The Challenge: Create your very own order 4 Permutohedron (see video below).
Materials Needed: Paper, pencil, patience, and if you want to go 3D, you sure can. Post-it notes work pretty well when playing with this.
Math concepts you could explore with this challenge: angles, combinations & permutations, geometric construction, geometry, graph theory, polyhedra, sequences, vertices/intersections.
The image that begins this post is a flattened version of the answer, but it’s really satisfying to find. If you were to build it in 3D you would get a truncated octahedron. Every permutation should be connected to 3 other permutations.
Depending on how you use this activity, you may engage with different mathematical standards. I’ve listed possible connected math content above. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!
1.) Make sense of problems and persevere in solving them. The solution here is not immediately obvious. This problem requires thoughtful processing of complex information, and a clear understanding of a final goal.
7.) Look for and make use of structure. In this, you get to make use of not just the structure of permutations of 4 numbers, but also the structure of polyhedra. Yay!