163998

Solution:
We noticed that you can only do a series of changes once. You can see the possibilities for 5 cubes:
1 and 4
2 and 3
You can only do a series of changes once therefore we are not going to count 3 and 2 or 4 and 1 as a possibility.
For 5 cubes there are 2 possibilities.

We noticed that when the number of cubes is odd, you take away one and then divide by two in order to find the number of breaks. If it is even you just divide by two.
Our prediction is that for all the even numbers of cubes you have to divide by 2 and for all the odd numbers of cubes you take away 1 and then divide by 2. This works because you can only change the amount of cubes on each side counting down to half way because otherwise you will be repeating numbers.