Discrete splittings of the necklace

This paper deals with, direct proofs and combinatorial proofs of the famous necklace theorem of Alon, Goldberg, and West. The new results are a direct proof for the case of two thieves and three types of beads, and an efficient constructive proof for the general case with two thieves. This last proof uses a theorem of Ky Fan which is a version of Tucker's lemma concerning cubical complexes instead of Simplicial complexes.