# Five screw-puzzles by George Hart

## by GeorgeHart, published Nov 18, 2013

Five screw-puzzles by George Hart Nov 18, 2013

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# Summary

These five puzzles challenge anyone who plays with them to think about combining the geometric transformations of translation and rotation in new ways. In a math class, they also provide inspiration to see that mathematics has fun and creative applications. Furthermore, they get progressively more challenging and so lead students to be comfortable with the important skill of exploring new problems that they have no idea how to solve. The five as a set will lead students to think about problem-solving strategies and provides a healthy "Aha!" when the solution to the last one is found, giving a feeling of enjoyment of mathematics.

1. Two-part bagel. This is trivial to solve, but calling it a "bagel" naturally leads the user to the next challenge of trying to understand the geometric form so they can reproduce it by cutting a real bagel in the same manner.

2. Four-part torus. Conveniently, you can never lose a piece, as every pair is linked. Slightly harder to solve. One approach is to put together two pieces and two pieces so it reduces by a divide-and-conquor approach to the easier two-part bagel. Or you can solve it one piece at a time. If you master this shape, you can also replicate it by cutting a real bagel.

3. Six-part torus. The next logical step. Advanced students will enjoy the challenge of proving that the six parts can be assembled in any sequence, e.g., if you label them when solved as ABCDEF you can also solve them as ABEDCF or any other permutation.

4. Four-part cube. These four identical parts assemble into a cube without any force required. It can be solved in an incremental manner, by adding one piece at a time, or a divide-and-conquor approach of first making two sub-assemblies that join.

5. Two part tetrahedron. Although there are only two pieces and they are identical, this is the most challenging of these five puzzles. Again, no force is required. It is OK if it takes you more than a half hour to solve it.

# Instructions

I've built all of these on a Replicator with ABS and they work well. (See photos.) When designing them, I applied a slight setback to the surfaces (relative to the mathematically exact place to slice them) so that a bit of room is available to account for imperfections in the 3D printing process. If your 3D printer is mathematically perfect, there will be a millimeter or so of space between the parts when assembled. If your 3D printer leaves ooze or rough patches on the surfaces, lightly sand with 100 grit sandpaper.

The first two puzzles are positioned so the parts are widely separated. If you build with full support, is should be easy to get between the parts and separate them cleanly. Puzzle 3 is built in the assembled position and you can carefully separate the parts with a small blunt screwdriver. For the last puzzle, build the file twice to make two pieces, ideally with two colors.

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Do any of them need supports when they are printed?

Jun 8, 2016 - Modified Jun 14, 2016

I feel so stupid, I can't figure out this damned tetrahedron. I'm starting to wonder if it's user error or printer error. The pieces look good though!

EDIT: Ended up coming back to this puzzle a few days later and finally solved it. I made a video for the solution in case anyone else needs a little help :) https://www.youtube.com/watch?v=2SLyZ-0kw_k

Thanks for the video!

tetrahedron works like a charm* with 10% infill, but if i use 100%, it won't fit together (and breaks if i try to make it). just figured folks might like to be aware of that

*still infuriatingly difficult to figure out the first time though

profhankd - in reply to tpchuckles

I'm building a bunch of these for a STEM group. Unfortunately, I agree: the more rigid these are, the tougher it is to fit them without breaking them. I think the key is that the tetrahedron could use a little larger tolerance; maybe 1.5-2X the supposed 1mm it has now. Any chance that could happen? I'd prefer not to do it from the STL file....

profhankd - in reply to profhankd

One of my students and I redid this from scratch using OpenSCAD to make the "Tunable Tolerance Tetrahedron Twist Timewasting Toy" -- http://www.thingiverse.com/thing:1542564

Our version allows use of the Customizer to set fit tolerance (gap), side length, thickness (to hollow the tetrahedron), inside/outside edge rounding, and even imprinting with a UK (University of Kentucky) logo. BTW, I don't know how you computed the tolerance, but a 1mm tolerance on ours gives a really loose fit without any sanding....

Tunable Tolerance Tetrahedron Twist Timewasting Toy

Congrats! I thought you model was one of the best choices for a 3D Printed Christmas gift! I made a link from here: http://3dprintingninja.blogspot.com/2014/12/top-20-free-3d-printed-christmas-gifts.html

Can anyone recommend a good instructional video for using supports? I have no idea where to begin with them and don't want to accidentally damage my machine.

These puzzles are tantalizing, but I'm afraid of going all FUBAR.

Nice puzzles. I printed the pyramid 2-piece puzzle in white and translucent orang, it took me a while but finally figured it out! Thanks for sharing it, I'll post a photo of it.

Great puzzles and great fun. I printed the Four-Part-Cube over the weekend using a MendemMax2, Slic3r, and Prinrtrun. I printed it with supports and zero fill in ABS. After the print I removed the supports and sanded some rough edges and it came out beautifully.
It took me about 40 minutes to solve it the first time. After discovering the solution it takes less than a minute.
Thanks, great work.

Brilliant work. As far as teaching goes, from my perspective based on what I am trying to learn having OpenSCAD source would be better for me.

I own a low-quality printer and I can't solve the two part tetrahedron. I don't know if it is caused by the bad print quality or if I just need to keep trying. Can you please include a solution video? :-)

superm - in reply to HaPe

what printer do you have?

Jun 7, 2015 - Modified Jun 7, 2015
hpederse - in reply to superm

I have a Micro 3D, and I had to spend some quality time with a file and some nippers to get it, after looking up the start position on Georges site to make sure I wasn't just totally off in how I thought it should go together (I was a bit off, but the print did need fixing as well)