Mathematically Interesting Mechanical Puzzles

Stan Issacs took the lead on this activity for the 2012 Circle on the Road festival.

AUDIENCE: Grades 4 – 12 plus teachers

This session will use a collection of mathematical puzzles to engage participants, and draw them into mathematical discovery. These are puzzles that I have been using at many festivals, along with developing activities around new puzzles. The main idea is to have some mechanical puzzles that the participants can play with at first, but can solve much easier by applying some basic mathematical principles that will also apply to more difficult variations of the puzzles. So the goal is to have the puzzles, but also variations that will test how well the participants have understood the solutions.

Instant Insanity (this is possibly the best for older kids, the graphics solutions are a little difficult for younger students, and there are lots of alternative colorings and variations that can test ones solution skills.)
Soma Cube: Straightforward mathematical helps towards a solution, without solving it.
Conway cubes: Also reasonable for younger students, with some good variations

I also use a nice tiling set of puzzles, that basically tile a square or right triangle with either integer or root-two sides, so they have to figure out which, and realize that the integer sides of the pieces must match up with the integer sides of the goal, and similarly for the root-two sides. Not as many good variations.

And I’m working on a set of puzzles demonstrating some of the relationships between a tetrahedron, cube, and octahedron by dissecting cubes.

These are mainly geared towards a student exploring the puzzles and learning the math through the exploration, in a JR type setting. For Math Circles, I think the Instant Insanity is good because it lends itself better to a central presentation rather than a simple exploration.


APPRENTICES: Anne Chen, Elizabeth Tarbutton, Yulan Qing

The Julia Robinson notes should be on the MSRI website, findable if you google for “Julia Robinson”.

And an old writeup follows:

Mathematically Interesting Mechanical Puzzles Selected for Youth
All MIMPSY were the puzzle troves…

I’m looking for mechanical puzzles that can be used with middle- to high-school students that will demonstrate and teach mathematical principles. The puzzles must be interesting enough for the students to want to play with them and solve them, but be difficult enough so thinking about them, instead of just fiddling with them, will be beneficial. In addition, the puzzles should have variations, so that once they have thought through and solved one puzzle, the variations will become much easier. Since I collect mechanical puzzles, that is the kind of puzzle I’m most interested in. Mechanical puzzles are ones that have a physical, three-dimensional component, so they can be fiddled with and examined by hand.

In summary, the puzzles should be:
Good mechanical puzzles, that can be played with and manipulated.
Have interesting mathematics associated with them, not too difficult (eg, Rubik’s Cube), nor too easy (eg, Tangrams), basically aimed at high school and junior high school level.
The mathematics should be discoverable by the students, via a series of hints and suggestions.
The puzzle should have variations, such that once the mathematics has been discovered for the original, it will help solve the variations


1. Instant Insanity
4 cubes with 4 colors stacked in a tower so that each side of the tower shows all 4 colors

2. Soma Cube
6 polycube pieces to fit in a 3x3x3 cube

3. WayCon Cube (Slothouber-Graatsma)
6 2x2x1 pieces plus 3 1x1x1 pieces fit in a 3x3x3 cube

4. Tower of Hanoi
6 disks of 6 sizes on a peg with 2 other pegs; move the disks to another peg, never putting a big disk on a smaller disk

Other possibilities:

Tower of Hanoi and/or Chinese Rings
To discover recursion; to count how many steps

Peg jumping
Discover proofs of possibility: can a shape be solved

Lights Out
Linear algebra


Rubik’s Cube
Group theory (too difficult, and the mathematics applicable to the cube isn’t interesting in group theory

— Stan

Thinking about new puzzles and puzzle variations is always good, and any (mechanical) puzzles or variations you can think of would be great, either before or during the meetings. This is a continually expanding set of topics, and I’ve just scratched the surface of what might be possible. The main thing (from my point of view) is to focus on mechanical puzzles, in the 8 ti 18 age group. But I’m finding more and more that there are also younger students coming to these, and so am exploring possibilities for younger students also. I think the mathematics involved in the four main puzzle-types I’ve working on so far (Soma, Conway, Instant Insanity, Square with Wonder) is more jr. high and high school then younger elementary school. The young ones may enjoy the puzzles, but are less likely to understand the underlying mathematics. So I’m looking at several geometrical puzzles (cube dissections, ball pyramids) that might be fun to play with and would help develop spacial intuition. But any ideas of puzzles along these lines would be greatly appreciated!

I find just as big a problem in getting the students to get from just playing with the puzzles (which is fun in itself) to getting an appreciation for the mathematics. That’s why the variations are important, but still, getting from a solution to an understanding remains a difficult problem. Getting the students to read the write-ups doesn’t often work (for most of them, anyway.) Encouraging them to think about the variations mathematically might work, but takes personal contact. I now have a couple of Soma variations that are impossible, and should be provably impossible by the students, giving the solving technique suggested. Proving something impossible might be the cleanest way to show the efficacy of the mathematics!

I’ve attached write-ups of the main 3 puzzles. They are similar to the website, but I’ve made some small changes. Any editorial suggestions you can make or errors you find would be gratefully accepted. I’m not sure what kind of computers anyone has, so I’ll send them as pdf files. (The last one is a “Show What You Know” page, where we give ways for the students to demonstrate their knowledge to the monitors.) (Josh: I haven’t changed the Instant Insanity or Square with Wonder.)

— Stan

Instant Insanity.pdf
Show What You Know.pdf 
Soma Cube.pdf