Friday Puzzler -- Ants

Re: Friday Puzzler -- Ants

Larry Barrett

Three Ant problem.

They never all meet at the same point or at the same time. However, if any one walks in a direction opposite to the other two then that one will meet first one other ant on the same leg, then meet the other ant at the next vertex.

Say A walks CW, B walks CCW, C walks CW. Then B meets A midway on leg AB, and B meets C at vertex A. If they continue walking, A and C will never meet since they are walking in the same direction, but B will continue to encounter both.

If all choose to walk in the same direction, either CW or CCW, there will be no meetings. The probability of this is 1/4.

Correct for Three-Ant problem

Re: Friday Puzzler -- Ants

Henry Higginbotham

When I first looked at it, the four-ant problem looked like they might be walking a catenary, or maybe a volute, but I'm too weak in math to do much more than speculate about such things.

So I'm going to try calling it an antenary, since I don't care much for cats, anyway.

Re: Friday Puzzler -- Ants

Alex Y

LOL!

Interestingly, determining the distance each ant travels is much easier than determining the path.

Four Ants hint

Alex Y

No calculus is needed, nor do you need to figure out the path to determine the distance each ant travels.

Four Ants hints 2 & 3

Alex Y

What pattern are the ants in after they move a certain distance?

Examine the motion vectors of one of the chasing ants and its target ant.

( I find it easier to think of the velocity vectors, then convert back to distances.)

Re: Four Ants hints 2 & 3

Henry Higginbotham

It seems each ant moves in the same pattern as the others, since there's no beginning or end, so they would presumably stay in a square pattern, becoming smaller and rotating and moving toward the center. But that doesn't help me with the description of a path (spiral, maybe?) or the distance traveled.

Re: Friday Puzzler -- Ants

Larry Barrett

Although the ants are walking CW and uniformly, look at their movements in slow motion as they move in A, D, C, B order, one ant-step at a time. A moves first and takes one step toward B and at this instant B, C, and D are all still on the corners. Now consider D; D takes a step directly toward A. Then C takes a step toward D. The vector for C is perpendicular to the vector that D took. Similarly, B takes a step toward C; B's vector is perpendicular to C's vector. And back to A who now takes a step toward B; A's vector is perpendicular to B's vector. From now on the motion vector for each ant will be perpendicular to the ant ahead of it.

Enough for now.

Re: Four Ants hints 2 & 3

Alex Y

It seems each ant moves in the same pattern as the others, since there's no beginning or end, so they would presumably stay in a square pattern, becoming smaller and rotating and moving toward the center.

Re: Friday Puzzler -- Ants

Alex Y

And back to A who now takes a step toward B; A's vector is perpendicular to B's vector. From now on the motion vector for each ant will be perpendicular to the ant ahead of it.

Enough for now.

Path of four ants

Alex Y

Henry and Larry have deduced the general path taken by the ants, and the fact that they stay in a square. Here is an illustration of their path (moving counterclockwise in this illustration). How long are those spiral paths? For extra credit, what direction is each ant facing when they meet?

Re: Path of four ants

Larry Barrett

I think the path length is equal to the length of the side of the square, and each ant is facing the same way it was facing when it started out.

Not sure I can prove this, but since their position remains perpendicular to each other, it seems equivalent to the path each would take if they walked along each side.

Correct Length!

Alex Y

Each ant moves 12" to the center on this spiral path.

Say Ant A is pursuing Ant B. We know that A's motion is always toward B, and that B's motion is always at a right angle to A's. So all of A's motion decreases the distance between them, while B's motion neither increases nor decreases that distance. Hence, the motion that closes the distance between them (12") is exactly that of A.

The path is very interesting, described as a logarithmic spiral (I need to look that up). What is interesting about it is that it has a finite length of 12", but it spins indefinitely. Without looking up the math for the spiral, consider this: after the ants have moved so that their direction is 45 degrees from where they started, they are in a square, and you have exactly the same problem you started with, just on a different scale.So if there were an answer to what direction they are facing when they meet, looking at the problem now would give a different final direction than you had at the start. In a sense, this infinitely spinning but finite length spiral is like the old "always walk half-way to the wall, will you ever get there" problem.

So a more accurate answer to the original problem is that they will get arbitrarily close together after each ant has walked arbitrarily close to 12".