Definitions

The definitions on this page introduce some basic terms used to describe and define burr-puzzles. The definitions are all numbered in a decimal classification scheme to simplify their referencing. The definitions are mostly trivial and kept brief for that matter.
A few basic lemmas have been interspersed where they follow immediately from the given definitions.

Rods

Rods are the basic building blocks from which burrs are made. There are quite a number of terms used to qualify the peculiarities of rods. Note, that we often use the term piece interchangeably with the term rod.

Burrs

Burrs are what this is all about. Using the definitions of the Rod from above we can now define what a Burr is.

Def 2.0:	Burr A burr can be roughly described as an interlocking geometrical puzzle with a high degree of external symmetry which is composed of notched, connected, rods (often made out of wood) [Cutler78].
Def 2.1:	Regular Burr A regular burr consists of three sets of regular rods intersecting each other at right angles. The length of the regular rods is depending on the particular burr. The burr must not show any external holes, but it can have internal voids.
Def 2.1.1:	Three-piece Burr A three-piece burr consists of three square rods intersecting each other at right angles. The rods have the dimensions 2*2*4. Note: The length used in this definition is 4, the minimum required. However, the length used most commonly is 6.
Lemma 2.1.1.1:	The three-piece burr is the simplest regular burr Proof: Each of the three sets consists of exactly one rod. This cannot be further reduced without making one of the sets empty.
Def 2.1.2:	Six-piece Burr A six-piece burr consists of three sets of two rods intersecting each other at right angles. The rods have the dimensions 2*2*6. Note: The length used in this definition is 6, but there are many six-piece burrs where the rods can be made longer if desired; many use 8 as a length to make real wood burrs for easier handling. Few use longer pieces of 10 or 12 units.
Lemma 2.1.2.1:	Not all six-piece burrs with a rod-length greater than 6 can be taken apart. Proof: Philipe Dubois' burr cannot be taken apart if the rods have a length greater than 6. [Slocum86].
Lemma 2.1.2.2:	For regular six-piece burrs, the pieces can only have the cubes 1 through 12 notched out. Proof: This is necessary to assure that there are no exterior holes in the burr.

Related terms and definitions

The following terms will be used in conjunction with regular Burrs to describe their properties.

Def 3.0:	Core The core of a regular burr is the volume taken up by the intersection of the rods.
Def 4.0:	Weight The weight of a regular burr is the number of 1*1 cubes of the core which are occupied by rods.
Def 4.1:	Solid A regular burr is said to be solid if it is of maximum weight. That is, if the core of the burr is completely occupied by the rods, and there are no interior holes.
Lemma 4.1.1:	A solid three-piece burr has a weight of 8. Proof: Looking at the pictures in the section about Three-piece Burrs will make it immediately evident, that the core consists of a 2 by 2 by 2 cube and its maximum weight is therefore 8.
Lemma 4.1.2:	A solid six-piece burr has a weight of 32. Proof: Slicing a six-piece burr into six layers reveals the 32 internal cubes.
Def 5.0:	Level The level of a burr is the number of moves needed to remove a piece. For example a level 2.3 burr means there are two moves needed to remove the first piece (or pieces) and three more for the second piece (or pieces). Note, that this definition as stated in several books is not very accurate, and has lead our calculation routines to label several puzzles with the wrong level. For a discussion and more accurate definition see the section about Six-piece Burrs.
Lemma 5.0.1:	Solid burrs are all of level 1. Proof: If there are no interior holes, then it must be possible to slide one or more key pieces out in the first move.
Def 6.0:	Type The type of a burr is the number of pieces which get removed at each consecutive move. For example a type 2.2.1.1 for a six-piece burr means that first 2 pieces are removed, then another 2, and the remaining 2 pieces are removed 1 by 1. Note: This definition is different from Bill Cutler's definition! He defines it as the way the first move can be made to take the puzzle apart [Cutler78]. In his definition there are the following types: 1.5 One piece can be taken out (this denotes all burrs with a solid key) 2.4 The burr can be separated into two sets of 2 and 4 pieces. 2.1.3 The burr can be separated by either pulling out 2 or 3 pieces 2.1.1.2 The burr comes apart by pulling out either set of 2 pieces. 3.3 The burr comes apart into two halves. 2.4-3.3, 2.1.3-3.3, 2.1.1.2-3.3, 3.3-3.3 The initial move can be made in one of two ways.
Def 7.0:	Fitable A set of rods is said to be fitable if they can be logically assembled into the shape of the burr. That is if there is a way to fit all rods into the burr.
Def 8.0:	Solvable A set of fitable rods is said to be solvable if the given burr can be taken apart by only moving the rods along the three perpendicular axes. No twisting or turning of the pieces is allowed.
Lemma 8.0.1:	There are fitable burrs which are not solvable. Proof: The following set of pieces fits in exactly one way but cannot be taken apart. See a java or a larger static version of this burr.