As you fill out a crossword, it can abruptly shift from fiendishly difficult to extremely simple. This is similar to the way some physical systems undergo a phase transition – but compared to them, these puzzles may be mathematically unique.

While solving a crossword one day, Alexander Hartmann at the University of Oldenburg in Germany came up with a personal challenge: he wanted to find enough words to create one large, connected island of letters. Then he realised that he had unintentionally stumbled upon a type of “percolation problem”, which was familiar to…