Author: GeorgeHart, published on 2013-11-18
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.
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.
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.
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.
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.
- 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.
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.
Tags: curriculum, MakerBotAcademyMath