On Langford’s Problem (II)

The problem is to arrange the numbers 1, 1, 2, 2, …, n, n in a sequence (without gaps) in such a way that for r = 1, 2, …, n the two r’s are separated by exactly r places; for example41312432.Priday has shown in the preceding paper that for every n there exists either such a perfect sequence (as he calls it) or else a hooked sequence, with a gap one place from one end, for example345131425*2.