Necklace Splitting Problem

In mathematics, and in particular combinatorics, the necklace splitting problem arises in a variety of contexts including exact division; its picturesque name is due to mathematicians Noga Alon and Douglas B. West.

Suppose a necklace, open at the clasp, has k ·n beads. There are k · a_i beads of colour i, where 1 ≤ i ≤ t. Then the necklace splitting problem is to find a partition of the necklace into k parts (not necessarily contiguous), each of which has exactly a_i beads of colour i; such a split is called a k-split. The size of the split is the number of cuts that are needed to separate the parts (the opening at the clasp is not included). Inevitably, one interesting question is to find a split of minimal size.

Alon explains that

the problem of finding k-splittings of small size arises naturally when k mathematically oriented thieves steal a necklace with k · a_i jewels of type i, and wish to divide it fairly between them, wasting as little as possible of the metal in the links between the jewels.

If the beads of each colour are contiguous on the open necklace, then any k splitting must contain at least k − 1 cuts, so the size is at least (k − 1)t. Alon and West use the Borsuk-Ulam theorem to prove that a k-splitting can always be achieved with this number of cuts. Alon uses these and related ideas to state and prove a generalization of the Hobby–Rice theorem.