Burr Puzzles

When I was a grad student, I used to keep a basket of burr puzzles on my coffee table for guests to fiddle with. Today we'll look at an activity where students design them before solving them!

Setup

I'm going to use Gram Unit Cubes for this write-up because they're cheap and plentiful. However, they are small, only have connectors on select sides, and the things we'll construct can be a little on the fragile side. This works for students with developed fine motor skills and patience to plan ahead. For easier construction and more robust creations, I'd recommend switching over to MathLink Cubes or Linking Cubes. You can also do this as a project using wooden cubes and glue.

In each of the puzzles, students will be given six pieces to construct that they must then assemble into this "knot" shape:

Example

Make and solve the puzzle with pieces below:

Here are the pieces created from Gram Unit Cubes:

If a puzzle has duplicates of the same piece, I like them to be the same color, but that's not necessary. Here is one way to go about assembling the puzzle pieces:

Challenges

Here is a collection of \(25\) puzzles. There are a number of well-known puzzles and puzzle sets in the recreation math community and I've included some of those. However, the emphasis is usually on coming up with fiendishly tricky puzzles, whereas I wanted a much lower floor, so I included a number that have similar assembly styles, symmetric assemblies, duplicate pieces, and other things that make solving them easier. The first five should be pretty straightforward. The last five are the fiendish ones. The fifteen between them hopefully increase in difficulty with a bit of a spike as you hit the difficult end, but the ordering comes from doing them by hand a few times, and you and I may disagree on relative difficulty!

The puzzles towards the back end require some delicate maneuvering, so will likely test the structural integrity of the pieces you've made. The goal is to be able to assemble the puzzle without taking the pieces apart, since some of the assemblies become much, much easier if you are allowed to take pieces apart and recombine them on the larger structure.

Notes

• I like this activity because students use 3D reasoning in a few different ways. First, they have to construct the pieces from a 2D spec sheet. Then they have to deal with the 3D constraints of assembling the knot, which can require some interesting maneuvers.

• Each of the pieces can be thought of as being carved from a \(2 \times 2 \times 6\) array of cubes. In order to give the semblance of a solid knot once assembled, there are \(6\) cubes on each end of a piece that will never be carved away. However, the middle \(12\) cubes can be carved out in many ways that leave a connected puzzle piece.

  

  If students are interested in designing their own puzzles, they can work out that the volume the completed knot occupies is \(96\) cubes, so the total number of cubes among their pieces can't exceed that. Since there can be voids inside the knot, the total volume of the pieces can be strictly less than \(96\) cubes, however.

• Since the solid piece must have a fully clear path to slide into place, it is the easiest piece to plan around, and a good way to go about a puzzle that features it is by arranging the other five pieces to accommodate it.

  

  Many of the puzzles in our set feature this piece.

• Must puzzles that do not feature the solid piece will require some interesting maneuvers to get the final piece in place. Puzzle 15 from our set requires the last two pieces to slide in at once to assemble with the other four, for example:

  
  
  

  There are a few other puzzles in the set that have this sort of structure.

• When a puzzle has voids, assembly can be a great deal more complicated. Burr puzzle designers call the number of moves required before the first piece (or pieces) can be pulled free the level of the puzzle. The level includes the movement that frees the piece, so all burr puzzles are at least level \(1\). Since these make for thornier puzzles, I've put them at the back end of the puzzle set.

• Bill Cutler did a computer-assisted analysis of six-piece burr puzzles and found all \(35,657,131,235\) ways to assemble a knot from these pieces. This spun out a few puzzle sets of pieces that allow a large number of distinct knots to be assembled relative to the number of pieces.

  Though Bruno Curfs' site seems to be down these days, Jim Storer saved a chunk of it as this document. On pages 14-19 Curfs gives a list of \(42\) pieces (\(25\) unique pieces with some duplication) that can be used in various combinations to create a slew of different knot assemblies. He has a table on pages 20-25 of the pieces for each assembly and a difficulty rating for each puzzle. I'm not sure if this was Cutler's original list of \(42\) pieces, since other \(42\)-piece sets have been created over the years.

  Darryl Adams wanted a smaller set that delivered a good number of assemblies, and created Darryl's Dense Dozen. He gives \(42\) examples of puzzles that can be made from the set, with a recommended list of \(10\). A few of these found their way into my puzzle set.

• My set features the following famous puzzles:

  • Darryl's Level-2 Burr (puzzle 20)

  • Coffin's Improved Burr (puzzle 21)

  • Cutler's Burr 305 (puzzle 22)

  • Cutler's Burr 306 (puzzle 23)

  • Bill's Baffling Burr (puzzle 24)

  • Philippe Dubois' Burr (puzzle 25)

I couldn't include them all without kicking the difficulty way up, but Jim Storer has a few other historically interesting burrs cataloged.

I won't count it as famous, but here's a six-piece burr puzzle that I must have picked up in high school or undergrad:



It appears as puzzle \(9\) in my set.

• I spent a chunk of the weekend 3D printing many of these pieces to make testing the puzzles easier on myself. They're a bit messy since I was just trying to knock things out, but if anyone wants some OpenSCAD files, let me know!

This post was sponsored by the Julia Robinson Mathematics Festival