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