The pieces which can be used in making standard 6-piece burrs are
constructed by removing some of the 12 cubes from the piece in
Figure 1 (below). Of the 2 ** 12 = 4096 ways to remove
the 12 cubes, 1871 result in disconnected pieces, leaving 2225
different connected configurations. These 2225 combinations are
produced by 837 distinct physical pieces. 59 of these pieces are
notchable, and they relate to 147 of the 2225 cube combinations.
When we refer to "pieces" or "physical pieces", we will be referring to one of the 837 pieces. The 2225 cube combinations will be referred to as "piece orientations".
The 59 notchable pieces are pictured in Figure 3. The first 25 pieces in the list correspond to the 25 pieces in JRM. The remaining 34 pieces cannot be used in notchable, solid solutions; but they can be used in general, solid solutions, so are used in the SCIAM analysis.
The 837 pieces are listed in Table 2.
The 2225 piece orientations are listed in Table 3.
Ambiguous pieces are ones which can be rotated about the long axis without creating external holes. These pieces must have either cubes 1&4 or 5&8 filled in Figure 1.
Examples: pieces #27 (cube codes: 1011 1011 11 11) and #228 (cube codes: 1011 0001 01 11) are ambiguous.
Piece #696 (cube codes: 1001 0000 00 11) might be called 'pseudo-ambiguous'. It can be rotated to form a new configuration, but it can also be flipped end-for-end to achieve the same configuration without using its 'ambiguity'.
Let us briefly consider how to manufacture un-notchable pieces made out of wood. One can either chisel out the un-notchable areas, or glue cubes back into the piece. The glue-in method is easier, but the glue joints may be visible from the outside of the assembled puzzle. For many un-notchable pieces, it is a simple matter to glue cubes on the part of the piece which will be inside the assembled burr. For un-notchable, ambiguous pieces, it may be impossible to glue the cubes in so that they are invisible for all assemblies using the piece, but one can choose the gluings to be invisible for the desired solution to the puzzle. This will give a clue to the solution, but this cannot be helped for some burrs when using the glue-in method.
Now we can consider one particularly unusual piece:
Piece # 616 (cube codes: 0000 1001 00 11)
This piece appears in many high-level solutions discovered by the computer, including Computer's Choice Unique-10. The easiest way to make it is by gluing in cubes # 4 and 8. However, in the high-level solutions, the piece appears in one of its ambiguous rotations, either 1111 0000 00 00 or 0000 1111 00 00. There is no easy way to construct the piece so that the glue joints are on the inside when oriented this way. Jerry McFarland makes this piece in the following way:
The result is a piece which appears to be glued from 3 separate parts, and has the glue joints on the inside when used in Computer's Choice Unique-10.