As a project with a certain 4-year-old relative of mine, we constructed the proof I described before that the outer vertices of George Hart’s 12-Card Star form a 4!-torsor. (I guess I didn’t say it that way before, but it’s true!) Here’s our proof:

Last time I suggested using a deck of 12 cards like this:

But instead, we used four solid colors, three cards of each. So, our “star” permutes the colors red, white, black, and silver:

You can get any permutation of these colors in our Star by *exactly one* symmetry taking outer vertices to outer vertices. The “exactly one” in this isomorphism is what makes the set of outer vertices a 4!-*torsor* rather than just a 4!-set.

Here’s what it looks like when you put three pieces together, from both sides: