Today’s problem is a classic puzzle and an excuse to post this picture of Melbourne’s annual sausage dog race, the Running of the Wieners.
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?
For bonus points, can you work out how far the dogs would travel if they started at the three corners of an equilateral triangle of side length 1, or the five corners of a regular pentagon of side length 1? Would the distance be further or shorter than in the case wih the square?
The dogs-in-pursuit problem was suggested to me by Steven Strogatz, author of the magnificent Infinite Powers: How Calculus Reveals the Secrets of the Universe, which is out this week. Strogatz, a professor at Cornell, is both a world class mathematician and one of the best popular mathematics authors writing today.
One of the aims of Strogatz’s book is to demystify calculus, so the general reader can appreciate what it is and how it is important it is to the modern world. He calls the big idea behind calculus the Infinity Principle:
To shed light on any continuous shape, object, motion, process or phenomenon – no matter how wild or complicated it may appear – reimagine it as an infinite series of simpler parts, analyse those, and then add the results back together to make sense of the original whole.
So what’s this got to do with the dogs? Well, they are running in such a way that their direction is changing continuously at every point. As each dog moves, the one chasing it must change its direction. You don’t need any technical ability at calculus to solve the problem, however, only a simple insight.
Pursuit problems, that is, problems about the curves you get when a point (the chaser) tracks a moving point (the chasee), usually do require calculus. Indeed, these problems were one of the things that got the teenage Strogatz excited about the power of calculus in the first place.
“I loved chase problems because of how visual they were,” he says. “I could picture a dog running after a fleeing postman, or a heat-seeking missile pursuing an enemy fighter plane. And with calculus I could solve such questions – it was thrilling!”
Pursuit is also of interest because it scampers into the area of mathematical art. Here’s a full look at the (logarithmic) spiral made by the four dogs. It looks like a piece of op-art, or an optical illusion.
When you arrange the 4-dog spiral with a 6-dog spiral (from a hexagon) and a 3-dog spiral (the triangle), you get an image like this. It is almost impossible to see the underlying shapes.
I’ll be back with the solutions at 5pm. Please NO SPOILERS. Instead discuss your favourite bits of calculus.
Or show me your pursuit-inspired art (coloured or not) and post it to me on social media.
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.
I’m the author of Football School, a book series for 7 to 12 year olds that opens up the curriculum through football. Football School Star Players: 50 Inspiring Stories of True Football Heroes is just out. It profiles 50 footballers who show that football can be a force for good both on and off the pitch.