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

     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


[1]  Cubic Circular 5&6, (Autumn & Winter 1982), p. 25.


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Updated May 24, 2005.