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Bill Cutler Puzzles, Inc.

### 6-Piece Burr Design Definitions

Pieces - each piece must be constructed by removing some of the 12 cubes as numbered in Fig. 1, and the piece must be connected. (Note that the numbering of the 12 cubes does not match the numbering in my article in the Journal of Recreational Mathematics . I apologize for this inconsistency in my notation. This is compounded by the problem of showing the numbering patterns by `layers.' It may not be clear if the bottom layer is looked at from the top or bottom.) Fig. 1 is for length-6 pieces. Length-8 pieces are one unit longer on each end. The representation of the pieces by which cubes are removed may not be unique. Two pieces are identical if one can be rotated and/or flipped end-for-end so as to match the other piece.

Symmetric Piece - a piece is symmetric if an end-for-end flip results in the same cube configuration. The usual flip would switch cube number pairs (1,8), (2,7), (3,6), (4,5), (9,12), and (10,11). For `ambiguous' pieces (see below), other flips are possible.

Assembly - an assembly, or `legal configuration,' is a way of arranging the six pieces into the 3-dimensional grid of cubes pictured in Fig. 2 without having two or more pieces occupying the same cube. One need not be able to physically do this with wooden (rigid) pieces. Two assemblies are the same if one can be rotated (in three dimensions) so that it matches the other exactly.

Internal Cubes - cubes in Fig. 2 that may belong to more than one piece.

External Cubes - cubes in Fig. 2 that belong to only one piece.

Internal Holes - internal cubes not occupied in an assembly. When discussing the number of `holes' this will be the count referred to.

External Holes - external cubes not occupied in an assembly. For most analyses in this paper, assemblies with external holes will not be allowed.

Ambiguous Piece - a piece which can be rotated around its long axis without creating external holes. In terms of cube numbers from Fig. 1, such a piece must have either both cubes 1 and 4 present, or both cubes 5 and 8 present. These pieces have been a thorn in the side of 6-piece burr analyzers and have led to miscommunications because of different methods of handling.

Solution - a solution is an assembly which can be physically constructed (or disassembled). The method of assembly or disassembly must be theoretically correct when used with rigid pieces. (It may not make use of slightly rounded corners, for example).

`Interesting' Solution - a solution which cannot be split into two sections on the first move.

### Definitions Related to BURR6 Disassembly Method

Rectilinear Solution - a method of assembling or disassembling an assembly which is carried out by a sequence of moves, each of which is a linear move of one or more pieces in one of the three directions. The distances the pieces are moved must be multiples of the unit cube in length. Solutions of this type are the only ones that the {\erm BURR6} program can discover. Whether other solutions for holey-6 burrs exist is, as far as I know, an open question. (See Challenge Questions.)

Displacement - three integers representing the number of units in each of the three directions that a piece has been shifted from its initial position. Since constants may have been added to `center' a set of pieces, these numbers are best viewed in relation to the displacements of the other pieces in the set.

State - a set of displacements for the pieces in an assembly. Two states for a set of pieces are the same if the displacements for the pieces, measured relative to the smallest numbered piece, are the same. Each new state represents a different geometrical arrangement to which the pieces of an assembly can be moved.

Subassembly - a set of from two to five pieces that are left together after an earlier assembly or subassembly has been separated.

Level - during a disassembly process, the number of times that the direction of movement has changed in order to reach a particular state. The computer treats all movement in the same direction as being the same level. The level can be looked on as an estimation of difficulty of the assembly. It is similar to what would be called `number of moves,' but is frequently smaller than the count that most people would get when disassembling a burr. More on this later.

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