Dancing alongside the more serious practitioners of mainstream mathematics are the purveyors of mathematical puzzles and problems. These go back at least as far as Diophantus (c. 200–284), the Alexandrian “father of algebra.” Alcuin of York (c. 735–804) produced a collection of problems that included the the wolf, the goat, and the cabbages (above); the three men who need to cross a river with their sisters; and problems similar to the bird puzzle published by Fibonacci a few centuries later. In more modern times, Martin Gardner (1914–2010) has done more than anyone else to popularise this offshoot of mathematics. It is often called “recreational mathematics,” because people do it for fun (in part because they are not **told** that it is mathematics).

Particularly popular in recent times have been Sudoku (which is really a network colouring problem in disguise) and the Rubik’s Cube (which illustrates many concepts of group theory, although it was not invented with that in mind). Sudoku puzzles have been printed in more than 600 newspapers worldwide, and more than 20 million copies of Sudoku books have been sold. The Rubik’s Cube has been even more popular: more than 350 million have been sold.

Recreational puzzles may be based on networks, as in *Hashi* (“Bridges”). They may be based on fitting two-dimensional or three-dimensional shapes together, as in pentominoes or the Soma cube. They may be based on transformations, as in the Rubik’s Cube. They may even be based on arithmetic, as in Fibonacci’s problem of the birds, or the various barrel problems, which go back at least as far as the Middle Ages.

In one barrel problem, two men acquire an 8-gallon barrel of wine, which they wish to divide exactly in half. They have an empty 5-gallon barrel and an empty 3-gallon barrel to assist with this. How can this be done? It is impossible to accurately gauge how much wine is inside a barrel, so that all that the men can do is pour wine from one barrel to another, stopping when one barrel is empty, or the other is full [highlight to show solution → (8, 0, 0) → (3, 5, 0) → (3, 2, 3) → (6, 2, 0) → (6, 0, 2) → (1, 5, 2) → (1, 4, 3) → (4, 4, 0)]. There is a similar problem where the barrel sizes are 10, 7, and 3.

Apart from being fun, puzzles of this kind have an educational benefit, training people to think. For this reason, Alcuin called his collection of problems *Propositiones ad Acuendos Juvenes* (Problems to Sharpen the Young). Problems like these may also benefit the elderly – the Alzheimer’s Association in the United States suggests that they may slow the onset of dementia. This is plausible, in that thinking hard boosts blood flow to the brain, and research supports the idea (playing board games and playing musical instruments are even better).