Puzzle time - Beetles edition

[AndyD]

@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?

[Klaus]

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.

[jon-nyc]

It’s a limit, not a discontinuity. The distance approaches infinity as n does.

[Klaus]

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

[jon-nyc]

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

[Klaus]

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?

[jon-nyc]

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

[Klaus]

:::

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.

:::

[jon-nyc]

That’s right!

Now generalize to N sides.

[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!

[Klaus]

Also, bonus question for Jon:

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

[jon-nyc]

@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.

[jon-nyc]

@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?

[Klaus]

yes

[jon-nyc]

@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

[jon-nyc]

infinite

[Klaus]

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

[Axtremus]

:::

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

:::

[jon-nyc]

:::

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.

:::

[Larry]

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