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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.