Bandaged Rubik's Cube / Siamese Cubes
This puzzle can be made from a standard Rubik's Cube by gluing the three pieces
along one edge of the cube together, i.e. there is now one large 1x1x3 moving
piece on the cube. The Siamese cubes consist of two cubes joined together along
one edge so that they share three pieces amongst them. As these pieces cannot
move, and the pieces of the joined cubes cannot intermingle, it can be
considered to be two bandaged cubes joined together. If you can solve a bandaged
cube, then you can solve the Siamese cubes separately in the same way.
There is only one bandaged piece on the cube, so by keeping that piece fixed and
moving the remainder (including the centres) the puzzle forms a group generated
by face and slice turns.
Notation:
The same notation is used as for the Rubik's cube. The letters U, D, L, R, F, B
are used to denote the faces of the cube. The cube is held so that the bandaged
piece lies along the bottom left edge, between the D and L faces. Only the R and
U face can be turned. These letters also denote a clockwise quarter turn of that
face. A letter followed by a 2 or an apostrophe denote a half turn and an
anti-clockwise quarter turn respectively. A combination of letters can be used
to denote a particular piece position on the cube, e.g. URF means the corner
contained in the three adjacent faces U, R, and F. The middle slices will be
denoted using lower case letters, r and u. Note that a slice turn does not move
any corner pieces.
The number of positions:
The easiest way to calculate this is to first consider the centres fixed and the
bandaged piece moving. There are 12 edges including the bandaged one which could
have 12! permutations, and 2 orientations each. The 6 unglued corners could then
have 6! permutations and 3 orientations each. This gives at most 12!*6!*2^12*3^6
positions. This limit is not reached because:
- The total permutation must be even (2)
- The number of flipped edges must be even (2)
- The total corner twist is fixed (3)
- If three corners are placed the other three are fixed too (6)
The last point is because the corners are moved only by turns of two adjacent
faces. The two generator group on the cube is well known, and its action on the
corners is as the PGL(2,5) group, of order 5!. For a good explanation, see David
Singmaster's "Notes on Rubik's 'Magic Cube'". This leaves 12!*5!*2^10*3^5 =
14,302,911,135,744,000 positions.
Note that the puzzle becomes somewhat more difficult if the bandaged piece has
no colours. It is then more difficult to find out where the pieces belong. Also,
if the bandaged piece is upside down (or rather the other pieces are placed
upside down w.r.t. the bandaged piece) then the puzzle cannot be solved because
the number of flipped edges is odd, and the permutation of pieces is odd.
Solution:
Phase 1: Place the two D corners correctly
a. Find the corner that belongs at DFR.
b. Turn R or U to bring it to the URF position.
c. Put it in place and in the correct orientation by one of the following:
1. To move URF->DFR, do U'R2.
2. To move RFU->DFR, do U'R'U'R2.
3. To move FUR->DFR, do RUR'.
d. Find the corner that belongs at DRB.
e. If it lies incorrectly positioned at DRB then do R'U'RU2, otherwise turn U
to bring it to the URF position.
f. Put it in place and in the correct orientation by one of the following:
1. To move URF->DRB, do U'R'URU2R'U'R.
2. To move RFU->DRB, do UR'U2R
3. To move FUR->DRB, do U'R'U'R.
Phase 2: Place the U corners correctly.
a. Count how many of the U corners are correctly oriented w.r.t. to U face.
b. If all 4 are twisted then put a corner needing a clockwise twist at the ULF
position. If three are twisted then put the correctly twisted corner at the
ULF position. If only two are twisted then put a corner needing an
anti-clockwise twist at the ULF position.
c. Do the sequence RUR'URU2R'.
d. Repeat b and c until all corners are oriented properly.
e. Turn U to put the corners in place.
Phase 3: Place the edges in their correct slices.
There are three sets of edges, those in the r slice, those in the u slice,
and the three edges in the other (immovable) slice. This phase places the
pieces in their own slice, though not correctly oriented or positioned. It
may move pieces that are correctly positioned already, but will keep them in
their slice. The centres are ignored.
a. Find a piece in the r slice that does not belong, and turn r to place it at
the BD position.
b. Find a second piece that belongs in the r slice but is not in that slice.
c. If the second piece is in the u slice, then turn u to bring it to the FR
position, and do RUr'U'R'.
d. If the second piece is at UR then do Ur'U'.
e. If the second piece is at DR then do R2Ur'U'R2.
f. If the second piece is at UL then do U'r'U'rU2r'.
g. Repeat a-f until the r slice has all its edges.
h. Turn the puzzle over, so that the D and L faces are swapped, and repeat
steps a-g until the new r slice is also correct.
Phase 4: Place the centres correct.
a. If the U/D centres need to be swapped then do r2.
b. If the U/D centres are not correct then turn u to bring the U centre to the
front, and do r.
c. Do u to place the remaining centres correct.
Phase 5: Place the r edges in position.
a. To swap two adjacent edges in the r slice, turn r to bring them to the U
face, do U2, and turn r to bring the centres back in position.
b. Repeat step a as often as necessary until the slice is correct (ignoring the
U corners).
c. If the U corners need to be moved back into position, do R2U2R2U2R2.
Phase 6: Place the u edges in position.
a. To swap two adjacent edges in the u slice, turn u to bring them to the R
face, do R2, and turn u to bring the centres back in position.
b. Repeat step a as often as necessary until the slice is correct (ignoring the
R corners).
c. If the R corners need to be moved back into position, then do the sequence
U R2U2R2U2R2 U R2.
Phase 7: Place the remaining edges in position.
a. If the remaining three edges are incorrectly positioned then there are only
two possibilities.
1. To move UL->UR->DR, do RUR2 U'R'U'R' U2RU.
2. To move DR->UR->UL, do U'R'U2 RURU R2U'R'.
Phase 8: Orient the edges.
All the sequences used below are conjugates of rUrUrUrU (which flips the
pieces UL, FD, BU, BD) or combinations thereof. If you are proficient at
cubing, you can probably solve this stage a lot quicker by using your own
conjugates. Note that the edges are not solved individually, so a step may
flip edges that are then solved in later steps.
a. Consider the edges in the F/B slice (i.e. UL, UR, DR), and see which ones
need to be flipped.
b. Do the appropriate sequence below:
1. To flip all three edges, do UR2r2 U'rU'rU'rU' r'R2U'.
2. To flip UR, UL then do Ur2 UrUrUrU r'U'.
3. To flip DR, UL then do R2Ur2 UrUrUrU r'U'R2.
4. To flip only UL, then do rUrUrUrU.
5. To flip only UR, then do rU'rU'rU'rU'.
In all other cases, you need to turn over the puzzle so that the D and L
faces are swapped and then you find yourself in one of the cases handled
above.
c. Count how many of the edges in the r and u slices still need to be flipped.
If there are more flipped edges in the u slice then turn over the puzzle, so
that the D and L faces are swapped.
d. If all 8 edges need to be flipped then do the sequence R rU'rU'rU'rU' R. Now
only 4 edges need to be flipped, which is explained below.
e. If 4 edges in the r slice and 2 in the u slice need to be flipped, then turn
u to bring an unflipped edge to FR and do R rU'rU'rU'rU' R'. Turn the u
slice back into position. Now 4 edges need to be flipped, which is explained
below.
f. If only the 4 edges of the r slice need to be flipped then do the sequence
U'rUr' UrUrUrU r2U'r'U.
g. If 3 edges in each slice need to be flipped, then turn r and u to bring the
unflipped edges at FU and FR and do R rU'rU'rU'rU' R'. Turn the u and the r
slices back into position. Now 4 edges need to be flipped, which is
explained below.
h. If 3 edges in the r slice and one edge in the u slice need to be flipped,
then turn r to bring the unflipped edge at FU, and u to bring the flipped
edge to FR, and then do R rU'rU'rU'rU' R'. Turn the u and the r slices back
into position.
i. If 2 edges in each slice need to be flipped then turn r and u to bring
unflipped edges to the FU and FR positions, and then do R rU'rU'rU'rU' R'.
Turn the u and the r slices back into position. Now there are still 4
flipped edges, but there are three in one slice, and this case is explained
above.
j. If only two edges in the r slice need to be flipped, then turn r to bring a
flipped edge to FU and do R rU'rU'rU'rU' R'. Turn the r slices back into
position. Now 4 edges need to be flipped, which is explained above.
k. If one edge in each slice need to be flipped, then turn r and u to bring the
flipped edges at FU and FR and do R rU'rU'rU'rU' R'. Turn the u and the r
slices back into position. Now 4 edges need to be flipped, which is
explained above.
Copyright 1999-2000 Jaap Scherphuis
Jaap's Puzzle Page: http://www.org2.com/jaap/puzzles