Did you solve it? Dogs in pursuit

Earlier today I set you the following puzzle:

Four dogs are in four corners of a square of side length 1. Each dog starts running towards the dog immediately anti-clockwise to it. The dogs start at the same time, they all run at the same speed, and at every moment each dog is running directly towards the neighbouring dog.

During the pursuit, the dogs will run in a spiral before they all meet in the centre. How far does each dog travel before the group collision?

Solution

Each dog runs exactly a side length.

The image of the four spirals, below, helps understand why. At each moment on their paths, the dogs are at four points of a square. The direction in which a dog is headed is therefore at all times perpendicular to the direction in which the dog that it is chasing (and which is chasing it) is headed.

Consider the view from one of the dogs (the chaser). It is chasing a dog (the chasee) that at every point is moving perpendicular to the line between them. In other words, there is no part of the velocity of the chasee that is taking it further away, or closer to, the chaser beyond the movement of the chaser. As far as the chasing dog is concerned, the chasee may as well be standing still. The length of the total path travelled therefore is the distance between chaser and chasee at the start, which is the side of the square.

Here’s another way to think about it, suggests Steven Strogatz, author of Infinite Powers, who suggested the puzzle. Imagine a dog has a GoPro on its head. The footage of its pursuit of its neighbouring dog along the spiral path would be exactly the same, and take the same time as, the footage of the dog if all it did was run directly along the edge of the square to a stationary dog. Lovely!

I also asked you what happens if the dogs were to start in the corner of a triangle, or a pentagon. In each of those cases, the dogs are not moving perpendicular to each other at any stage. For the triangle, the distance travelled is less than a side, because the dogs are getting closer faster, and for the pentagon, the dogs are getting closer slower, so the distance travelled is more than a side.

To work it out exactly requires some geometry, algebra and trigonometry. We need to work out how much of the velocity is taking the chasee towards or away from the chaser. Strogatz kindly wrote out the solution here:

Thanks to Steven Strogatz for suggesting today’s puzzle, and for sketching the proof. I recommend his book Infinite Powers, a New York Times bestseller, which is out in the UK on Thursday.

For those who prefer to solve the dogs-in-pursuit problem using the medium of wool, check out these blankets by the @matheknitician:

I’ll be back in two weeks.

I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.