On Permutations Whose Adjacent-Sum Reciprocals Sum to One

Abstract

In this paper, we give a constructive proof that for any integer n>7 (resp. n>6), the numbers 1,2,...,n can be arranged in a circle (resp. in a line) such that the sum of the reciprocals of all adjacent sums equals 1. The above conjecture was originally posed by Zhi-Wei Sun. Partitioning the integers according to n mod 12 into nine residue classes, we prove a recursive proposition and inductively construct new examples. Main lemmas and propositions in introduction are proved in Section 2. The rest of the cases are handled by explicit constructions in appendix.