The process of solving crossword puzzles is mathematically similar to well-studied physical systems – but there is one property that makes the game unique.
Crosswords. Who doesn’t do them? Some cause “bits” of pain. Others not so much.
Tony de Matos aside, crosswords are very similar to the way some physical systems go through a phase transition – but compared to them, this game is mathematically unique.
The story is told by: “One day, while solving crosswords, Alexander Hartmannfrom the University of Oldenburg, in Germany, set himself a personal challenge: he wanted to find enough words to create a large island of letters linked together“.
It was then that he realized that he had unintentionally stumbled upon some kind of “percolation problem”, which was familiar to him from his work as a physicist. The crossword version of this problem, however, appears to be unique.
Percolation refers to processes in which pieces of matter connect to form currents or flows. Imagine water filling the neighboring pores of a sponge until the liquid starts to flow out; or, in the same sense, two words connecting to fill a blank in a crossword puzzle.
Since this process is well studied, Hartmann knew it must present an abrupt transition – called phase transition – in which the crossword goes from very few words to many words filled in.
In a recent study in Physical Review EHartmann deduced a formula for this transition, based on a mathematical model of the crossword puzzle he devised.
Surprisingly, graphing this formula revealed a form unlike all comparable percolation problems.
According to the mathematician, this was due to the fact that crosswords have a new ingredient – each word you write makes it easier to search for the next one.
According to Hartmann’s calculations, the number of words needed for the puzzle going from almost empty to almost filled depends exactly on the Use of each word to find the next one.
Hartmann’s goal is to understand how, as a puzzle approaches this phase transition, finding each new word leads to a cascade of more words. He still wants to see if these waterfalls can be compared to more physical ones – like the way grains of sand accumulate in an avalanche.
