The following article appeared in
*Cubism For Fun*
**36** (February 1995) pp. 18-20.

A Czech Check Problem by Michael Reid In [1] David Singmaster gives the following problem, attributed to the Czech weekly Mlady Svet. Find positions of the cube that have exactly 8 squares correct on every face. Since I haven't seen a solution to this problem, I thought I'd share mine. First let us distinguish between "pattern" and "position". A "position" refers to one of the 43252003274489856000 states of the cube, while a "pattern" is an equivalence class of positions under the group of symmetries (rotations and reflections) of the cube. For example, "cube within a cube" describes a single pattern, whereas there are 8 positions representing this pattern. We will show that there are 56 patterns and 2248 positions which have exactly 8 squares correct on every face, and for each pattern we'll give a representative. Consider the 6 squares that are on the wrong face. We'll first consider the various combinations of centers, edges and corners to which these 6 squares belong. First note that if any centers are out of place, then either 4 or 6 are. Therefore the centers contribute either 0, 4 or 6 squares to the total of 6. Also, any corner that is out of place (or in place, but in the wrong orientation) contributes at least 2 squares. Furthermore, if any corner is out of place, then at least two are, so the corners contribute either 0 or at least 4 squares. Now we see that the only combinations possible are: 1) centers 2) centers and edges 3) corners 4) corners and edges 5) edges The first four cases are easy to handle. For case 1, the only pattern is 6 dots: 1. R L' F B' U D' R L' C_ufr (U, F, R) (D, B, L) For case 2, we must have 4 centers and 2 edges out of place. Again, there is only one suitable pattern. 2. R2 L' B' R' U F D' R F2 B L' F L D' C_d (F, R, B, L) (UR, DR) For case 3, there are two possibilities: two corners rotated in place, or a three cycle of corners. Each possibility yields exactly one suitable pattern. 3. F D2 F' L U2 L' F D2 F' L U2 L' (UFR-) (DLB+) 4. R' F2 R F2 R U2 R' F2 R' F2 R U2 (ULF, URB, DRF) For case 4, the only possible cycle structure is a two cycle of corners and a two cycle of edges. Then it is easy to see that there is only one suitable pattern. 5. F2 R' U2 R2 F2 R F2 U2 R2 F2 U2 R U2 R2 (UFR, UBL) (UR, DR) Case 5 is the most interesting and accounts for most of the patterns. Consider different types of cycles of edges. Here (n) denotes an n-cycle and (n+) denotes a flipped n-cycle of edges. cycle # of squares with wrong color (1+) 2 (2) 2 or 4 (2+) 3 or 4 (3), (3+) 3, 4, 5 or 6 (4), (4+) 4, 5, 6 or more (5), (5+) 5, 6 or more (6), (6+) 6 or more These may be combined in many ways to give a possible total of 6 squares, but only a few have both the correct parity and correct edge flip. These combinations are: (1+)(3+), (2)(2), (2)(4), (2+)(2+), (3)(3), (3+)(3+), (3) and (5). We list the patterns here without going into detail about their enumeration. Cycle structure (1+)(3+): 6. L F' L U B L2 B' L' U' F L2 F2 L (LB+) (FD, FU, FR+) 7. inverse of 6 8. F D' B' U' L U2 L2 U' L B D L' (LB+) (FU, FR, FD+) 9. R B D2 B R' L U' B2 R2 L2 D' R2 L (LB+) (UR, UF, DF+) 10. inverse of 9 11. B L' D' F2 L' F' D L D F' B R' B2 L (LB+) (UR, FR, FD+) Cycle structure (2)(2): 12. R2 B R2 L2 F2 R2 L2 B R2 (UF, DF) (RF, LB) 13. B' R' D L2 F2 R2 U L2 B2 R' B (UF, DF) (RF, BL) 14. R2 U' R2 F2 B2 L2 F2 B2 U R2 (UF, UR) (UB, DL) 15. L' U F U' F' L2 U' B' U B L' (UF, UR) (UB, LD) Cycle structure (2)(4): 16. R2 F R2 U2 D2 L2 B' U2 D2 R2 (UF, DF) (FR, BR, BL, FL) 17. B2 U' R2 L2 D R L D2 L2 U2 R' L' (UL, UB) (UF, DF, DR, UR) Cycle structure (2+)(2+): 18. B R' U R' L F' R F' R L' U B' (UL, UR+) (DF, DB+) 19. L' F2 D F' B R' F R' F B' D F L (UF, UR+) (DL, DB+) 20. L2 U' R L' B' R' L U R L' B R' L' (UF, UR+) (DB, DL+) 21. U L' U' D F' U D' L U' D F D' (UF, RF+) (LB, LD+) 22. R' D' R L' F' L' F' R' L D L D2 R (UF, RF+) (LD, LB+) 23. inverse of 22 Cycle structure (3)(3): No solutions. Cycle structure (3+)(3+): 24. B R F B' D' F' D' R' L' D' R L' F L2 D (UF, RF, RU+) (DB, LB, LD+) 25. D R F U D' L' F' R2 U B U' D L' U' B2 D' (UF, UR, FR+) (DB, LB, LD+) Cycle structure (3): 26. L' U' D B D B' U D' R D' R' L (RB, FD, UL) 27. D' L' U D' B2 U' D L' D (RB, FD, LU) Cycle structure (5): 28. U B2 R2 B2 U' B2 R2 B2 U2 (UF, UB, UL, UR, DR) 29. inverse of 28 30. L2 U' F2 B2 D F B' D2 F B' (UB, UL, UF, UR, DR) 31. inverse of 30 32. F2 B2 D B2 L2 B2 D' F2 R2 B2 U2 (UL, UF, UB, UR, DR) 33. inverse of 32 34. R L B2 R' L U' R2 L2 D' R2 (UB, UR, UF, DF, DL) 35. inverse of 34 36. L2 U L2 F2 L2 U' L2 F2 L2 U2 L2 (UR, UB, UF, DF, DL) 37. inverse of 36 38. B' L U B' L' B' U' B' L U B2 (UB, UR, UF, LF, LD) 39. inverse of 38 40. R' L' U L U R B2 L' F B2 U' F' (UR, UB, UF, LF, LD) 41. inverse of 40 42. L2 B' L2 D F2 L2 B2 U' L2 F R2 D2 (UR, UB, UF, LF, DF) 43. inverse of 42 44. F' L2 U' L2 F' L2 U L2 F L2 U L2 (UB, UR, UF, LF, DF) 45. F U' F' U' F U2 F2 U' F U F U' (UR, UB, UF, DF, LF) 46. D R2 F' L2 U2 R2 B R2 U' B2 L2 F2 (LF, UF, UB, UR, DR) 47. inverse of 46 48. R2 D R2 F2 L2 U' R2 B' R2 U2 L2 F (DB, UB, UR, UF, LF) 49. inverse of 48 50. U F2 B2 D R2 D F2 B2 U' L2 U2 (DL, DR, UR, UF, UB) 51. inverse of 50 52. R2 U F2 R U2 B2 D2 L' F2 D' L2 B2 (DL, DF, UF, UR, BR) 53. inverse of 52 54. B' L B2 D2 F2 R F2 L2 U2 F R2 U2 L2 (UL, UR, FR, FD, BD) 55. L U2 F U2 L2 D2 B' U2 R' U2 B2 D2 (DL, FL, FR, BR, BU) 56. L' U' L U D' F' D F B' R' B R (BR, DR, DF, LF, LU) A given position gives (up to) 48 different positions by applying symmetries of the cube. However, these need not be distinct. The original position is fixed by some subgroup H of the group G of symmetries of the cube. Now the positions equivalent to the original position are in one-to-one correspondence with the cosets G / H , and thus their number is |G| / |H| = 48 / |H|. For example, position number 1 has six symmetries, generated by the rotation C_ufr and central reflection. The same is true for positions 3 and 24, so each of the corresponding patterns is represented by 8 different positions. Positions 2, 12, 13 and 16 each have 2 symmetries, generated by the reflection through the U-D plane. Positions 4 and 26 each have three symmetries, generated by the rotation C_ufr. Positions 18, 19, 22 and 23 each have 2 symmetries, generated by the rotation C_fl. Position 20 has 2 symmetries, generated by central reflection. Position 25 has 6 symmetries, generated by the rotations C_ufr and C_fl. The remaining 41 positions have only the identity symmetry. Therefore, we have 3 * 8 + 4 * 24 + 2 * 16 + 4 * 24 + 1 * 24 + 1 * 8 + 41 * 48 = 2248 positions. Reference [1] Cubic Circular 5&6, (Autumn & Winter 1982), p. 25.

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Updated May 24, 2005.