Abstract
A sequence { d, d+1,…, d+ m−1} of m consecutive positive integers is said to be perfect if the integers {1, 2,…, 2 m} can be arranged in disjoint pairs {( a i, b i): 1⩽ i⩽ m} so that { b i − a i : 1⩽ i⩽ m}={ d, d+1,…, d+ m−1}. A sequence is hooked if the set {1, 2,…, 2 m−1 2 m+1} can be arranged in pairs to satisfy the same condition. Well known necessary conditions for perfect sequences are herein shown to be sufficient. Similar necessary and sufficient conditions for hooked sequences are given.
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