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Now that this computer analysis has been completed, in which all 35.5 billion 6-piece burr assemblies were analyzed, is there anything left to analyze in the way of 6-piece burrs? Are there any more interesting designs to discover? The answer is, of course, "Yes". In some areas the computer analysis is incomplete. There are also some situations which are beyond the capabilities of the current computer programs.

Interesting 6-piece burr designs which were overlooked or not included in the computer analysis are broken into 3 categories:

### Standard Pieces, Standard Moves

In this category are questions which could be answered by computer programs similar to the ones used in the analysis, but were not included because of time constraints or other reasons.

1. Complete solution analysis - only high level solutions were completely disassembled. How many of the 5,951,254,866 partial solutions can be completed?
2. Different definitions of level - other definitions of level are frequently made. What assemblies have the highest level first move with other definitions of level?
3. Instead of considering only the level of the first move, what about considering the total number of moves to disassemble the burr? The GENDA program handles complete disassembly of a burr and minimizes the level at each stage of the disassembly, but this does not necessarily minimize the total number of moves made. Although solving such a problem can definitely be done with a program, there is not an easy way to add this logic to the current program.
4. What if external holes are allowed? Using long pieces with notches repeated every few units and 'back-and-forth' movement techniques, one can design 6-piece burrs with as high a level as one desires.

### Standard Pieces, Non-Standard Moves

In this category we consider 6-piece burr assemblies which are made up of standard pieces, and so are included in the 35.5 billion assemblies analyzed, but can come apart in unusual ways. The program is limited to movements which are linear in one of the orthogonal directions and are a multiple of the cube width in length. What about 1/2 moves, slant moves and twists? Can these non-standard moves be used to advantage to disassemble one of the standard 6-piece burr assemblies?

The "Weave" in the examples section is an example of a cubic build-up assembly which cannot be disassembled with the "standard" moves above, but which will fall apart easily if the pieces can be moved in other ways. The computer programs used in the analysis are incapable of determining that this puzzle comes apart. Are such designs possible within the confines of a 6-piece burr?
Diagram of The Weave

I posed this question in 1986 (see [5]). A key restriction was that any such move be completely "legal". By this I mean that if the pieces were cut perfectly from a rigid material with no tolerance or extra space whatsoever, then the pieces can be physically separated using the rotations.

At the time I asked this question regarding the 6-piece burr, I thought the answer was probably "no", but I had not really tried to construct such a design. When the Computer's Choice 4-Hole design was discovered in 1988, I noticed there was a twist movement in the puzzle. The twist is equivalent to rotating a square within 3 squares the same size in the shape of an 'L'. See figure below. In theory, the twist is illegal. In practice, the twist can be made even with well-made pieces because the sides of the 'L' hole are different pieces, so that the 'play' in the pieces is magnified. Although the move is illegal and does not help the disassembly process if it is made, this discovery made me realize the possibilities for similar, legal movements, and I started investigating assemblies with such movements.

The first design I came up with required a fully-legal twist for disassembly, but unfortunately the solution was not unique (there were different ways to assemble the puzzle which had standard solutions). Imagine my surprise when, about two weeks later, I received a design from Peter Rosler (GERMANY) with the identical twist move in a very similar design. This design was also not unique. By this time, I was running programs on all similar assemblies looking for one with a unique twist solution.

After doing an exhaustive search of about 2,000 assemblies, I found 2 unique designs, one of which is now called "The Programmer's Nightmare". Running BURR6 on this design produces "102 assemblies with no solution". One of the assemblies has the twist move which was originally planned. However, there are 3 other assemblies which also appear to have twist moves that will result in disassembly. In two of these assemblies the twist moves are equivalent to the illegal twist move described above, and are therefore not legitimate solutions. With well-made pieces, these moves cannot be made without noticing that the pieces 'jam'. On first glance, the twist move in the third assembly appears to be every bit as legal as the original solution. I was disappointed when I first found this out, because an equivalent additional assembly appears in all of the 6-piece burr designs with similar twist moves. However, on closer examination both moves have additional restrictions in another layer of the assembly. The original solution turns out to be completely legal, and the third assembly illegal, but even with well-made pieces there is little evidence of the 'jam'. See the following pages for detailed pictures:
Programmer's Nightmare Solution
Programmer's Nightmare Three Close Solutions
Programmer's Nightmare Rotation in Legal Solution

What about 1/2 moves, in which the moves are linear in the orthogonal directions, but the lengths of the moves are not integral multiples of the cube width? Can they be used to advantage?

In the Programmer's Nightmare, a 1/2 move was combined with a twist to disassemble the puzzle. However, I do not see how 1/2 moves, or other fractional moves, can be used by themselves to advantage. I cannot conceive of an assembly, built up from cubes in a standard 3-d lattice, in which fractional moves can accomplish anything that full moves cannot. However, I have been wrong before!

What about slant moves, that is, linear moves in a direction other than an orthogonal direction?

In similar fashion to the 1/2 move argument, I cannot see how slant moves can allow disassembly of a cubic assembly where standard moves cannot do the same. However, a slant move, or a combination of a 1/2 move and a slant move, can replace a number of repeated standard moves that 'jog' a piece in a slanted direction. Hence, the slant move can be used to decrease the number of 'moves' in the solution, but disassembly can still be achieved with standard moves.

### Non-Standard Pieces

In this category are 6-piece burrs in which the pieces are not restricted to the cubic-cut pieces of Figure 2. We can use slant cuts, which may require that two pieces be moved simultaneously in different directions. We can use rounded cross-sections, resulting in twisting moves which are completely legal for these pieces. There are an infinite number of possible piece shapes, and perhaps an infinite variety to the possible uses of such pieces.

A few examples:

• U-Nam-It Burr - A long time ago, perhaps 1965, I tried to design a 6-piece burr in which all 6 pieces had to move at the same time in different directions. This was necessitated by slant cuts in pairs of pieces. Unfortunately, when I resurrected the design in 1989, it turned out not to work at all. I modified the design so that 3 pieces must move simultaneously in different directions.
Diagram of U-Nam-It
• Explode-A-Burr - About the same time as the original U-Nam-It design, I crossed the familiar 'star burr' design with the 6-piece burr shape. The result is a standard-looking 6-piece burr in which all 6 pieces are identical and there are no internal holes. The disassembly technique requires all pieces to be moved simultaneously away from the center of the puzzle - a technique which is sometimes referred to as 'explosion'. There are other designs in which all 6 pieces are identical and there are no holes, but I believe all of these are modifications of this design.
Diagram of Explode-A-Burr

Stewart Coffin has recently been experimenting with 'inclined' 6-piece burr designs. In some of these designs the cross-sections of the pieces are rhombic and in other designs the pieces do not meet each other at right angles. Some of these designs can be analyzed by the same computer programs used in standard 6-piece burr analysis, but for other types of designs these programs are useless.

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