Arranging numbers on circles to reach maximum total variations (Q2372892)

Summary: The dartboard problem is to arrange \(n\) numbers on a circle to obtain maximum risk, which is the sum of the \(q\)th power of the absolute differences of adjacent numbers, for \(q\geq 1\). \textit{S. A. Curtis} [Darts and hoopla board design, Inf. Process. Lett. 92, 53--56 (2004)] showed that the dartboard problem admits a greedy algorithm. We generalize the dartboard problem by considering more circles and the goal is to arrange \(kn\) numbers on \(k\) circles to obtain the maximum risk. In this paper, we characterize an optimal arrangement for \(k=2\) and show that the generalized dartboard problem also admits a greedy algorithm.