Block partitions of sequences

Given a sequence A = (a1, …, an) of real numbers, a block B of A is either a set B = {ai, ai+1, …, aj} where i ≤ j or the empty set. The size b of a block B is the sum of its elements. We show that when each ai ∈ [0, 1] and k is a positive integer, there is a partition of A into k blocks B1, …, Bk with |bi−bj| ≤ 1 for every i, j. We extend this result in several directions. © 2015, Hebrew University of Jerusalem