This week I listened a great conversation between Steven Strogatz and Federico Ardila on the Joy of X podcast:

We have played with the shape a few times before – see these blog posts of you are interested in seeing other ways that kids can explore the shape:

https://mikesmathpage.wordpress.com/2017/03/09/a-fun-shape-for-kids-to-explore-the-permutohedron/

https://mikesmathpage.wordpress.com/2019/10/27/labeling-each-vertex-of-a-permutahedron-is-a-terrific-mathematical-exercise-for-kids/

Today we explored the and idea that Ardila discussed in the podcast – finding paths on the permutahedron.

We started by just reviewing what the shape is and what it represents:

Next we tried an easy example of finding a path on the permutahedron going from a random permutation back to the correct order:

For the last part of the project we tried a more complicated scramble of the cards and found that walking back to the unscrambled state would take a minimum of 3 steps:

I love playing with this shape with kids. It is a great way to get them talking about fairly advanced mathematical ideas and also allows them to see a really neat 3d shape that research mathematicians find interesting!