The UDSlice Coordinate |
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
The problem reduces to the following: There are 12 positions, 4 of them are occupied. Assign a unique number c from 0...494 to each possible configuration. In Cube Explorer we have the following assignment between the position numbers and the edges:
If the UD-Slice edges are in their home positions, the UDSlice coordinate is 0. Here are a few examples:
The binomial coefficients C(n,k) written under the free places add up to the coordinate. n is the position, k depends on how much occupied places are to the right of the position. The maximal value for k is 3, each occupied place to the right increases k by one. The minimal value for k is 0, so the free places at the leftmost positions do not count. C(2,0) + C(3,0) + C(5,1) + C(6,1) + C(7,1) + C(10,3) + C(11,3) = 1 + 1 + 5 + 6 + 7 + 120 + 165 = 305 C(4,3) + C(5,3) + C(6,3) + C(7,3) + C(8,3) + C(9,3) + C(10,3) + C(11,3) = 4 + 10 + 20 + 35 + 56 + 84+ 120+ 165 = 494 |