Another Five Numbers [Simon Singh's Numbers]

Not long ago auctions seemed to be the preserve of either the mega-rich, bidding for Van Goghs at some plush auction house, or the shady car-dealer, paying cash-no-questions-asked for vehicles of dubious provenance. However, the advent of the Internet and David Dickinson has changed this. Auction web-sites allow the average punter to buy and sell pretty much anything, whilst an army of Bargain Hunt devotees can now happily tell their Delft from their Dresden.

But this auctioneering is just the tip of the iceberg. In 2000, the UK government received a windfall of around £23 billion from its auction of third generation (3G) mobile phone licences. This astronomical sum wasn't the result of corporate bidders 'losing their heads', but a careful strategy designed to maximise proceeds for the Treasury.

Its architect was Professor Ken Binmore of University College London. He devised an auction, seeped in a branch of mathematics called game theory. Game theory deals with player's tactics. In any given game, a participant develops a strategy that incorporates their own strengths and goals, and those perceived in their opponent(s). Incomplete information and 'bluff' can make things more complicated, and the balance shifts from a purely mathematical approach to one involving greater psychology. But game theory doesn't just apply to cards and Tiddlywinks. In the 1950s, mathematicians started to use these principles to study the economy. One such proponent was Princeton University's John Nash. Immortalised by an Oscar-winning Russell Crowe in the film 'A Beautiful Mind', Nash helped revolutionize game theory with the 'Nash equilibrium'.

Nash focussed on 'non-zero sum' games. These occur when all sides can win or lose, unlike traditional 'zero-sum' games like poker, where one person's victory simultaneously heralds the opponent's defeat. 'Nash equilibrium' occurs when competing strategies achieve a win-win compromise. All participants realise that the end result might not be in their best individual interest, but collectively it suits all. Applied to the real world, many economic transactions fall into the 'non-zero sum' category.

The practical application of these principles came into their own for the UK 3G licence auction. Binmore's remit was to devise a mechanism that would leverage the government's goals of maximising income and encouraging new blood into the industry. Traditionally, such licences had been tendered arbitrarily, based on intuitive rather than mathematical considerations. This meant low revenues, and licences going to the wrong companies.

Binmore's application of game theory ensured this didn't happen. He devised an auction with game rules engineered to achieve the government's objectives, but which would also generate a 'Nash equilibrium' or win-win for all. Critics have argued that the phone companies paid over the odds but Binmore is more circumspect, arguing that they paid what they knew they could recoup from future profits.

In the case of mobile phones, game theory has proved a big win for the government. But for the rest of us, it just means more annoying ring tones on the bus journey home.

When 3G phone licences were sold, game theory was used to boost proceeds for the Treasury.

Simon Singh takes a quirky look at some of the most important numbers in mathematics.

Sir Walter Raleigh was a poet, adventurer and all-round Elizabethan scallywag. In between searching for El Dorado and harrying the Spanish fleet, he is credited with introducing the humble potato to England. He was also the first Brit to seriously go over their Duty Free tobacco allowance on his return from the Americas.

One of his more obscure contributions to posterity however, lies in mathematics. Raleigh wanted to know if there was a quick way of estimating the number of cannonballs in a pile.

In 1606 this problem was presented to German astronomer, Johannes Kepler, who took it on but adapted it significantly. His concern wasn't with how many cannonballs, but with how to pack them in the most efficient way.

Kepler experimented with different ways of stacking spheres. He concluded that the 'face-centred cubic lattice' was the most efficient. If you arrange 100 oranges (cannonballs were replaced by oranges for convenience's sake) in a flat layer of 10x10 and then place a similar layer directly on top, you have created a 'simple cubic lattice'. Provided your oranges haven't rolled apart, your pile has a packing efficiency of only 52% - you're effectively stacking as much air as oranges.

With Kepler's 'face-centred cubic lattice' the first layer of oranges is formed in the same way you would spread penny coins on a desk to cover it leaving the least amount of gaps. Nature seems to dictate that a penny, surrounded by 6 others in a honeycomb arrangement, is best. Replicate this with your oranges. Then for the second layer, place your fruit in the 'dimples' created by the honeycomb beneath. Each successive layer is then built in the same way so the pile forms a pyramid. Using this method, Kepler calculated that the packing efficiency rose to 74%, constituting the highest efficiency you could ever get. But, how to prove it? There are untold ways that oranges can be stacked and any of these might yield a higher percentage.

The conjecture dogged mathematicians for centuries. The general feeling was that a proof might never appear.

Then in 1998, American Professor Thomas Hales stunned the world of mathematics. Aided by his research student Samuel P. Ferguson, Hales devised a monster equation based on a cluster of 50 spheres. The equation and its 150 variables expressed every conceivable arrangement of these spheres. They then used computers to confirm that no combination of variables led to a packing efficiency higher than 74%. 250 pages of argument and 3 gigabytes of computer files proved them and Kepler right.

So, 400 years after posing his question, Sir Walter can finally be given an answer of sorts. It might not have helped with estimating how many cannonballs he had left to face down the Spanish, but it would have allowed him to pack his spuds most efficiently.

Is the 'face-centred cubic lattice' the most efficient way of stacking spheres?

Simon Singh takes a quirky look at some of the most important numbers in mathematics.