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{\bf Necklace Bisection with One Cut Less than Needed}
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{\bf G\'abor Simonyi}
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A well-known theorem of Goldberg and West states that two thieves can
always split a necklace containing an even number of beads from each
of $k$ types fairly by at most $k$ cuts. We prove that if we can use
at most $k-1$ cuts and fair splitting is not possible then the thieves
still have the following option. Whatever way they specify two
disjoint sets $D_1, D_2$ of the types of beads with $D_1\cup
D_2\neq\emptyset$, it will always be possible to cut the necklace
(with $k-1$ cuts) so that the first thief gets more of those types of
beads that are in $D_1$ and the second gets more of those in $D_2$,
while the rest is divided equally. The proof combines the simple proof
given by Alon and West to the original statement with a variant of the
Borsuk-Ulam theorem due to Tucker and Bacon.
\bye