Six-piece burrs

Six-piece burrs are the simplest regular burrs. To analyze the six-piece burrs, we first discuss the pieces used to build them. Then we introduce a few interesting facts about six-piece burrs. Finally we present some statistics from a (mostly) complete analysis made by Bill Cutler.

Six-piece burr pieces

Before an analysis of six-piece burrs can be conducted, the usable pieces need to be computed. William Cutler was the first to completely analyze six-piece burr pieces used to make solid six-piece burrs [Cutler78].

To get a complete analysis we need to find a way to enumerate the pieces. To be able to refer to particular pieces, we introduce the following numbering scheme:

Def 9.0:	Binary Piece Number As we have seen in Lemma 2.1.2.2 a six-piece burr piece may have only the cubes 1 through 12 notched out. If we write down a 1 or 0 for the presence or absence of any of the twelve cubes we get the binary number of a piece.
Lemma 9.0.1:	The solid six-piece burr piece has a binary number of 111111111111. Proof: The solid piece does not have any cubes removed, hence all of the twelve cubes are present.
Lemma 9.0.2:	The lightest usable piece has the binary number 000000000011. Proof: The following image shows the cubes 1 through 10 are absent and 10 through 12 are present. Removing any more cubes would leave the piece disjoint.
Def 9.1:	Decimal Piece Number The decimal piece number is defined as 4096 minus the binary piece number, expressed as a decimal number as shown below: 4096 - 111111111111b |||||||||||+- *2048 ||||||||||+-- *1024 + |||||||||+--- * 512 + ||||||||+---- * 256 + |||||||+----- * 128 + ||||||+------ * 64 + |||||+------- * 32 + ||||+-------- * 16 + |||+--------- * 8 + ||+---------- * 4 + |+----------- * 2 + +------------ * 1 + Note: the reason for this seemingly reverse definition from normal mathematical notations of binary numbers was chosen for a graphical reason. If a program lists all the pieces sorted by decimal numbers, then this definition will yield generally better visible 3D representations of the pieces.
Lemma 9.1.1:	The solid six-piece burr piece has the decimal number 1. Proof: 4096 - 111111111111b = 4096 - 4095 = 1. Note: The fact that the solid piece comes out as number 1 is the main reasons for choosing to subtract the binary number from 4096.
Lemma 9.1.2:	The lightest usable piece has the decimal number 1024. Proof: 4096 - 000000000011b = 4096 - 2048 - 1024 = 1024. Note: If you rotate the piece by 180o you get the piece 000000001100 which has the decimal number 4096 - 000000001100b = 4096 - 512 - 128 = 3456.

We have already introduced notchable pieces in Def 1.2.1.1.2. now we are going to define the type of a six-piece burr piece with a finer grain of detail.

Def 9.2:	internal corner A piece is said to have an internal corner, if the sides of three cubes meet inside the piece in a concave fashion. The piece shown to the right has one internal corner.
Lemma 9.2.1:	The maximum number of internal corners a piece can have is 8. Proof: The piece on the right shows 8 internal corners, trying to remove any more cubes e.g from the bottom either reduces the number of internal corners or makes the piece disjoint.
Def 9.3:	millable A piece is called millable if it can be produced by a milling machine. This means the piece may not have any internal corners. For example the piece shown in the picture is millable.
Lemma 9.3.1:	Notchable pieces are millable. Proof: Since notchable pieces have no concave crosscuts perpendicular to the axis, no internal corners may be formed. Hence they are millable.
Lemma 9.3.2:	There are millable pieces which are not notchable. Proof: The piece shown in the definition 9.3 above, is millable but not notchable.
Def 9.4:	Piece Type Piece types are introduced as to whether a piece is notchable, millable, or neither. And within those three categories they are divided as to whether they can be used to build solvable, solid six-piece burrs. Type Description notchable32 notchable piece that can be used for solid burrs notchable notchable piece that can be used for general burrs millable32 millable piece that can be used for solid burrs millable millable piece that can be used for general burrs general32 general piece that can be used for solid burrs general general piece that can be used for general burrs disjoint unusable piece Note: the numeral 32 signifies the fact that a solid burr has a weight of 32 (see Lemma 4.1.2).
Theorem 9.4.1:	A notched piece is usable for solid burrs (of type notched32) if it meets the following criteria: The piece is solid The following cubes may only be removed as a pair: 1 and 5, 4 and 8, 9 and 10, 11 and 12. Cubes 2, 3, 6, and 7 may be removed individually. Proof: see [Cutler78].
Theorem 9.4.2:	A general piece is usable for solid burrs (of type general32) if it meets the following criteria: The piece is solid Cubes 1, 2, 5 and 6 removed; others may vary Cubes 3, 4, 7 and 8 removed; others may vary Cubes 2, 3, 9 and 10 removed Cubes 6, 7, 11 and 12 removed Cubes 1 and 5 removed and either (a) 3 and 7 removed, (b) 4 and 8 removed, or (c) 4 and 8 not removed Cubes 2 and 6 removed and either (a) 4 and 8 removed, or (b) 4 and 8 not removed Cubes 1 and 5 not removed and (a) 3 and 7 removed, or (b) 4 and 8 removed Proof: see [Cutler78].
Theorem 9.4.3:	For each type there exist the following number of usable, unique pieces. That is duplicates (from rotations) have been removed. Type Nbr of Pieces List notchable32 25 (20KB) notchable 59 (44KB) millable32 78 (57KB) general32 369 (250KB) takes long to load general 837 (554KB) takes forever to load Proof: enumeration by computer programs, using Theorems 9.4.1 and 9.4.2.

Now we have enumerated all the pieces and we could move on to analyze what six-piece burrs can be built using these pieces. But before we do so, we want to mention a few more facts about six-piece burr pieces:

Def 9.5:	Piece Weight The weight of a six-piece burr piece is 12 minus the number of 1*1*1 cubes which have been removed.
Lemma 9.5.1:	The solid six-piece burr piece has a weight of 12. Proof: The solid piece does not have any cubes removed, hence the weight is 12-0 = 12.
Lemma 9.5.2:	The lightest usable piece has a weight of 2. Proof: The following image shows the piece with all but 2 cubes removed. Removing any more cubes would render the piece disjoint and thus unusable.
Lemma 9.5.3:	The weight of a piece is equal to the number of the 12 center cubes which have not been removed. Proof: This follows immediately from the definition.

In Def 4.0 we have defined the weight of a regular burr as the number of 1*1 cubes of the core which are occupied by pieces. Given the above definitions on piece weights, we can now formulate the relationship between piece weights and burr weights:

In Def 5.0 we have defined a level as the number of moves required to release a piece. This definition, however, leaves a couple of open questions. First, what constitutes a move, and second, is the move taking the piece out counted or not. If we assume that a move is considered as moving a piece in a particular direction, and we count the removing of the piece as a move as well, then we arrive at the conclusion that the levels published for the Fearsome Four in Puzzles Old & New [Slocum86] are correct. However the level 9 for Peter Marineau's is not. It would be level 10. We have to further add the constraint that, if two independent pieces get moved in the same direction then we consider this only one move. Note that there is a difference between being able to move two pieces independently in the same direction versus two pieces which can only be moved as a unit. Since our computation algorithms favored the behavior to move pieces independently if possible, special code had to be added to the calculation routines to compact the moves if possible. With this further constraint we arrive at the following more exact definition of a level.

This definition will however reclassify Philippe Dubois' burr from a level 7.4 to a level 6.4. But we decided to stay with this definition since it gets more of the published burr levels right.

Cutler's analysis of solid six-piece burrs

William Cutler was the first to completely analyze solid six-piece burrs in 1978 [Cutler78]. He showed that there are exactly 25 of the possible 59 notchable pieces which can be used to build solid six-piece burrs. With these 25 pieces, a total of 314 solvable burrs can be built. 158 of them use a solid piece as the key.

Cutler further published that there are 369 general pieces which can be used for solid burrs. With these, a total of 119'979 solvable solid burrs are possible. The following table summarizes his findings.

Cutler's analysis of holey six-piece burrs

From the late 1980s to the mid 1990 Bill Cutler and others undertook a complete analysis of all six-piece burrs [Cutler94]. Original estimates by Cutler assumed it would take about 62.5 years to run the analysis on a PC AT. With some help from others who run Cutlers programs on faster and bigger machines, total calculation time was cut down to 2.5 years. From this analysis we now know that there are roughly 35.65 billion ways to assemble burr puzzles pieces (71.3 billion if mirror images are counted also). Of these 35.65 billion logical assemblies 5.95 billion can be taken apart. The highest level found was a level 12 puzzle. Unfortunately it is not unique, i.e . it has more than one assembly, most of which are of a much lower level. The highest level unique six-piece burr is of level 10 if the pieces are 8 units long and level 9 if the pieces are 6 units in length. If all pieces are notchable, the highest level is 5 for a unique burr.

This analysis has completely explored all assemblies for the first piece to be taken out. From there only high level burrs have been completely analyzed. For lower level burrs the programs where not run to complete the disassembly. So the number of total solvable burrs is a statistical estimate.

The following table summarizes some of the findings for the high level burrs: