# 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:

 0 1 2 3 4 5 6 7 8 9 10 11 UR UF UL UB DR DF DL DB FR FL BL BR

If the UD-Slice edges are in their home positions, the UDSlice coordinate is 0.

Here are a few examples:

 0 1 2 3 4 5 6 7 8 9 10 11 c = 0 x x x x

 0 1 2 3 4 5 6 7 8 9 10 11 c=1 x x x x C(8,0)

 0 1 2 3 4 5 6 7 8 9 10 11 c = 62 x x x x C(4,0) C(5,0) C(7,1) C(8,1) C(10,2)

 0 1 2 3 4 5 6 7 8 9 10 11 c = 305 x x x x C(2,0) C(3,0) C(5,1) C(6,1) C(7,1) C(10,3) C(11,3)

 0 1 2 3 4 5 6 7 8 9 10 11 c = 494 x x x x C(4,3) C(5,3) C(6,3) C(7,3) C(8,3) C(9,3) C(10,3) C(11,3)

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 reduces 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