Ranking the Puzzles by Number of Combinations
---------------------------------------------

  In general, puzzles without a known mechanism exist as an "idea"
only. Ideas for puzzles are welcome, and with GAP's help it is
not difficult to find the number of combinations. The list is
admittedly mixing puzzles of different types, e.g. Rubik's type
and sliding tile type, but we are also interested in the
computibility of finding God's Algorithm. Hopefully this list
will provide a checklist of the feasibility of using a computer
to sift through all the combinations.

    Name                     Combinations   Mechanism
    ----                     ------------   ---------

 1. Rubik's Wahn (5x5x5)     2.8*10^74      Udo Krell
 2. Megaminx                     10^68      Kersten Meier, Ben Halpern
 3. Rubik's Revenge (4x4x4)  7.4*10^45      Unknown
 4. Alexander's Star         7.2*10^34      Adam Alexander
 5. Pyraminx Hexagon (A)     2.9*10^30      No known mechanism
 6. VIP Sphere               4.4*10^26      Unknown
 7. Impossi-ball             2.4*10^25      Wolfgang Kuppers
 8. Picture Cube (3x3x3) (E) 8.8*10^22      Erno Rubik, Dan Hoey
 9. Calendar Cube (3x3x3)(F) 4.4*10^22      Marvin Silbermintz
10. Rubik's Cube 4th Dim.(D) 1.1*10^22      Erno Rubik
11  Rubik's World (G)        2.7*10^21      Erno Rubik
12. Rubik's Cube (3x3x3)     4.3*10^19      Erno Rubik
13. Pyraminx Octahedron      8.2*10^18      Unknown
14. Octagon                  5.4*10^18      Unknown
15. Christoph's Jewel (B)    2.0*10^15      Christoph Bandelow
16. Master Pyraminx (C)      4.5*10^14      Uwe Meffert
17. Barrel                   2.7*10^14      Gumpei Yokoi
18. Square 1                 1.2*10^13      Dr. Vojtech Kopsky
19. 15 Puzzle                1.0*10^13      Sam Lloyd
20. Missing Link             8.2*10^10      Marvin Glass & Associates
21. Trillion                 1.0*10^9       Unknown
22. Rubik's Domino (3x3x2)   4.0*10^8       Erno Rubik
23. Picture Skewb            1.0*10^8       Tony Durham, Uwe Meffert
24. Pyraminx                 7.6*10^7       Uwe Meffert
25. Dino Cube #1 (H)         1.9*10^7       Erno Rubik??
26. Halpern's Tetrahedron    3.7*10^6       Ben Halpern, Kersten Meier
27. Pocket Cube (2x2x2)      3.6*10^6       Erno Rubik
28. Skewb                    3.1*10^6       Tony Durham
29. Snub Pyraminx            9.3*10^5       Uwe Meffert
30. Simple Octahedron        5.0*10^4       No known mechanism
31. Dino Cube #2 (I)         4.2*10^4       Erno Rubik??
32. Rubik's Layer (3x3x1)         192       No known mechanism

(A) This assumes 90 degree turns for the faces adjacent to the top face
(B) This is a snub Pyraminx Octahedron (Octahedron minus the tips)
(C) This assumes a Pyraminx visually the same as a regular pyraminx
     with rotations about the 4 vertices AND 6 edges.
(D) Yet another picture cube that does not have 4 orientations for
     each of it's 6 centres.
(E) This assumes a cube with centres which can show 4 distinct
     orientations for all 6 centres, and the only example I know
     of is Dan Hoey's Tartan Cube.
(F) Interestingly, due to the 'O' character on one of the centres
     of the Calendar Cube having only 2 distinct orientations,
     this picture cube has only half of the number of combinations
     of the Tartan Cube.
(G) This one has 3 blank centres and 3 centres with 4 orientations.
(H) Standard Dino Cube colouring with each side with a different colour.
(I) Easy Dino Cube colouring has each tetrad with a different colour.

Collated by Mark Longridge, March 22, 1996.