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1. Does there exist an assembly using six pieces as defined at the beginning (must be constructable by removing a subset of the 12 cubes) which has a solution, but not a rectilinear solution? Such a design must make use of twisting or slanted movement of some kind. Such designs certainly exist in larger burrs, but can such a design exist in a 6-piece burr? You may use any length pieces. Stewart Coffin's `Convolution' comes to mind as a puzzle with a twist move. The move is actually theoretically impossible to make, but that such movement can take place quite legally in other designs is clear.

2. In this paper are many designs which have solutions when the pieces are of length-6, but have no solutions when using length-8 pieces. One can also construct assemblies which are solvable with length-8 pieces, but are unsolvable with length-10. What is the largest length (even), for which this can happen? That is, determine the even value of n for which (a) there is an assembly which is solvable using length-n pieces but not solvable with length-(n+2) pieces (give example); and (b) every assembly which is solvable using length-(n+2) pieces is also solvable using length-(n+4) pieces (give proof).

3. The BURR6 program analyzes movements that are linear and are a multiple of the unit cube in length. The disassembly portion of the program works on larger burrs or constructions that are `build-ups' of a single cube size. As mentioned in item 1 above, rotational moves can be necessary in some constructions of this type. Here we ask if allowing only moves a multiple of the cube width also causes us to miss some solutions. The problem is to prove or disprove the following:

Suppose we have an assembly of two or more pieces which satisfies:

• The assembly can be thought of as occupying a particular set of the cubes in a regular 3-dimensional grid of cubes all the same size. Moreover, each piece is composed of a particular set of these cubes, and is connected. (All pieces are constructed by gluing a number of cubes to each other, face-to-face; and the assembly has these fitted together so that all pieces are subsets of the same regular grid.) The assembly may have any number of holes.
• There is a way to disassemble the pieces by a finite sequence of linear moves, each of which involves one or more pieces and is in one of the three major axis directions, but need not be an even multiple of the unit cube size in length.

Does it necessarily follow that there is a way to disassemble the pieces in which all moves are not only linear as above, but are all integral multiples of the unit size in length?

4. The following records (using BURR6 terminology for `level') are waiting to be challenged:

• level-9 for length-6 designs with a unique solution - Peter J. Marineau Burr `B.'
• level-10 for length-6 designs with multiple solutions - Edward Hordern's modification of the Marineau burr.
• level-5 for length-8 (or longer) designs with a unique solution - Bruce Love B-4.

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