In 1966, Claude Berge proposed the following sorting problem. Given a string
of n alternating white and black pegs on a one-dimensional board consisting
of an unlimited number of empty holes, rearrange the pegs into a string
consisting of β2nββ white pegs followed immediately by
β2nββ black pegs (or vice versa) using only moves which
take 2 adjacent pegs to 2 vacant adjacent holes. Avis and Deza proved that the
alternating string can be sorted in β2nββ such {\em Berge
2-moves} for nβ₯5. Extending Berge's original problem, we consider the
same sorting problem using {\em Berge k-moves}, i.e., moves which take k
adjacent pegs to k vacant adjacent holes. We prove that the alternating
string can be sorted in β2nββ Berge 3-moves for
nξ β‘0(mod4) and in β2nββ+1 Berge 3-moves for
nβ‘0(mod4), for nβ₯5. In general, we conjecture that, for any k
and large enough n, the alternating string can be sorted in
β2nββ Berge k-moves. This estimate is tight as
β2nββ is a lower bound for the minimum number of required
Berge k-moves for kβ₯2 and nβ₯5.Comment: 10 pages, 2 figure