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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
    #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 Offline
      jon-nycJ Offline
      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 Offline
          jon-nycJ Offline
          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.

            :::

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            • jon-nycJ Offline
              jon-nycJ Offline
              jon-nyc
              wrote on last edited by
              #14

              That’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! 😖

                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! 😖

                    jon-nycJ Offline
                    jon-nycJ Offline
                    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! 😖

                    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 Offline
                      jon-nycJ Offline
                      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! 😖

                          jon-nycJ Offline
                          jon-nycJ Offline
                          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! 😖

                          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 Offline
                            jon-nycJ Offline
                            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
                            • KlausK Offline
                              KlausK Offline
                              Klaus
                              wrote on last edited by Klaus
                              #22

                              I think so, too. But how can they turn around each other infinitely often while only traveling a finite distance? (pinging @Zeno!)

                              1 Reply Last reply
                              • AxtremusA Offline
                                AxtremusA Offline
                                Axtremus
                                wrote on last edited by
                                #23

                                :::

                                “Orthogonality” (90°) in the n-polygon case ==> inside angle of the polygon (n-2)*180°/n

                                :::

                                1 Reply Last reply
                                • jon-nycJ Offline
                                  jon-nycJ Offline
                                  jon-nyc
                                  wrote on last edited by jon-nyc
                                  #24

                                  :::

                                  Along the lines of Ax’s post: You can calculate how much distance is traveled for every unit reduction of R, with R being the distance from a bug to the center of the rotating polygon.

                                  In the case of a square:

                                  Think of the initial square rotated and superimposed on an x,y coordinate system such that each bug is sitting on one of the axes. Infinitesimally, the bug on the positive x axis moves in a direction 45° off the vertical. You can draw the infinitesimal right triangle formed by the x axis, the distance he traveled, and the line from his new position back to the x axis. As he moved along the hypotenuse of that infinitesimal triangle, he covers sqrt(2) distance for every unit of radius reduction.

                                  When you frame the orthogonality that way, you can see how it generalizes - that 45° angle changes as does the ratio of distance traveled to radius reduction.

                                  :::

                                  Only non-witches get due process.

                                  • Cotton Mather, Salem Massachusetts, 1692
                                  1 Reply Last reply
                                  • LarryL Offline
                                    LarryL Offline
                                    Larry
                                    wrote on last edited by
                                    #25

                                    If you walk them backwards it says Paul is dead..

                                    1 Reply Last reply
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