Burr Notation and How Solvers Talk About Pieces

Two people who have never held the same puzzle can agree precisely on what it is, because burr collectors describe pieces with numbers. A design is written as a short list of integers plus a level, and that is enough to reconstruct the object exactly. This is not decoration. Without it, the computational work that mapped the entire six-piece burr design space would have been impossible to state, let alone check.

Why notation exists

Three practical reasons.

• Reproducibility. A photograph shows you the outside of an assembled burr, which is the one view that tells you nothing. The interior is the puzzle.
• Cataloguing. There are far too many valid designs to name individually. Numbers scale, names do not.
• Computation. A search program needs pieces as data. Notation is the interface between the woodshop and the algorithm.

Numbering a burr piece

A standard burr stick is two units square. The section that sits inside the assembled knot is divided into small cubic cells, and every possible piece is defined by which cells are present and which have been cut away.

The convention assigns each cell a fixed position in an agreed order. A piece then becomes a pattern of present and absent cells, which can be written as a single number by treating the pattern as binary digits. Two makers using the same cell ordering will always produce the same number for the same physical stick, and can cut an identical stick from someone else's number.

A full puzzle is then written as a list of piece numbers, one per stick, in a defined order of positions and orientations. Six numbers, and the design is fully specified.

This is exactly what made exhaustive analysis feasible. The summary of Bill Cutler's work hosted by Brandeis University computer scientist James Storer records that 35,657,131,235 possible standard six-piece burrs were analysed, that the maximum level found was 12, that no burr of level 11 exists, and that 5 is the highest level for a standard six-piece notchable burr with a unique solution. Numbers on that scale are only checkable because each design has a canonical encoding.

What level actually means

Level counts moves, not pieces. A level 5 burr requires five distinct moves before the first piece or subassembly comes free. It does not mean five pieces, five minutes, or a five out of ten difficulty rating.

Extended level notation uses several numbers separated by dots. The first is the moves to release the first piece, the second the moves to release the next, and so on. A puzzle written as 7.3.2 gives up its first piece after seven moves, then needs three more, then two more.

What counts as a move is a convention that has to be agreed. Usually one move means sliding one piece or one rigid group of pieces by any distance in a straight line until something blocks it. Rotations are often excluded from the count, or flagged separately, because many solvers and most classical solving programs assume translation only.

Notation as the basis for design software

Once pieces are data, you can generate them rather than only catalogue them. The SIGGRAPH 2011 paper Making Burr Puzzles from 3D Models by Xin, Lai, Fu, Wong, He and Cohen-Or takes the classic six-piece orthogonal knot and generalises it, embedding a network of knots into an arbitrary 3D shape. Its stated design constraint is the single key property: when the puzzle is assembled, every piece is notched except one mobile key component, and the result is stable and interlocked without glue or screws.

The wider engineering literature formalises the solving side. A preprint accepted in IEEE Transactions on Automation Science and Engineering on task and path planning for maintenance automation surveys how disassembly sequences are represented, listing directed graphs, AND/OR graphs, diamond diagrams and Petri nets, alongside methods such as the directional and non-directional blocking graph. It also notes that computing an optimal disassembly sequence is NP-complete, which is why heuristics matter so much.

Comparing puzzles you have never touched

The payoff for a collector is concrete. Given a piece list and a level, you can tell before buying whether a puzzle uses only notchable pieces, whether it is solid or holey, whether the solution is unique, and roughly how long the opening sequence runs. You can also tell whether two puzzles sold under different names are actually the same design, which happens more often than you would expect.