Dartboard arrangements (Q5942558)

Summary: 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 \(p\geq 1\). 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 \(p=4\).