜–

                              jon-nycJ 2 Replies Last reply
                              • KlausK Offline
                                KlausK Offline
                                Klaus
                                wrote on last edited by
                                #16

                                Also, bonus question for Jon:

                                How many times do the beetles spin around each other before they meet?

                                jon-nycJ 1 Reply Last reply
                                • KlausK Klaus

                                  But then the "orthogonality" thing doesn't hold anymore and things get complicated. Please don't tell me you want me to write down some complicated trigonometric functions and differential equations! ๐Ÿ˜–

                                  jon-nycJ Online
                                  jon-nycJ Online
                                  jon-nyc
                                  wrote on last edited by jon-nyc
                                  #17

                                  @Klaus said in Puzzle time - Beetles edition:

                                  But then the "orthogonality" thing doesn't hold anymore and things get complicated. Please don't tell me you want me to write down some complicated trigonometric functions and differential equations! ๐Ÿ˜–

                                  I didn't have to.

                                  Only non-witches get due process.

                                  • Cotton Mather, Salem Massachusetts, 1692
                                  1 Reply Last reply
                                  • KlausK Klaus

                                    Also, bonus question for Jon:

                                    How many times do the beetles spin around each other before they meet?

                                    jon-nycJ Online
                                    jon-nycJ Online
                                    jon-nyc
                                    wrote on last edited by
                                    #18

                                    @Klaus said in Puzzle time - Beetles edition:

                                    Also, bonus question for Jon:

                                    How many times do the beetles spin around each other before they meet?

                                    Define 'spin around each other'. You mean that the square formed by the four of them rotates 360 degrees?

                                    Only non-witches get due process.

                                    • Cotton Mather, Salem Massachusetts, 1692
                                    1 Reply Last reply
                                    • KlausK Offline
                                      KlausK Offline
                                      Klaus
                                      wrote on last edited by
                                      #19

                                      yes

                                      1 Reply Last reply
                                      • KlausK Klaus

                                        But then the "orthogonality" thing doesn't hold anymore and things get complicated. Please don't tell me you want me to write down some complicated trigonometric functions and differential equations! ๐Ÿ˜–

                                        jon-nycJ Online
                                        jon-nycJ Online
                                        jon-nyc
                                        wrote on last edited by
                                        #20

                                        @Klaus said in Puzzle time - Beetles edition:

                                        But then the "orthogonality" thing doesn't hold anymore and things get complicated. Please don't tell me you want me to write down some complicated trigonometric functions and differential equations! ๐Ÿ˜–

                                        I could give you a hint. Really a way to reframe the orthogonality that will generalize to other n

                                        Only non-witches get due process.

                                        • Cotton Mather, Salem Massachusetts, 1692
                                        1 Reply Last reply
                                        • jon-nycJ Online
                                          jon-nycJ Online
                                          jon-nyc
                                          wrote on last edited by jon-nyc
                                          #21

                                          infinite

                                          Only non-witches get due process.

                                          • Cotton Mather, Salem Massachusetts, 1692
                                          1 Reply Last reply
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