This week’s puzzle puts you at the mercy of an unjust torturer. Explaining why he is unjust can help you make sense of a daunting mathematical proof that last year made headlines for being “bigger than Wikipedia.”

Illustration by Jim Cooke

We’re out of the country for the next couple of weeks, so the next two installments of the Sunday Puzzle will look a little different than normal. This week, mathematician James Grime—who you may recognize from singingbanana, Numberphile, and elsewhere—has permitted us to feature the video below, in which he uses a puzzle to explain a mathematical proof. (The video was recommended to us by Sunday Puzzle reader Aaron Baker—thanks Aaron) Unlike previous columns, where we’ve presented the brainteaser one week and the solution the next, this week’s puzzle is entirely self-contained; in this video, Grime presents the puzzle and the solution all in one go. Next week’s puzzle will, most likely, be similarly comprehensive in its presentation.

## Sunday Puzzle #32: Explaining A Wikipedia-Sized Proof

We’ll be back next week with a new puzzle (or puzzles!). Got a great brainteaser, original or otherwise, that you’d like to see featured? E-mail me with your recommendations. (Be sure to include “Sunday Puzzle” in the subject line.)

## SOLUTION To Sunday Puzzle #31: The Perilous Bridge Crossing

Last week I asked you to determine the smallest window of time in which four people could cross a perilous bridge to safety. The conditions of the puzzle were as follows:

The bridge can only support two people at once. The four people share one torch, which must be used to safely traverse the bridge. Each person can cross the bridge in a different amount of time: Person A can cross the bridge in one minute, person B in two minutes, person C in five minutes, and person D in ten minutes. When two people cross together, they must do so at the slower person’s pace.

The most obvious solution to this puzzle is to have person A, who is the fastest, chaperone every other person across the bridge, returning alone after each crossing with the torch in hand. By this method, all four people can cross the bridge in 19 minutes.

But they can actually cross even faster than this, and not by breaking the torch in half. (Assume for the purposes of this puzzle, that we’re dealing with a flashlight, not a burning branch.)

The crux of this puzzle, it turns out, is in realizing that more time is saved by having the two slowest people cross together than having them cross individually. But Persons C and D can’t just cross at any time. If, for example, they are the first two people to cross the bridge, one of them must make the return trip to bring the torch back. This is obviously an enormous waste of time. So how do you get persons C and D across the bridge AND get the torch back to the other side? Commenter Quint gives a possible scenario:

A and B cross - [Elapsed time:] 2 minutes

B returns - [Elapsed time:] 4 minutes

C and D cross - [Elapsed time:] 14 minutes

A returns - [Elapsed time:] 15 minutes

A and B cross - [Elapsed time:] 17 minutes

By having the slowest persons cross on the second cross, and using persons A and B to return the torch on the two occasions required, it’s actually possible to shave two minutes off the total time.