Date: 18 May 1981 1046-EDT (Monday)
From: Dan Hoey at CMU-10A
To: Cube-Lovers at MIT-MC
Subject: Drum info
Message-Id: <18May81 104634 DH51@CMU-10A>
Most of this note is pretty straightforward application of
the known cube properties, but if you want to know about the
drum....
The drum shows everything you see on a regular cube except
the orientation of the four truncated edges, or wedges. Because the
(invisible) edge parity is preserved, each visible position of the
drum corresponds to 2^4/2 cube positions. Thus there are
5.406x10^18 drum positions.
To count the number of solutions, note that as in the
normal cube, the face centers force each edge to its home position
and orientation. In addition, each corner has a facelet that says
whether it is top or bottom and fixes the corner's orientation.
This means that solved positions are obtained from each other by
permuting top corners, bottom corners, and wedges. But the three
cubies on a diagonal face must match, and so the three permutations
are the same. Only even permutations are achievable in this way
(since the cube of an odd permutation is odd) and there are 4!/2=12
of these. One easy process that goes from one solved position to
another is FF RR FF BB RR BB.
I asked Ole Jacobsen what he meant when he said of the
wedges that "as you will discover when using your edge moves: the
orientation matters." It turns out this is because he solves by
layers: top-middle-bottom, and doesn't know which way to orient the
edges in the middle so that the edges on the bottom will have the
right parity. There are several ways out of that problem; one is to
turn the drum sideways and solve left-middle-right.
The problem of solving without knowing the order of the
wedges is trickier. Solving sideways is one method: do the left
side any way; on the right side there two possibilities, one of
which will work. (This is Bernie Greenberg's suggestion, modified
so you don't need to memorize the whole map.)
One interesting thing to do with a drum is to turn it into
baseball. Using colored tape and disassembly, change the colors and
positions so that the wedges appear in the UF, DF, BL, and BR
positions when the colors match. On a baseball, there are only two
solved positions.