It was the moment of which every schoolboy dreams. The call-up to play football for England against the Germans. The publication of my recent book, Finding Moonshine, qualified me last year for inclusion in the illustrious England writers football team. But a 6-1 drubbing at the hands of the Germans has left us licking our wounds . . . me in particular, as I had my writing hand broken by a hulking German playwright.

He certainly knew where to hit a writer where it hurts most. So we are all waiting for the chance to make amends at the rematch, which will take place in England on June 13. This time we have a secret weapon. Aidy Boothroyd, the manager who took Watford to the Premier League and the semi-final of the FA Cup, has volunteered to help the English cause and last week gave our team two full days of training. The sessions were a revelation. Boothroyd exposed the game for what it is: geometry in motion. We learnt a whole range of shapes to apply at different points in the game: the banana, the diamond, the flower and the hedgehog. The more philosophical among the team were disappointed not to have a fox to go with the hedgehog, although I don’t think Boothroyd’s bedside reading includes the essays of Isaiah Berlin.

I’m certain it must include books on higher mathematics, though, because the shapes that were flying across the pitch left my mathematical brain racing. Kids are often taught to “make triangles” when playing football, so that the player on the ball always has two passing options, but under Boothroyd’s tutelage when one player had the ball the whole team had to adjust its formation to allow the ball to move freely through the team. It was, I realised, the sporting equivalent of a mathematical network: the 11 players represented the hubs, while the passing options that didn’t risk being intercepted by opposing players were the links between them.

William Hamilton, the 19th-century mathematician, was intrigued by such networks, so much so that he patented a game based on them. The Irishman’s puzzle took a dice-like shape made from 12 pentagonal faces, but with one of the faces removed, so that it could be flattened on to a two-dimensional board. Starting at one corner, or vertex, of the shape, players had to trace a circuit round the edges of it that visited all the other 19 vertices only once and returned to the starting point.

Hamilton sold the game to a dealer in 1859 for £25, but it didn’t capture the public’s imagination and flopped.

The puzzle, though, intrigued mathematicians, who began to investigate which networks had these Hamiltonian circuits. It turned out that you can trace a path round the vertices of any Platonic solid such as the dodecahedron and even the Archimedean solids, shapes that have a selection of different symmetrical faces.

More intriguingly, it was discovered that 11 is the smallest number of points for which there exist “polyhedral” networks that have no Hamiltonian circuit (the word polyhedral here refers to a network that can actually be wrapped around to make the faces of a ball). Transferring such a network to the field of play, there is no way to pass the ball around all the players without one team member making two passes.

So, from a mathematical point of view, there is something intrinsically interesting about the 11-a-side game that doesn’t occur for five-a-side or seven-a-side matches. Boothroyd probably knows this already, but it was an exciting revelation for someone who has spent years with little connection between my right foot and my right brain. The great Dennis Bergkamp was right when he said that every kick of the ball requires a thought. Then again, as Cristiano Ronaldo has shown with his careful inspections of the ball before placing it for a free kick, the leather sphere itself is also worthy of contemplation.

During our break for orange segments, I began obsessively trying to trace a Hamiltonian circuit round the football’s network of seams. The classic football is constructed from 12 pentagons and 20 hexagons. What Boothroyd had been trying to get our team to knock gracefully around the park was, perhaps, the most famous Archimedean solid of all.

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