Solution:
There are four general types we can make: (6,12),(7,11),(8,10),(9,9).
First type: (6,12), there are 18 possible ways to choose those 6, which are going to be cut from the necklace: choose a direction to count the beads, and choose a starting position (between 2 beads). There are exactly 18 starting positions, since there are 18 spaces between the beads. One cut is equivalent for a pair we can make.
Second type: (7,11), with the same reasoning, there are 18 possible ways to cut the necklace.
Third type: (8,10), with the same reasoning, there are 18 possible ways to cut the necklace.
Fourth type: (9,9), The same reasoning cannot be applied again, since half of the cuts would be exactly the same as the other half. So there are 9 possible cuts, exactly one or each axis of symmetry for the necklace.
The solution will be the sum these values: 18+18+18+9=63
