THE RAINBOW MASTERBALL
Some rules for the rainbow masterball (referred to simply as "masterball"
in the following): A masterball sphere has 32 tiles of 8 distinct colors.
We shall assume that the masterball is in a fixed position in space, centered
at the origin. A geodesic path from the north pole to the south pole is called
a "longitudinal line" and a closed geodesic path parallel to the equator is
called a "latitudinal line". There are 8 longitudinal lines and 3 latitudinal
lines. In spherical coordinates, the longitudinal lines are at the angles
which are multiples of Pi/4 (i.e., at theta = nPi/4, n=1,..,8) and the
latitudinal lines are at phi = Pi/4, Pi/2, 3Pi/4. (Here Pi is the usual
3.141592...) The sphere shall be oriented by the right-hand rule - the thumb
of the right hand wrapping along the polar axis points towards the north
pole. We assume that one of the longitudinal lines has been fixed once and
for all. This fixed line shall be labeled "1", the next line (with respect
to the orientation above) as "2", and so on.
Allowed moves: One may rotate the masterball east-to-west by multiples of Pi/4
along each of the 4 latitudinal bands or by multiples of Pi along each of the 8
longitudinal lines.
A "facet" will be one of the 32 subdivisions of the masterball created by
these geodesics. A facet shall be regarded as immobile positions on the sphere
and labeled either by an integer i in {1,...,32} or by a pair (i,j) in
[1,4]\times [1,8], whichever is more convenient at the time. If a facet has
either the north pole or the south pole as a vertex then we call it a "small"
or "polar" facet. Otherwise, we call a facet "large" or "middle". A "coloring"
of the masterball will be a labeling of each facet by one of the 8 colors
in such a way that
(a) each of the 8 colors occurs exactly twice in the set of the 16 small
facets,
(b) each of the 8 colors occurs exactly twice in the set of the 16 large
facets.
A "move" of the masterball will be a change in the coloring of the masterball
associated to a sequence of manuevers as described above.
If we now identify each of the 8 colors with an integer in {1, ...,8} and
identify the collection of facets of the masterball with a 4x8 array of
integers in this range. To "solve" an array one must, by an appropriate
sequence of moves corresponding to the above described rotations of the
masterball, put this array into a "rainbow" position so that the matrix
entries of each column has the same number. Thus the array
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
is "solved". The array
6 7 8 1 2 3 4 5
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
corresponds to a rotation of the north pole facets by 3Pi/4.
NOTATION We use "matrix" notation to denote the 32 facets of the masterball.
The generators for the latitudinal rotations are denoted r1, r2, r3, r4, so,
for example, r1 sends
11 12 13 14 15 16 17 18
21 22 23 24 25 26 27 28
31 32 33 34 35 36 37 38
41 42 43 44 45 46 47 48
to
12 13 14 15 16 17 18 11
21 22 23 24 25 26 27 28
31 32 33 34 35 36 37 38
41 42 43 44 45 46 47 48
As you look down at the ball from the north pole, this move rotates the ball
clockwise. The other moves r2, r3, r4 rotate the associated band of the
ball in the same direction - clockwise as viewed from the north pole.
The generators for the longitudinal rotations are denoted f1, f2,...,f8,
so for example, f1 sends
12 13 14 15 16 17 18 11
21 22 23 24 25 26 27 28
31 32 33 34 35 36 37 38
41 42 43 44 45 46 47 48
to
44 43 42 41 16 17 18 11
34 33 32 31 25 26 27 28
24 23 22 21 35 36 37 38
15 14 13 12 45 46 47 48
With these rules, one can check the relation
f5=r1^4*r2^4*r3^4*r4^4*f1*r1^4*r2^4*r3^4*r4^4.
Exerccise: Find similar identities for f6, f7, f8.
Also, one can check that
r1=(f3*f7)^{-1}*r4^{-1}*f3*f7.
Exercise: There are similar identities for r2, r3, r4. Find them.
Identify the facets of the masterball with the entries of the array
8 7 6 5 4 3 2 1
16 15 14 13 12 11 10 9
24 23 22 21 20 19 18 17
32 31 30 29 28 27 26 25
(there is a reason for labeling the facets "backwards" like this but it's
not important). We may express the generators of the masterball group in
disjoint cycle notation as a subgroup of S_{32} (the symmetric group on 32
letters):
r1^{-1} = (1,2,3,4,5,6,7,8),
r2^{-1} = (9,10,11,12,13,14,15,16),
r3^{-1} = (17,18,19,20,21,22,23,24),
r4^{-1} = (25,26,27,28,29,30,31,32),
f1 = (5,32)(6,31)(7,30)(8,29)(13,24)(14,23)(15,22)(16,21),
f2 = (4,31)(5,30)(6,29)(7,28)(12,23)(13,22)(14,21)(15,20),
f3 = (3,30)(4,29)(5,28)(6,27)(11,22)(12,21)(13,20)(14,19),
f4 = (2,29)(3,28)(4,27)(5,26)(10,21)(11,22)(12,23)(13,24),
f5 = (1,28)(2,27)(3,26)(4,25)(9,20)(10,19)(11,18)(12,17),
f6 = (8,27)(1,26)(2,25)(3,32)(16,19)(9,18)(10,17)(11,24).
f7 = (7,26)(8,25)(1,32)(2,31)(15,18)(16,17)(9,24)(10,23),
f8 = (6,25)(7,32)(8,31)(1,30)(14,17)(15,24)(16,23)(9,22),
Exercise: Verify that the properties of a permutation puzzle are satisfied
for this puzzle.
2x2 RUBIK'S CUBE
The "pocket" Rubik's cube has 6 sides, or "faces", each of which has 2x2=4
"facets", for a total of 24 facets:
______________
/______/u_____/ |
/______/______/| | 6 sides: front (f), back (b),
| | | | | left (l), right (r),
| | | r/| up (u), down (d)
|______f______ /| |
| | | | |
| | | |/
|______|______|/
Fix an orientation of the Rubik's cube in space. Therefore, we may label the
six sides as f, b, l, r, u, d, as in the picture. It has 8 subcubes. Each
face of the cube is associated to a "slice" of 4 subcubes which share a facet
with the face. The face, along with all of the 4 cubes in the "slice", can
be rotated by 90 degrees clockwise. We denote this move by the upper case
letter associated to the lower case letter denoting the face. For example,
F denotes the move which rotates the front face by 90 degrees to clockwise.
______________
/______/u_____/ | |
/______/______/| | |
/ \ | _|_ | | | |
| | / | \ | r/| | The move F
| |__ /__f__\___ /| | |
| | __ | | | | | |
| | |\__|__/ | |/ \ /
| |______|______|/
<-------------------
As in chapter 2, we label the 24 facets of the 2x2 Rubik's cube as follows:
+--------------+
| 1 2 |
| u |
| 3 4 |
+--------------+--------------+--------------+--------------+
| 5 6 | 9 10 | 13 14 | 17 18 |
| l | f | r | b |
| 7 8 | 11 12 | 15 16 | 19 20 |
+--------------+--------------+--------------+--------------+
| 21 22 |
| d |
| 23 24 |
+--------------+
The 24 facets will be denoted by xyz where x is the face on which the facet
lives and y, z (or z, y - it doesn't matter) indicate the 2 edges of the
facet. Written in clockwise order:
front face: fru, frd, fld, flu
back face: blu, bld, brd, bru
right face: rbu, rbd, rfd, rfu
left face: lfu, lfd, lbd, lbu
up face: urb, urf, ulf, ulb
down face: drf, drb, dlb, dlf
Exercise: Verify that the properties of a permutation puzzle are satisfied
for this puzzle.
For future reference, we call this system of notation (which we will also
use for the 3x3 and 4x4 Rubik's cube) the "Singmaster notation".
3x3 RUBIK'S CUBE
Much has been written on the Rubik's cube (see, for example, [Ru] or[Si]).
In this section we shall, for the most part, simply introduce enough basic
notation to allow us to check that the puzzle is in fact a permutation
puzzle.
The Rubik's cube has 6 sides, or "faces", each of which has 3x3 = 9
"facets", for a total of 54 facets. We label these facets 1, 2, ..., 54 as
follows:
+--------------+
| 1 2 3 |
| 4 u 5 |
| 6 7 8 |
+--------------+--------------+--------------+--------------+
| 9 10 11 | 17 18 19 | 25 26 27 | 33 34 35 |
| 12 l 13 | 20 f 21 | 28 r 29 | 36 b 37 |
| 14 15 16 | 22 23 24 | 30 31 32 | 38 39 40 |
+--------------+--------------+--------------+--------------+
| 41 42 43 |
| 44 d 45 |
| 46 47 48 |
+--------------+
then the generators, corresponding to the six faces of the cube, may be
written in disjoint cycle notation as:
F:= (17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11),
B:= (33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27),
L:= ( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35),
R:= (25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24),
U:= ( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19),
D:= (41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40).
Exercise: Check this.
The notation for the facets will be similar to the notation used for the
2x2 Rubik's cube. The corner factes will have the same notation and the
edge facets will bve denoted by xy, where x is the face the facet lives
on and y is the face the facet borders to. In clockwise order, starting
with the upper right-hand corner of each face:
front face: fru, fr, frd, fd, fld, fl, flu, fu
back face: blu, bl, bld, bd, brd, br, bru, bu
right face: rbu, rb, rbd, rd, rfd, rf, rfu, ru
left face: lfu, lf, lfd, ld, lbd, lb, lbu. lu
up face: urb, ur, urf, uf, ulf, ul, ulb, ub
down face: drf, dr, drb, db, dlb, dl, dlf, df
Exercise: Verify that the properties of a permutation puzzle are satisfied
for this puzzle.
4x4 RUBIK'S CUBE
The 4x4 Rubik's cube has 6 sides, or "faces", each of which has 4x4 = 16
"facets", for a total of 96 facets. As usual, we fix an orientation of the
cube in space, so we may pick a front face, back face, ... . We label these
facets 1, 2, ..., 96 as follows:
+-----------------+
| 49 50 51 52 |
| 61 62 63 64 |
u
| 73 74 75 76 |
| 85 86 87 88 |
+------------------+-----------------+-----------------+-----------------+
| 53 54 55 56 | 1 2 3 4 | 5 6 7 8 | 9 10 11 12 |
| 65 66 67 68 | 13 14 15 16 | 17 18 19 20 | 21 22 23 24 |
l f r b
| 77 78 79 80 | 25 26 27 28 | 29 30 31 32 | 33 34 35 36 |
| 89 90 91 92 | 37 38 39 40 | 41 42 43 44 | 45 46 47 48 |
+------------------+-----------------+-----------------+-----------------+
| 57 58 59 60 |
| 69 70 71 72 |
d
| 81 82 83 84 |
| 93 94 95 96 |
+-----------------+
Notation: We need notation for the facets and for the moves.
Facets: To label the facets, we must pick an orientation of each face, say
clockwise. For example, the the front face may be labeled as
+---------------------+
| flu fu1 fu2 fru |
| fl1 f4 f1 fr1 |
f
| fl2 f3 f2 fr2 |
| fld fd2 fd2 frd |
+---------------------+
The labeling of the other faces is similar.
Exercise: Label the other 5 faces.
Moves: Parallel to each face x are 4 slices of 16 subcubes each labeled
X1, X2, X3, X4, in order of their distance from the face. For example,
the front face f has 16 subcubes comprising the F1 slice, the two inner
slices are F2, F3, and the last slice F4 is actually the same as the
first slice B1 associated to the back face.
The 12 generators (written in disjoint cycle notation), corresponding 2
each to the six faces of the cube are given by:
U1=(49, 52, 88, 85)( 62, 63, 75, 74)( 50,64,87,73)
(51,76,86,61)(5,1,53,9)(6,2,54,10)(7,3,55,11)(8,4,56,12),
U2=(17, 13, 65, 21)( 18, 14, 66, 22)( 19,15,67,23)(20,16,68,24),
L1=(96,48,49,1)(84,36,61,13)(72,24,73,25)(60,12,85,37)
(89,53,56,92)(90,65,55,80)(91,77,54,68)(66,67,79,78),
L2=(59,11,86,38)(71,23,74,26)(83,35,62,14)(95,47,50,2),
F1=(89,5,93,92)(77,17,81,80)(65,29,69,68)(53,41,57,86)
(1,4,40,37)(2,16,39,25)(3,28,38,13)(14,15,27,26),
F2=(73,6,81,91)(74,18,82,79)(75,30,83,67)(76,42,84,55),
R1=(40,88,9,57)(28,76,21,69)(16,64,33,81)(4,52,49,93)
(41,5,8,44)(42,17,7,32)(43,29,6,20)(18,19,31,30),
R2=(39,87,10,58)(27,75,22,70)(15,63,34,82)(3,51,46,94),
B1=(52,53,44,60)(51,65,32,59)(50,77,20,58)(49,89,8,57)
(9,12,48,45)(10,24,47,33)(11,36,46,21)(22,23,35,34),
B2=(54,72,43,64)(66,71,31,63)(78,70,19,62)(90,69,7,61),
D1=(57, 60, 96, 93)( 58, 72, 95, 81)(59, 84, 94, 69)
(70,71,83,82)(45,89,37,41)(46,90,38,42)(47,91,39,43)(48,92,40,44),
D2=(33, 77, 25, 29)( 34, 78, 26, 30)(35, 79, 27, 31)(36,80,28,32).
Exercise: Verify that the properties of a permutation puzzle are satisfied
for this puzzle.