Month: August 2002

Buying a specific set of 27 tickets for the UK National Lottery will mathematically guarantee that you win something.

Buying 27 tickets ensures a win in the UK National Lottery

You can guarantee a win in every draw of the UK National Lottery by buying just 27 tickets, say a pair of mathematicians – but you won’t necessarily make a profit.

While there are many variations of lottery in the UK, players in the standard “Lotto” choose six numbers from 1 to 59, paying £2 per ticket. Six numbers are randomly drawn and prizes are awarded for tickets matching two or more.

David Cushing and David Stewart at the University of Manchester, UK, claim that despite there being 45,057,474 combinations of draws, it is possible to guarantee a win with just 27 specific tickets. They say this is the optimal number, as the same can’t be guaranteed with 26.

The proof of their idea relies on a mathematical field called finite geometry and involves placing each of the numbers from 1 to 59 in pairs or triplets on a point within one of five geometrical shapes, then using these to generate lottery tickets based on the lines within the shapes. The five shapes offer 27 such lines, meaning that 27 tickets bought using those numbers, at a cost of £54, will hit every possible winning combination of two numbers.

The 27 tickets that guarantee a win on the UK National Lottery

Their research yielded a specific list of 27 tickets (see above), but they say subsequent work has shown that there are two other combinations of 27 tickets that will also guarantee a win.

“We’ve been thinking about this problem for a few months. I can’t really explain the thought process behind it,” says Cushing. “I was on a train to Manchester and saw this [shape] and that’s the best logical [explanation] I can give.”

Looking at the winning numbers from the 21 June Lotto draw, the pair found their method would have won £1810. But the same numbers played on 1 July would have matched just two balls on three of the tickets – still a technical win, but giving a prize of just three “lucky dip” tries on a subsequent lottery, each of which came to nothing.

Stewart says proving that 27 tickets could guarantee a win was the easiest part of the research, while proving it is impossible to guarantee a win with 26 was far trickier. He estimates that the number of calculations needed to verify that would be 10¹⁶⁵, far more than the number of atoms in the universe. “There’d be absolutely no way to brute force this,” he says.

The solution was a computer programming language called Prolog, developed in France in 1971, which Stewart says is the “hero of the story”. Unlike traditional computer languages where a coder sets out precisely what a machine should do, step by step, Prolog instead takes a list of known facts surrounding a problem and works on its own to deduce whether or not a solution is possible. It takes these facts and builds on them or combines them in order to slowly understand the problem and whittle down the array of possible solutions.

“You end up with very, very elegant-looking programs,” says Stewart. “But they are quite temperamental.”

Cushing says the research shouldn’t be taken as a reason to gamble more, particularly as it doesn’t guarantee a profit, but hopes instead that it encourages other researchers to delve into using Prolog on thorny mathematical problems.

A spokesperson from Camelot, the company that operates the lottery, told New Scientist that the paper made for “interesting reading”.

“Our approach has always been to have lots of people playing a little, with players individually spending small amounts on our games,” they say. “It’s also important to bear in mind that, ultimately, Lotto is a lottery. Like all other National Lottery draw-based games, all of the winning Lotto numbers are chosen at random – any one number has the same and equal chance of being drawn as any other, and every line of numbers entered into a draw has the same and equal chance of winning as any other.”

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Matthew Sparkes*

Knot patterns (which might not actually represent knots, but just braids or plaits), like the one shown here, are common in Celtic design.

One way to create knot patterns is to use a set of tiles that are put together following a few basic rules. This is not, as far as I know, a standard way to create these patterns – the book by Aidan Meehan, provides a technique for drawing the patterns by hand, rather than laying them out as tiles.

A wide range of knot patterns, including the one shown above, can be created using the set of six tiles shown below. These were constructed using Geometer’s Sketchpad.

The rules for putting these together are reasonably clear, but can be formalized. You can even come up with a notation for the tiles and formal rules for assembling them into knot patterns.

When laying down the tiles, you have the option of rotating them – some of the tiles don’t alter with rotation, some of them can be placed in two ways, some in four.

A nice property of this set is that you can’t paint yourself into a corner when using it – you can always find a tile that will compose with the pattern that you have started. This property means that this set of tiles is closed, in a certain sense.

A wider range of patterns can be created if you add tiles like the ones below to your set.

Unfortunately, when you include these tiles, your set is no longer closed. An open question (at least as far as I know) is – how many more tiles would you need to add in order to create a closed set of tiles that includes these? Also, what would these tiles look like? Do they lead to “reasonable” looking knots?

The knot pattern below (better described as a woven set of three links) was made using tiles including the ones from the “extended” set above.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to dan.mackinnon*