Running puzzle: the solution

This is a fantastically counter-intuitive puzzle because it seems impossible for Daphne, who runs every mile in 8 minutes and one second, to beat Constance, who runs every mile in 8 minutes flat.

But, of course, she can – otherwise there would be no puzzle. The runner who is slower over every mile can surprisingly devise a way to be faster over 26.2 miles.

Let’s first think about how Daphne runs the race. She does not run at a constant speed, but even so she covers every mile interval in the same time. How does she do this?

Look at the graph below, which tracks a runner’s speed over the distance of the course. It shows a racing strategy of running the first a of each mile at a high constant speed, and the remaining section of the mile at a low constant speed. If this strategy is repeated every mile all the way through the race, the runner is not running at a constant speed overall, but every mile interval will be run in the same time. This is because whichever mile you take, a of that mile is run at high speed and the remainder of the mile at low speed.

Graph for Alex Bellos running puzzle

If Daphne runs a marathon using this strategy, then let’s adapt it to best suit her needs. The marathon is 26.2 miles. We know – since she loses a second every full mile to Constance – that she will be 26 seconds behind at the 26-mile mark. So Daphne needs to make up 26 seconds on the remaining 0.2 miles.

Let Daphne’s strategy be to run her high speed for the first 0.2 of each mile and her low speed for the rest. We now need to switch to talking about time rather than distance. Let Daphne run the first 0.2 miles in x seconds, and the remaining 0.8 miles in y seconds. She runs every mile in x + y seconds.

The question states that Daphne runs every mile in 8 minutes 1 second, which is 481 seconds. This gives us the equation:

[1] x + y = 481

Constance runs every mile in 8 minutes, so she runs the entire marathon in 26.2 × 8 × 60 seconds, which is 12,576 seconds.

Let’s now assume that Daphne wins the marathon by a single second, completing it in 12,575 seconds. Since she runs the first 0.2 miles of her mile cycle 27 times, and the 0.8 mile section only 26 times, we get the following equation:

[2] 27x + 26y = 12,575

We can solve these simultaneous equations without too much trouble. From [1] y = 481 – x. Substitute this value of y in [2]. The numbers thankfully cancel out nicely to leave x = 69 seconds and y = 412 seconds. So, if Daphne sprints the first 0.2 of each mile in 69 seconds, and takes 412 seconds for the rest of the mile, she will lose every single mile versus mile comparison, but pip Constance to the final post by a second.