Two thieves steal a necklace with 2 kinds of jewels — split it fairly with the fewest cuts
A necklace has beads of k different colors, strung in any order, with an even number of beads of each color. Two thieves want to split it so each gets exactly half the beads of every single color — but cutting between beads destroys the links, so they want to use as few cuts as possible. How many cuts does it always take, no matter how the colors are arranged?
Reveal the answer
Exactly k cuts are always enough, and sometimes necessary — just one cut per color, however tangled the arrangement looks. Noga Alon proved the general version, for any number of thieves, in 1987 using an extension of the Borsuk-Ulam theorem, a tool from topology, applied to a combinatorics problem. It's a striking example of 'fair division' math: a guarantee that perfect equity is always achievable with surprisingly little cutting.
— Noga Alon, Splitting Necklaces — Advances in Mathematics, 1987