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Piece Orientation - a subset of the 12 cubes pictured in Figure 1 which results in a physically connected piece. Of the 2**12 such combinations, 2225 are connected and 1871 are disconnected.

Piece (or physical piece) - one of the 837 physically different shapes which are represented by the 2225 piece orientations. One piece may produce as many as 8 piece orientations.

Cube Configuration - an assignment of each of the 32 cubes in Figure 2 to one of the 2 or 3 pieces which may occupy that cube or to empty space. The number of cube configurations is:

4^8 * 3^24 = 18,509,302,102,818,816

Assembly - A cube configuration in which each of the 6 piece orientations is a physically connected piece. The term 'legal configuration' has also been used this way.

Fully-Rotated Assembly - for counting purposes, an assembly in the sense above. An orientation is implied in this definition.

Physical Assembly - for counting purposes, two physical assemblies are the same if one can be rotated to match the other.

Logical Assembly - two assemblies are the same logical assembly if one can be rotated and/or reflected to produce the second. If a logical assembly has no internal symmetry, then it is equivalent to 2 physical assemblies and 24 fully-rotated assemblies.

Move - the linear movement of one or more pieces as a group. The direction must be parallel to one of the 3 axes and the distance must be some multiple of the cube width in length.

Solution - an assembly which can be physically disassembled into the 6 individual pieces by a series of moves.

Level of Solution - the minimum number of moves required in order to separate the assembly into at least two parts.

Level-type - solution levels for an assembly using length-6, length-8, length-10 and length-12 pieces. (e.g. level-type 2-3-0-0 means level 2 with length 6 pieces; level 3 with length 8 pieces, and no solution with length 10 or longer.)

Partial Solution - an assembly which can be split into at least two parts with moves (described above). Complete disassembly may or may not be possible.

Unique Solution - the physical pieces which are used to make a particular assembly can frequently be used to make other, different assemblies. If only one of the assemblies which can be formed from these pieces has a solution (can be taken completely apart), then this solution is called a unique solution.

Assembly with Movement - some moves with the pieces are possibly, but no disassembly is possible.

State - an arrangement of the pieces which can be reached by moves as described above. A state is uniquely described by the displacements of each piece in the 3 directions from its original position, with one of the 6 pieces being held fixed.

Notchable Piece - a piece which can be cut from a square rod of wood by making cuts perpendicular to the axis of the rod with repeated passes of a regular blade or with a dado blade. Generally speaking, notchable pieces have no "inside" corners or other areas that would require chiseling parts out or gluing cubes in.

Notchable Assembly (Solution) - an assembly (solution) in which each of the 6 pieces is notchable.

Ambiguous Piece - a piece which can be rotated about its long axis in more than one way for placement in an assembled 6-piece burr without creating external holes.

Apart Code - a shorthand notation for denoting the method for removing the first piece from an assembly. This allows for easy comparison of moves between two different assemblies.

Computer-Associated Definitions

GENDA (GENeral DisAssembly program) - used to analyze interlocking puzzles built up from cubes to see if they can be taken apart.

FDA (Fast Disassembly Analysis) - a fast version of the GENDA program which finds partial solutions of 6-piece burrs only.

BURR6 - an assembly & disassembly program used for 6-piece burrs. The program determines all ways 6 given pieces can be assembled into a 6-piece burr, and then uses the GENDA routines to complete the analysis.

MATR - formula for converting a physical 6-piece burr piece with some of its 12 cubes removed into a number from 1-4096. The resulting matrix can then be used to quickly rule out disconnected pieces, identify duplicate pieces, and other tasks. See section III.C.

LL - a simple, compact way of representing a 6-piece burr assembly. It consists of 32 1-digit numbers from 0-6. See Section IV.D. and the Love's Dozen and Computer's Choice Unique-10 examples.

JRM (for Journal of Recreational Mathematics) - earliest 6-piece burr analysis which was restricted to solid, notchable solutions. See [3].

SCIAM (for Scientific American) - analysis of solid, general (un-notchable) 6-piece burr solutions. See [2], [4] and [7].

NOTC - analysis of holey, notchable 6-piece burrs.

HB6 - analysis of holey, general 6-piece burrs.

GB6 - first phase of HB6 analysis which was used to analyze holey 6-piece burrs with from 0 to 5 holes.

IB6 - programs used in the HB6 analysis for assemblies with from 16 to 20 holes.

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