Dartboard Arrangements

Abstract

This note considers possible arrangements of the sectors of a generalised dartboard. The sum of the $p$th powers of the absolute differences of the numbers on adjacent sectors is introduced as a penalty cost function and a string reversal algorithm is used to determine all arrangements that maximise the penalty, for any $p\ge1$. The maximum value of the penalty function for $p=1$ is well known in the literature, and has been previously stated without proof for $p=2$. We determine it also for $p=3$ and $p=4$.