# The optimal dartboard

To many people, darts is already a fiendishly difficult game, but now David Percy, Professor of Mathematics at the University of Salford, has made it even harder.

He has created the ‘Optimal Dartboard’ by re-arranging the numbers in such a way that they minimise the chances of a high-scoring random throw. As well as alternating odd and even numbers, Percy’s board also groups the numbers in similar scoring clusters.

“People have been looking at this problem for the last 30 years, but only as a bit of fun,” Professor Percy explains.“ But they were using the wrong optimality measure and also ignored the importance of having alternating odd and even numbers. The idea is to make the dartboard as hard as possible.

“What they were looking at was trying to make the difference between adjacent numbers as large as possible, so that if you miss a big number, you get a small number. But what you actually need to do is look at the total score of two adjacent numbers and try and make that as similar as possible all the way around the board.”

Percy analysed the traditional board by totting up the average scores for different sections of the board and laid out the plans for the new board in a paper published in the journal Mathematics Today.

“The average of all the numbers on a dartboard is 10.5, so put simply, you can gain by targeting those sections where the average is higher than that,” he explains. He showed that it was easier for a lucky throw to score highly in the left-hand side of the board, where there is a section containing 12-9-14, with an average of 11.7, as well as 16-7-19, with an average score of 14.

With players traditionally trying to get from 301 or 501 down to zero, the layout of the optimal board also makes a big difference at the end of the game, when players have to go out on a double. If they’re on an odd number, they’ll need to hit another odd number before being able to go for a double. On the old board, even the worst player had a good chance of hitting one close to the bottom of the board, with its cluster of four adjacent odd numbers 7, 19, 3 and 17.

Percy’s new alternating arrangements has put pay to this, along with another favourite, double 16. On the traditional board 16 and 8 are next to each other, so if you miss double 16 and hit single 16 instead, you then require double 8, theoretically making it easier to hit as you already have your sights set on that part of the board. Similarly, the 10 and 6 are no longer adjacent, which makes finishing harder when players are on an even number.

The dartboard was trialled at the BDO World Professional Darts Championships in January by darts legend Bobby George, and several prototypes have been developed by leading dartboard manufacturer Winmau. However, there are no plans to replace the traditional board, the numbering of which is thought to date back to the 19th century.

Modestly, Percy adds: “Maths is often about finding the optimal solution to a problem and this is the optimal solution to that particular problem. I lay no further claims – it was just an interesting piece of work!”