Fairly dividing a mixed necklace seems to require separating beads one by one. The question is whether the continuity of the circle itself can do the work.
The Necklace Split Fairly
Riddle statement
Two thieves steal a circular necklace with red and blue beads.
There is an even number of red beads and an even number of blue beads.
They want to cut the necklace with as few cuts as possible and divide the pieces between them, so that each thief receives:
half of the red beads;
and half of the blue beads.
How many cuts are always enough?
Show solution
Solution
Two cuts are always enough:
cuts.
With two cuts in a circular necklace, you get two pieces: an arc and the complementary arc.
The question is to prove that there is some arc containing exactly half of the red beads and exactly half of the blue beads.
Since the number of red beads and the number of blue beads are both even, the total number of beads is also even.
Imagine a window that covers exactly half of the beads of the necklace.
That window contains some number of red beads.
Now slide the window one position at a time around the necklace, always covering half of the beads.
If at some moment it contains exactly half of the red beads, then it also contains exactly half of the blue beads, because it contains half of the total number of beads.
Then we are done.
If that does not happen at the start, observe this:
when the window is in one position, the opposite arc contains the beads that the window does not contain.
Therefore, if one window contains more than half of the red beads, the opposite window contains less than half.
As the window slides step by step, the number of red beads inside it can only change gradually: when one bead leaves and another enters, the number of red beads changes by at most 1.
So, to go from “more than half” to “less than half,” at some point it must pass through “exactly half.”
Cut the necklace at the two ends of that arc.
One thief receives that arc. The other receives the rest of the necklace.
Both receive half of the red beads and half of the blue beads.
Answer: two cuts are always enough.