The following is a hypertext version of an article that appeared in Cubism For Fun 36 (February 1995) pp. 18-20. I have enhanced it with images of the positions, and also added minimal maneuvers for all positions. You can also see the original text version. I have also calculated all minimal maneuvers, using my optimal cube solver.

Note that some of the cubes appear the same, but differ on the three hidden faces.

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:

[Czech Check pattern #1] 1. R L' F B' U D' R L' Cufr (8q*, 8f*) (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.

[Czech Check pattern #2] 2. F R2 L U B' R' B R2 L' F' U R' D' F Cd (16q*, 14f*) (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.

[Czech Check pattern #3] 3.
F' U B' U' F U L F L' B L F' L' U' (14q*, 14f)
F D2 F' L U2 L' F D2 F' L U2 L' (12f*, 16q)
(UFR-) (DLB+)
[Czech Check pattern #4] 4.
F R U2 L' U' R2 U L U' R2 U' R' F' (16q*, 13f)
F2 R U2 R' U2 R' F2 R U2 R U2 R' (12f*, 18q)
(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.

[Czech Check pattern #5] 5.
F U' R F' U F B U B' R' F D' F D F (15q*, 15f)
F R' U B U2 L2 D F D' L2 U B' R F' (14f*, 17q)
(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+):

[Czech Check pattern #6] 6.
L F' L U B L2 B' L' U' F L2 F2 L (16q*, 13f*)
(LB+) (FD, FU, FR+)
[Czech Check pattern #7] 7. (inverse of 6)
[Czech Check pattern #8] 8.
F U D' L' B D' B' U' D L D L' (12q*, 12f*)
(LB+) (FU, FR, FD+)
[Czech Check pattern #9] 9.
U' F2 U F' L' F' D' B' U L B L' D (14q*, 13f*)
(LB+) (UR, UF, DF+)
[Czech Check pattern #10] 10. (inverse of 9)
[Czech Check pattern #11] 11.
B L' D' F2 L' F' D L D F' B R' B2 L (16q*, 14f*)
(LB+) (UR, FR, FD+)

Cycle structure (2)(2):

[Czech Check pattern #12] 12.
R L F B U2 F' B' R2 F2 R L' (14q*, 11f)
R2 F U2 D2 B2 U2 D2 F R2 (9f*, 16q)
(UF, DF) (RF, LB)
[Czech Check pattern #13] 13.
F D R' U2 F2 D2 L' U2 B2 D F' (16q*, 11f*)
(UF, DF) (RF, BL)
[Czech Check pattern #14] 14.
F' B' U D' F' U' D R2 F B' D' B2 (14q*, 12f)
U' R2 F2 U B2 L2 F2 D B2 U (10f*, 16q)
(UF, UR) (UB, DL)
[Czech Check pattern #15] 15.
L B' U' B U L2 F U F' U' L (12q*, 11f*)
(UF, UR) (UB, LD)

Cycle structure (2)(4):

[Czech Check pattern #16] 16.
R L F B D2 F B R2 B2 R' L (14q*, 11f)
R2 U D R2 B2 U' D' R2 F2 U2 (10f*, 16q)
(UF, DF) (FR, BR, BL, FL)
[Czech Check pattern #17] 17.
F' B' U2 F B' U R L U2 R L D L2 (16q*, 13f)
F B D2 B2 U2 F' B' D F2 B2 U' L2 (12f*, 18q)
(UL, UB) (UF, DF, DR, UR)

Cycle structure (2+)(2+):

[Czech Check pattern #18] 18.
F U' R L' B L' B R' L U' L F' (12q*, 12f*)
(UL, UR+) (DF, DB+)
[Czech Check pattern #19] 19.
F' L2 U R L' B' L B' R' L U L F (14q*, 13f*)
(UF, UR+) (DL, DB+)
[Czech Check pattern #20] 20.
F B R' F B' U' R' L B R L' U F2 (14q*, 13f*)
(UF, UR+) (DB, DL+)
[Czech Check pattern #21] 21.
U R' F' B D' F B' R U' D F D' (12q*, 12f*)
(UF, RF+) (LB, LD+)
[Czech Check pattern #22] 22.
R' D' F' U D' L D2 L U' D F' D' R (14q*, 13f*)
(UF, RF+) (LD, LB+)
[Czech Check pattern #23] 23. (inverse of 22.)

Cycle structure (3)(3): No solutions.

Cycle structure (3+)(3+):

[Czech Check pattern #24] 24.
F D B D' B U D' L' F L U' D B2 D' F' (16q*, 15f*)
(UF, RF, RU+) (DB, LB, LD+)
[Czech Check pattern #25] 25.
F D F2 B2 D' F D F B' R' D' F' R F B' D' (18q*, 16f*)
(UF, UR, FR+) (DB, LB, LD+)

Cycle structure (3):

[Czech Check pattern #26] 26.
F R' U D' F' U' D R F' B U B' (12q*, 12f*)
(RB, FD, UL)
[Czech Check pattern #27] 27.
R' U R' L F2 R L' U R (10q*, 9f*)
(RB, FD, LU)

Cycle structure (5):

[Czech Check pattern #28] 28.
F' L U' F' B2 D' F B' L B' U L' F (14q*, 13f)
U F2 R2 F2 U' F2 R2 F2 U2 (9f*, 16q)
(UF, UB, UL, UR, DR)
[Czech Check pattern #29] 29. (inverse of 28.)
[Czech Check pattern #30] 30.
L2 U' F2 B2 D F B' D2 F B' (14q*, 10f*)
(UB, UL, UF, UR, DR)
[Czech Check pattern #31] 31. (inverse of 30.)
[Czech Check pattern #32] 32.
R' U' R U' R U R U R2 U2 R' U' (14q*, 12f)
F2 R2 B2 D F2 L2 F2 D' F2 B2 U2 (11f*, 20q)
(UL, UF, UB, UR, DR)
[Czech Check pattern #33] 33. (inverse of 32.)
[Czech Check pattern #34] 34.
F' L' F R' L F' R L' U F' L F (12q*, 12f)
R L B2 R' L U' R2 L2 D' R2 (10f*, 14q)
(UB, UR, UF, DF, DL)
[Czech Check pattern #35] 35. (inverse of 34.)
[Czech Check pattern #36] 36.
U' R L U' D L U D' R L' D' R2 U (14q*, 13f)
F B' U2 F B L2 U F2 L2 F2 U (11f*, 16q)
(UR, UB, UF, DF, DL)
[Czech Check pattern #37] 37. (inverse of 36.)
[Czech Check pattern #38] 38.
B' L U B' L' B' U' B' L U B2 (12q*, 11f*)
(UB, UR, UF, LF, LD)
[Czech Check pattern #39] 39. (inverse of 38.)
[Czech Check pattern #40] 40.
R' L' U L U R B2 L' F B2 U' F' (14q*, 12f*)
(UR, UB, UF, LF, LD)
[Czech Check pattern #41] 41. (inverse of 40.)
[Czech Check pattern #42] 42.
F R' B' D' F' U' D R U B R F' (12q*, 12f*)
(UR, UB, UF, LF, DF)
[Czech Check pattern #43] 43. (inverse of 42.)
[Czech Check pattern #44] 44.
F' L' U' R U R' L F' L' U L F (12q*, 12f*)
(UB, UR, UF, LF, DF)
[Czech Check pattern #45] 45.
F U' F' U' F U2 F2 U' F U F U' (14q*, 12f*)
(UR, UB, UF, DF, LF)
[Czech Check pattern #46] 46.
D' F D F U R U' D R' U D' R' U' F' (14q*, 14f)
D R2 F' L2 U2 R2 B R2 U' B2 L2 F2 (12f*, 20q)
(LF, UF, UB, UR, DR)
[Czech Check pattern #47] 47. (inverse of 46.)
[Czech Check pattern #48] 48.
D' R F' R2 F' R L' U R L F R' D (14q*, 13f)
R2 D R2 F2 L2 U' R2 B' R2 U2 L2 F (12f*, 20q)
(DB, UB, UR, UF, LF)
[Czech Check pattern #49] 49. (inverse of 48.)
[Czech Check pattern #50] 50.
F' B R2 F B' U F B' R2 F' B U D2 (16q*, 13f)
U F2 B2 D R2 D F2 B2 U' L2 U2 (11f*, 18q)
(DL, DR, UR, UF, UB)
[Czech Check pattern #51] 51. (inverse of 50.)
[Czech Check pattern #52] 52.
D2 R B U B R U' R' B' U' B' D2 (14q*, 12f*)
(DL, DF, UF, UR, BR)
[Czech Check pattern #53] 53. (inverse of 52.)
[Czech Check pattern #54] 54.
F D R U2 D2 R U D' F' U D' R' D' F' (16q*, 14f)
B' D2 F2 R U2 F' L2 U2 R2 B U2 L' B' (13f*, 20q)
(UL, UR, FR, FD, BD)
[Czech Check pattern #55] 55.
F D' F2 L B R L' D' R' F2 D F' (14q*, 12f*)
(DL, FL, FR, BR, BU)
[Czech Check pattern #56] 56.
L' U' L U D' F' D F B' R' B R (12q*, 12f*)
(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 Cufr 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 Cufr. Positions 18, 19, 22 and 23 each have 2 symmetries, generated by the rotation Cfl. Position 20 has 2 symmetries, generated by central reflection. Position 25 has 6 symmetries, generated by the rotations Cufr and Cfl. 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] David Singmaster, Cubic Circular 5&6, (Autumn & Winter 1982), p. 25.

Addendum

Herbert Kociemba has confirmed my hand count of these Czech check patterns using his fabulous pattern explorer.

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Updated October 28, 2006.