1. If the diamonds and rubies are evenly spaced, then we will only
have to cut once. This is the trivial case.

The necklace above only needs
1 cut to equally cut up the necklace! 
2. Now suppose that all the diamonds are on one side
one side of the necklace and all the rubies are on the other. To
make sure that each thief received an even number of each, we would
have to cut the necklace twice.

The above necklace needs to
be cut twice to evenly distribute the diamonds and rubies. 
3. Will we ever have to cut it more than twice?
NO!
The answer lies in the Ham Sandwich Cut
theorem!
Suppose we drape the necklace over a parabola.
We know that any line that cuts a parabola cuts it at most two
places. Therefore, if we find the ham sandwich cut of the
necklace, while it is draped over the parabola, the most we will ever
have to cut it is twice!

Necklace draped over a
parabola, with a hamsandwich cut through it! 