The following is a hypertext version of an article that appeared in
Cubism For Fun
36 (February 1995) pp. 1820.
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:
 centers
 centers and edges
 corners
 corners and edges
 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}

(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.

2.

F R^{2} L U B' R' B R^{2} L' F' U R' D' F C_{d}

(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.

3.

F' U B' U' F U L F L' B L F' L' U'

(14q*, 14f)

F D^{2} F' L U^{2} L' F D^{2} F' L
U^{2} L'

(12f*, 16q)


(UFR) (DLB+)


4.

F R U^{2} L' U' R^{2} U L U' R^{2} U' R' F'

(16q*, 13f)

F^{2} R U^{2} R' U^{2} R' F^{2} R
U^{2} R U^{2} 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.

5.

F U' R F' U F B U B' R' F D' F D F

(15q*, 15f)

F R' U B U^{2} L^{2} D F D' L^{2} 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
ncycle and (n+) denotes a flipped ncycle 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 L^{2} B' L' U' F L^{2} F^{2} L

(16q*, 13f*)


(LB+) (FD, FU, FR+)


7.

(inverse of 6)


8.

F U D' L' B D' B' U' D L D L'

(12q*, 12f*)


(LB+) (FU, FR, FD+)


9.

U' F^{2} U F' L' F' D' B' U L B L' D

(14q*, 13f*)


(LB+) (UR, UF, DF+)


10.

(inverse of 9)


11.

B L' D' F^{2} L' F' D L D F' B R' B^{2} L

(16q*, 14f*)


(LB+) (UR, FR, FD+)

Cycle structure (2)(2):

12.

R L F B U^{2} F' B' R^{2} F^{2} R L'

(14q*, 11f)

R^{2} F U^{2} D^{2} B^{2} U^{2}
D^{2} F R^{2}

(9f*, 16q)


(UF, DF) (RF, LB)


13.

F D R' U^{2} F^{2} D^{2} L' U^{2}
B^{2} D F'

(16q*, 11f*)


(UF, DF) (RF, BL)


14.

F' B' U D' F' U' D R^{2} F B' D' B^{2}

(14q*, 12f)

U' R^{2} F^{2} U B^{2} L^{2}
F^{2} D B^{2} U

(10f*, 16q)


(UF, UR) (UB, DL)


15.

L B' U' B U L^{2} F U F' U' L

(12q*, 11f*)


(UF, UR) (UB, LD)

Cycle structure (2)(4):

16.

R L F B D^{2} F B R^{2} B^{2} R' L

(14q*, 11f)

R^{2} U D R^{2} B^{2} U' D' R^{2}
F^{2} U^{2}

(10f*, 16q)


(UF, DF) (FR, BR, BL, FL)


17.

F' B' U^{2} F B' U R L U^{2} R L D L^{2}

(16q*, 13f)

F B D^{2} B^{2} U^{2} F' B' D F^{2}
B^{2} U' L^{2}

(12f*, 18q)


(UL, UB) (UF, DF, DR, UR)

Cycle structure (2+)(2+):

18.

F U' R L' B L' B R' L U' L F'

(12q*, 12f*)


(UL, UR+) (DF, DB+)


19.

F' L^{2} U R L' B' L B' R' L U L F

(14q*, 13f*)


(UF, UR+) (DL, DB+)


20.

F B R' F B' U' R' L B R L' U F^{2}

(14q*, 13f*)


(UF, UR+) (DB, DL+)


21.

U R' F' B D' F B' R U' D F D'

(12q*, 12f*)


(UF, RF+) (LB, LD+)


22.

R' D' F' U D' L D^{2} L U' D F' D' R

(14q*, 13f*)


(UF, RF+) (LD, LB+)


23.

(inverse of 22.)

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

24.

F D B D' B U D' L' F L U' D B^{2} D' F'

(16q*, 15f*)


(UF, RF, RU+) (DB, LB, LD+)


25.

F D F^{2} B^{2} D' F D F B' R' D' F' R F B' D'

(18q*, 16f*)


(UF, UR, FR+) (DB, LB, LD+)

Cycle structure (3):

26.

F R' U D' F' U' D R F' B U B'

(12q*, 12f*)


(RB, FD, UL)


27.

R' U R' L F^{2} R L' U R

(10q*, 9f*)


(RB, FD, LU)

Cycle structure (5):

28.

F' L U' F' B^{2} D' F B' L B' U L' F

(14q*, 13f)

U F^{2} R^{2} F^{2} U' F^{2}
R^{2} F^{2} U^{2}

(9f*, 16q)


(UF, UB, UL, UR, DR)


29.

(inverse of 28.)


30.

L^{2} U' F^{2} B^{2} D F B' D^{2} F B'

(14q*, 10f*)


(UB, UL, UF, UR, DR)


31.

(inverse of 30.)


32.

R' U' R U' R U R U R^{2} U^{2} R' U'

(14q*, 12f)

F^{2} R^{2} B^{2} D F^{2}
L^{2} F^{2} D' F^{2} B^{2}
U^{2}

(11f*, 20q)


(UL, UF, UB, UR, DR)


33.

(inverse of 32.)



34.

F' L' F R' L F' R L' U F' L F

(12q*, 12f)

R L B^{2} R' L U' R^{2} L^{2} D'
R^{2}

(10f*, 14q)


(UB, UR, UF, DF, DL)


35.

(inverse of 34.)


36.

U' R L U' D L U D' R L' D' R^{2} U

(14q*, 13f)

F B' U^{2} F B L^{2} U F^{2} L^{2}
F^{2} U

(11f*, 16q)


(UR, UB, UF, DF, DL)


37.

(inverse of 36.)



38.

B' L U B' L' B' U' B' L U B^{2}

(12q*, 11f*)


(UB, UR, UF, LF, LD)


39.

(inverse of 38.)


40.

R' L' U L U R B^{2} L' F B^{2} U' F'

(14q*, 12f*)


(UR, UB, UF, LF, LD)


41.

(inverse of 40.)



42.

F R' B' D' F' U' D R U B R F'

(12q*, 12f*)


(UR, UB, UF, LF, DF)


43.

(inverse of 42.)


44.

F' L' U' R U R' L F' L' U L F

(12q*, 12f*)


(UB, UR, UF, LF, DF)


45.

F U' F' U' F U^{2} F^{2} U' F U F U'

(14q*, 12f*)


(UR, UB, UF, DF, LF)



46.

D' F D F U R U' D R' U D' R' U' F'

(14q*, 14f)

D R^{2} F' L^{2} U^{2} R^{2} B
R^{2} U' B^{2} L^{2} F^{2}

(12f*, 20q)


(LF, UF, UB, UR, DR)


47.

(inverse of 46.)



48.

D' R F' R^{2} F' R L' U R L F R' D

(14q*, 13f)

R^{2} D R^{2} F^{2} L^{2} U'
R^{2} B' R^{2} U^{2} L^{2} F

(12f*, 20q)


(DB, UB, UR, UF, LF)


49.

(inverse of 48.)



50.

F' B R^{2} F B' U F B' R^{2} F' B U D^{2}

(16q*, 13f)

U F^{2} B^{2} D R^{2} D F^{2}
B^{2} U' L^{2} U^{2}

(11f*, 18q)


(DL, DR, UR, UF, UB)


51.

(inverse of 50.)



52.

D^{2} R B U B R U' R' B' U' B' D^{2}

(14q*, 12f*)


(DL, DF, UF, UR, BR)


53.

(inverse of 52.)



54.

F D R U^{2} D^{2} R U D' F' U D' R' D' F'

(16q*, 14f)

B' D^{2} F^{2} R U^{2} F' L^{2}
U^{2} R^{2} B U^{2} L' B'

(13f*, 20q)


(UL, UR, FR, FD, BD)



55.

F D' F^{2} L B R L' D' R' F^{2} D F'

(14q*, 12f*)


(DL, FL, FR, BR, BU)



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
onetoone 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 UD 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] 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.