## Routes on a Cube

If you can't visit a corner more than once in a route, how many routes exist on a cube between:
• One corner and an adjacent corner?
• One corner and a corner diagonally opposite on the same face?
• One corner and the corner diagonally opposite through the cube?
• The middle of one side and the middle of the side diagonally opposite through the cube?
Source: Based on a puzzle in the recent PQRST Puzzle Competition.
Soultions were received from Alan O'Donnell, Kirk Bresniker, Joseph DeVincentis, Jozef Hanenberg, Dan Chirica.  Kirk's solution lists each route below.
```     1--------5
/|       /|
8 |      / |
/  |     /  |
0--------4   |
|   2----|---6
|  /     |  /
| /      | 9
|/       |/
3--------7

1. 0 -> 1 = 15 routes
2. 0 -> 5 = 16 routes
3. 0 -> 6 = 18 routes
4. 8 -> 9 = 28 routes

KMB

p.s. here are the details:

1--------5
/|       /|
/ |      / |
0--------4  |
|  2-----|--6
| /      | /
|/       |/
3--------7
01
0321
032651
03267451
037451
03745621
037621
037651
0451
045621
04567321
047321
04732651
047621
047651
routes 15 from 0 to 1
1--------5
/|       /|
/ |      / |
0--------4  |
|  2-----|--6
| /      | /
|/       |/
3--------7
01265
0126745
0123745
0123765
015
03215
03265
0326745
03745
0376215
03765
045
0473215
0473265
0476215
04765
routes 16 from 0 to 5
1--------5
/|       /|
/ |      / |
0--------4  |
|  2-----|--6
| /      | /
|/       |/
3--------7
0126
01237456
012376
01547326
015476
0156
03215476
032156
0326
03745126
037456
0376
045126
04512376
0456
04732156
047326
0476
routes 18 from 0 to 6
1--------5
/|       /|
8 |      / |
/  |     /  |
0--------4   |
|   2----|---6
|  /     |  /
| /      | 9
|/       |/
3--------7
803215479
80321569
803265479
803269
8037451269
80374569
80379
80451269
804512379
804562379
804569
8047321569
80473269
80479
8126540379
81265479
81269
812304569
81230479
812374569
812379
815403269
81540379
815473269
815479
8156230479
81562379
81569
routes 28 from 8 to 9

```

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