This puzzle is nearly impossible – but working out why is its own brain-teaser

You and a friend are prisoners with one shot at escape. A peculiarly mathematics-oriented warden has set up a puzzle that, if solved, will set you free. The rules are as follows: while the other prisoner is locked away elsewhere, you watch the warden hide a key under a single square of a chessboard. Each square has a coin flipped to heads or tails on top. You must flip just one coin to help indicate the key’s location to your puzzle partner, who will receive it in another room. Before you witness the key’s placement, you and your friend are given an opportunity to concoct a strategy together – although the warden is able to listen in, and can arrange the coins ahead of time to try and disrupt it. Is it possible to hatch a foolproof plan? The answer, in most cases, is no. But figuring out why, as guided by the mathematical wizardry of the US YouTuber Grant Sanderson, is a brain-bending exercise all of its own. In this video, part of Sanderson’s 3Blue1Brown video series, he uses the twisty prisoner puzzle as a springboard to explore how mathematical visualisation can be a useful tool for methodical problem-solving.