Dartboard Arrangements
Several authors have considered the question of how best to arrange the numbers around a dartboard, for example see Eiselt and Laporte[2]. Bearing in mind that the game of darts requires players to aim at particular sectors of the board, and that reasonably good players tend to hit either the desired sector or the adjacent one, then it seems reasonable to direct attention to numbers on adjacent sectors.
Suppose we decide that the bigger the difference between numbers of adjacent sectors, the bigger the risk to the player, and so the more exciting the game of darts. Then one reasonable measure of the "excitement level" of a dartboard is the sum of the differences between all pairs of adjacent numbers. Another alternative measure that particularly emphasises bigger differences is the sum of the squares of the differences between all pairs of adjacent numbers. The standard dartboard (below left) does pretty well with respect to both of these measures, but is not the best possible. The arrangement illustrated below right, is an optimal arrangement for both measures of excitement.
![[Original dartboard]](original-colour.gif)
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![[Dartboard as produced by the greedy algorithm]](greedy-colour.gif)
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The arrangement above right was produced by a greedy algorithm. This algorithm will work for any collection of numbers (not just the standard 1 ... 20, and it doesn't matter whether some numbers are duplicated), and is performed as follows:
- Imagine you have a physical dartboard in your hands, with spaces for the numbers to go round the edge, and some numbers to glue on. Take the largest number you have, and glue it onto one of the sectors. Now at each stage of the algorithm, consider all the numbers you have left, and choose the number remaining that makes the biggest difference when you place it next to a number already on the board. Glue it in place there. Then repeat until all the numbers are used up.
Hmm. Well. It may be the most exciting dartboard mathematically, but somehow the symmetry of the whole thing is a bit boring, really, compared to the original!
This work has been written up as a paper [1], where you can get the full details.
References
[1] S. A. Curtis:
Darts and Hoopla Board Design
[2] H. A. Eiselt and Gilbert Laporte:
A Combinatorial Optimization Problem Arising in Dartboard Design
Journal of the Operations Research Society, Vol. 42, No.2,
1991, pp. 113--118