Necklace splitting is a division problem for a finite sequence of colored beads. Cuts are made between beads,
and each resulting contiguous piece is assigned to one of 
recipients. A fair split gives every recipient the same number
of beads of each color. The bead order is fixed. The rotational equivalence used
to define a necklace is not part of this problem (Alon
1987, p. 247; Numberphile 2019).
Suppose there are 
colors and 
beads of color 
,
where 
,
,
and the 
are positive integers, 
, and 
. The necklace-splitting theorem
guarantees a fair split using at most 
cuts (Alon 1987, p. 247). Every recipient receives
exactly 
beads of color 
.
The pieces assigned to one recipient need not be consecutive.
The bound is sharp. If each color occupies one contiguous block, then giving all 
recipients some of that color requires at least 
cuts inside its block. The 
color blocks therefore require at least 
cuts in total. The general existence proof
uses an extension of the Borsuk-Ulam theorem
(Alon 1987, p. 247).
