Ed Pegg Jr., August 2, 2004

In 1803, the Bestelmeier Toy catalog listed a 6-piece puzzle. In 1899, Scientific American introduced a three piece puzzle by Wilhelm Segerblom. In 1917, US patent 1225760 described a 6 piece burr puzzle. All of these puzzles were difficult to take apart -- interlocking pieces held the puzzle together. Over the next 60 years, millions of similar puzzles were made. The designs were so overused that they were considered somewhat trite, as a puzzle type.

The rebirth of interlocking puzzles began with columns in *Scientific American*, in particular about Miguel Berrocal and Burr puzzles. One burr puzzle, by Bill Cutler, appeared in the October 1985 issue. Due to an printing error, Bill's Baffling Burr was impossible to solve, which prompted many letters. The Bill Cutler Puzzles site contains more of this history, available puzzles, books, and software. The **burr puzzles site** by IBM Research explains some the method of computer analysis undertaken by Bill Cutler.

Another person involved in the rebirth of puzzles was Stewart Coffin. His book, The Puzzling World of Polyhedral Dissections, is available online at John Rausch's PuzzleWorld, and should be considered a must-read for any recreational mathematician. Where possible, Stewart has put his designs into the public domain. One might suspect that this would dilute the collectability of his puzzles. Not true. Puzzles made by Stewart are the most sought after of all modern puzzles. I highly encourage the reader to build one of his best puzzles - The Three Piece Block. All it requires is 10 cubes. I built one myself with a $1 set of wooden cubes from a craft store and some wood glue during a wait at mechanics shop. I first built 10 side-2 cubes, then followed the directions. Easy. But not easy to solve.

As a result of these searches, puzzles of higher order were found. Just to take a 6-piece puzzle apart might take considerable fiddling. Bill's Baffling Burr, for example, requires 5 moves to remove the first piece, and is thus considered an order-5 puzzle. There are also 24 assemblies that might work, but it turns out that only one of them truly works. Hence, the solution is unique. After the analysis was completed, the Burr Puzzle was considered settled, for a few years.

In 1991, Junichi Yananose investigated 4-piece burr puzzles with a frame, and found one with a unique order-9 solution. Minimizing the pieces while maximizing complexity is considered a good thing. Juno followed up with a 17-move 6-piece framed burr with a unique solution, and then the search was on again for new burr puzzles.

In 1999, Dic Sonneveld and Frans de Vreugd used Bill Cutler's program to analyze a class of 12-piece burrs. The "Dic's Dozen" analysis found an order-60 puzzle. In 2001, they looked at framed burrs, and found a 4-piece puzzle with a order-47 solution. "It has baffled both Dic and myself that such extremely high levels can be reached with only so few pieces." One of these extreme puzzles is available very inexpensively at Bits&Pieces, the Boxed Burr. In 2002, Dic found an order-98 framed burr with 6 pieces.

In 2001, Bill Cutler and Frans de Vreugd got stuck in traffic in Tokyo, and discussed an attack on the 6-board burr. They identified an order-13 puzzle, and the order-11 puzzle below, which has a unique assembly.

Figure 1. Frans de Vreugd's Irregular Board Burr, with an order 11-3-3-2-3 solution.

Extending this, Frans looked at 6 board burrs where parallel boards were a unit space apart. This is where things got really interesting.

The above was built with the LiveCube system, which has been a great boon to puzzle designers.

If you haven't tried Burr Puzzles, you might want to start with the Aluminum Rainbow Brainteaser Puzzle by Paul Eibe. From there, you can move up to the Boxed Burr. Alternately, you can obtain a puzzle from a craftsperson. For that, I feel I must explain a few things. Most craftspeople make their puzzles for the sheer enjoyment of building them, and not for money. The puzzles are cut and assembled by hand. Usually, they will make the puzzles in batches of a dozen or so, and that might be all! They usually have a request list, and build a particular puzzle when there is enough demand for it. That said, you can obtain excellent puzzles from Josef Pelikan, the PuzzleWorld designers, Puzzle Palace, Puzzle Craft, Cubic Dissection, GarE Maxton, and Mr. Puzzle.

In the past few years, thousands of excellent designs

**References:**

Stewart Coffin, The Puzzling World of Polyhedral Dissections, http://www.johnrausch.com/PuzzlingWorld/default.htm.

Bill Cutler, Bill Cutler Puzzles, Inc. http://home.comcast.net/~billcutler/.

Eric Fuller, Cubic Dissection.com. http://www.cubicdissection.com/html/purchase/purchaseindex.html.

André van Kammen, Puzzlesolver3D, http://www.puzzle.cx/.

Jürg von Känel, "the burr puzzles site", IBM Research, http://www.research.ibm.com/BurrPuzzles/index.html.

Ishino Keiichiro, "Puzzle will be played", http://www.asahi-net.or.jp/~rh5k-isn/Puzzle/.

GarE Maxton, Maxton's Conundrum, http://www.maxton.com/max5.html.

George Miller and Frans de Vreugd, "Extreme Torture", Puzzle Palace. http://puzzlepalace.com/puzzle.php?catalogNum=200201.

Robert C. Morgan, "The Puzzle of Existance", Sculpture Magazine, April 1999, Vol.18 No. 3 (online version).

Josef Pelikan, The Pelikan Company, http://www.hlavolam.cz/homeeng.htm.

John Rausch, "Miguel Berrocal", PuzzleWorld. http://www.johnrausch.com/PuzzleWorld/des/mb000.htm.

Steve Strickland, "Burr Puzzles", Puzzlecraft.com, http://puzzlecraft.com/.

Frans de Vreugd, "Some Notes on Dic's Dozen", Cubism For Fun 52, June 2000, p 16-19.

Frans de Vreugd, "Extreme Boxed Burrs ", Cubism For Fun 56, October 2001, p 32-37.

Frans de Vreugd, "Cracking the Six-Board Burr", Cubism For Fun 62, November 2003, p 18-23.

Junichi Yananose, Juno's World, http://www.ne.jp/asahi/j/yananose/puzzle/zpuzzle01.html.

Brian Young, Gallery of Interlocking Puzzles, http://www.mrpuzzle.com.au/webcontent69.htm.

Comments are welcome. Please send comments to Ed Pegg Jr. at ed@mathpuzzle.com.

Ed Pegg Jr. is the webmaster for mathpuzzle.com.
He works at Wolfram Research, Inc. as the administrator of the
*Mathematica*
Information Center.