This section covers interlocking puzzles - wherein multiple pieces fit together such that the puzzle does not fall apart and presents a challenge to disassemble and re-assemble. Use the links in the panel below to jump to a section of interest. Interlocking Misc. Metal Misc. Figural The Puzzle Sculptures of Miguel Berrocal Traditional 6-Piece Burrs Any story about interlocking puzzles has to start with the traditional six-piece burr puzzle.
The term "burr" is thought to have been first used by Edwin Wyatt in Puzzles in Wood , but Wyatt seems to use the term as if it was already commonly understood to apply. Supposedly whoever coined the term did so because the puzzle resembles the clinging burrs of some plants. Like other well-known vintage puzzles, the burr has acquired a probably-fanciful backstory, and details of its history are lost.
According to the literature, the earliest relevant U. Patent seems to be - filed by O. Brown on June 27, and granted on May 15, But take a look at U. Patent , filed by J. Keiser on March 16, , and granted on April 2, Keiser seems to have filed earlier but his patent was granted later. Keiser's pieces are the Chinese Cross set; those pieces are shown in an book so Keiser did not invent them. Jerry Slocum and Dieter Gebhardt put together a compendium of puzzle advertisements found in the catalogue of the merchant Peter Friedrich Catel , who established a retail store in Berlin in The catalogue contains an ad for a traditional six-piece burr puzzle called "The Small Devil's Hoof" in addition to an ad for the Large Devil's Hoof which is a piece cage burr , but the individual pieces are not shown.
Brown's Patent filed June Keiser's Patent filed March Hoffmann's "Nut" Puzzle One early depiction of the six-piece burr puzzle and specific pieces occurs in a Spanish book, primarily on the topic of magic, from by the many-talented Pablo Minguet y Irol b. Read about this engraving at the University of Oxford.
Martin Gardner discusses burrs briefly as an introduction to the puzzle sculptures of Miguel Berrocal in his book Penrose Tiles to Trapdoor Ciphers , and most of the key puzzle authors mention the puzzle. There have been sporadic fits of research into the six-piece burr, including an extensive analysis by hand by the Dutch mathematician J.
One can visualize an individual burr piece as being composed of unit cubes arranged in a 2 x 2 x 2 n prism where n is greater than or equal to and usually 3. A solid piece will contain 24 unit cubes, and other piece types will have some of the cubes removed, resulting in notches.
See diagram above left. The burrs in this section are composed of six such pieces, often but not always distinct, selected from the overall set of possible such pieces of a given length , and interlocked in a characteristic 2x2x2 pattern along 3 orthogonal axes.
The burr shape is tricky to envision without an example in front of one, but it gets easier with practice. See diagram above right. One may only remove up to 10 of 12 specific cubes from a 2x2x6 prism before it becomes disjoint or improperly notched for this type of puzzle for example, showing notches on the outside where they shouldn't be visible. Overall, this results in distinct physical pieces. Cutler determined that there are 35,,, ways that six pieces drawn from the universe of fit together in the requisite shape allowing dups of pieces within a set, but discarding rotations and mirror image assemblies of sets , but of those 35 billion, "only" about 5.
There is a distinction made between burr puzzles that contain no internal "holes" or voids - termed solid burrs, and those that do contain one or more - termed holey burrs. There are , solid burrs , and there are piece types needed to produce them. Of those , are used in duplicate and 2 in triplicate, making a useful set of pieces to make all the solid burrs. The rest of those 5. A holey burr can contain from 1 to 20 holes. The weight of a burr relates to the number of internal holes it has, and can range from 32 no internal holes , down to 12 the maximum of 20 holes.
The weight of a piece refers to the number of cubies not removed from it, and can range from 12 the key down to 2 the Y. If the sum of the weights of six pieces exceeds 32, it is impossible to construct a valid burr from that set. Also, there is a distinction made among the pieces which can be produced without hard-to-manufacture blind or internal corners i. Any piece without any such blind corner can be made using a milling machine and is millable , otherwise it is a general type piece.
In a millable piece, any cut parallel to the long axis of the piece is bounded on both ends by a cut perpendicular to the long axis. There are 78 millable pieces. However, to produce pieces on a table saw with a dado blade , or by hand without resorting to a chisel, one must also avoid internal edges that run parallel to the piece's long axis, and employ only cuts running perpendicular to the long axis.
These pieces are called notchable , and there are only 59 of them they're all millable, too. Only 25 of those 59 pieces are useful to build solid burrs, and only solid burrs can be made from that set of 25 some dups are required, so you need a set of 42 pieces with dups.
Overall, the 59 notchable pieces can be used to make 13,, assemblies. The level of a burr puzzle is the number of distinct linear moves a shift of one or more pieces together, sometimes by one unit but usually by an arbitrary number of units, in just one direction that must be performed to remove the first piece or pieces - there can be a concatenation of figures usually separated by dots - these are the numbers of steps to remove successive pieces.
All solid burrs are level 1 - they come apart without any preliminary shifting. Burrs with internal holes can achieve higher levels, and one goal of research has been to delimit what is possible in terms of level complexity.
Bill Cutler has done extensive analysis on both the "holey" six-piece burr and all six-piece burrs in general , and Bill offers several burrs for sale.
Ed Pegg wrote a good survey article about burrs. Peter Roesler's site also discusses burr puzzles, and has an interesting history of Willem van der Poel's Grandfather 6x6x6 burr. You can see some burrs at John Rausch's Puzzleworld. Bruno Curfs' site now defunct? In lieu of the applet, you can use BurrTools to analyze burr puzzles. If you're interested in collecting 6-piece burrs, I suggest you first check out Ishino's "Puzzle Will Be Played" site to get some idea of the variety available.
Look under "Interlocking 6 piece burr: traditional. That site also provides a comprehensive catalogue of burr pieces. Note that when discussing traditional burrs, twists or rotations of pieces typically are not required or allowed. It is possible, however, to design burrs that appear traditional but require such moves and frustrate the usual computer analysis - for example, see Bill Cutler's Programmer's Nightmare burr.
For some burr designs, twisting a piece might be possible and might offer a shortcut, but isn't strictly required. It is also possible to mimic the outer appearance of a traditional burr but use different internal notchings - but such designs are outside the scope of this section e. Cutler's Explode-A-Burr. See U. Patent - Dykstra for an interesting burr that can be assembled in two distinct ways. I admit that, early on, I didn't like burr puzzles. But as I read more about them, and tried various designs, my appreciation for them grew.
I put together the diagram below to try to summarize and organize some of the facts I learned about this category of puzzle. Check out a nice writeup on how to go about solving 6-piece burrs, written by Guillaume Largounez, over at the Puzzle Place Wiki. Identifying Burr Pieces Over the years, different researchers and writers have employed different schemes to identify the pieces.
Some have used rather arbitrary letters or numbers; others have devised more systematic schemes employing a mathematical calculation based on assignments of binary values to "cubies" or "cubelets" to be removed from the unnotched basic block.
For symmetric pieces without a mirror image, this gives Ishino's ID. For pieces that have a mirror image, the result gives Ishino's ID for the mirror image piece. My piece ID is 1 plus the value, shown below, of each cubie removed. The cubies behind cubies and can be removed, too, and have respective values and Such pieces appear infrequently.
Sometimes a piece could be assigned more than one number - use the smaller number. This entails orienting it so that cubies and are present if possible. The ASCII character-based renderings rely on fixed-width fonts and won't display well on some devices, particularly phones.
These are depicted as length-6; for longer pieces simply extend the 2x2 solid burr equally on each end. The first number is the piece ID as described above.
The first letter, in bold , is the "standard" letter ID for the piece, and is used in Pentangle's set. The second letter is as assigned by Curfs and is also used in Wayne Daniel's i. The third letter, if present, is that assigned by Edwin Wyatt in Puzzles in Wood.
A [p n ] suffix, if present, indicates the piece is included in the Professor burr set and n is the count 1 if absent. The next number is the count of this piece in a piece set that allows you to construct solid burrs. The last number is the count of this piece in the Level-5 set.
I have lately given names to some of the pieces, which I find more helpful than the letters or numbers when trying to remember sets of pieces I have seen before. Piece 1 is the "key" piece. No more than one Key appears in any puzzle.
Also, when the key 1 is used, neither 18 nor 35 can be used in the same puzzle with it. Can you tell why? Piece Y is the "minimal" piece - no more material can be removed without the piece falling apart. I have located some of the pieces out of numerical sequence, to show related pieces together. Three symmetric pieces The pieces are in numerical order from top down left to right, but I show mirror image pairs together using an arbitrary color.
Notchable pieces will have an N after the ID , non-notchable but millable pieces will have an M. Non-millable and therefore non-notchable pieces have internal corners and are more difficult to manufacture.
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