I've made some good explanation diagrams and animations, and I've labelled that map I linked to earlier according to what card type goes where. I've been trying to put together something of a careful explanation of the math, but I keep getting distracted just trying to understand it better myself.
So I'll post more in a bit.
I wanted to make a note of a certain realization, though, for anyone who might be following along. I tried putting the cards in sets of three just according to simple shared ideas, just to see what it would look like. But it didn't add up - my choices were contradictory! It turns out there's more structure than I thought, and now I understand the difference between the "A sets" and the "B sets" that I was talking about in my first post.
Firstly: The A sets are the points of a Fano Plane. The B sets are the lines. (Or, the other way around - it makes no difference which is which.) What this means is, if you pile the cards into seven stacks, one containing each A set, then you can arrange the stacks like the triangular Fano plane diagram, and if three A-sets are on the same line, then there's a B-set which contains one card from each pile on the line.
Once I've chosen the A-sets, I can lay them out in the diagram and decide on three B-sets that I like; after that, the math gives me everything else. I think regardless of whether they're A-sets or B-sets, 10 decisions are all that's needed.
Secondly: The Fano Plane formed of the major arcana sorts the cards naturally into 3 sets overall. (The math is a bit harder to explain here, or, I haven't found the simplest explanation yet.) Basically each stack of cards in the arrangement will have a top card, a bottom card, and the card in the middle. Each A-set obviously contains one top, one middle, and one bottom card. Well it turns out, each B-set *also* contains one top, one bottom, and one middle card.
Much more interestingly, all the "top" cards form a circle; their order in the circle is determined by our choice of sets! And the same is true of the "middle" cards and the "bottom" cards! So by choosing sets of 3 related cards we're actually choosing an ordering. Well, not quite; three circular orderings, each of 7 cards. And the three circles can be lined up concentrically so that the "A" sets or "points" are all aligned, or so that the "B" sets or "lines" are all aligned.
I *think* that it's natural to call the top/middle/bottom cards "beginning", "middle" and "end" cards, and to walk each "line" set from "beginning" to "end" and then find out which "point" you're in and switch back to "beginning" and keep going. This gives a natural ordering to trumps.
There's still a lot of wiggle room to choose a correspondence between points and lines, but I'm much closer to having a single correspondence.
You might like the Tarot of Ceremonial Magic by Duquette. It's full of all kinds of stuff that I haven't studied yet. Mathematics, including algebra, are probably among the intriguing aspects of the deck, although I wouldn't guarantee it.
Thanks for the recommendation!
Algebra and I had a lover's quarrel many, many moons ago when mostly what I learned in Algebra class was that if we could get the teacher (who was also a football coach) to talking about football, we didn't have to bother with the rest of the gibberish of algebraic equations. I could figure out the answers but he insisted we had to do equations to show how we got to the answer and that's where I flopped on my face.
Sounds familiar. I ended up having a couple pretty good teachers but I've heard stories of exactly this before; being a coach and a math teacher seems to be a bad combination.
I do want to mention, algebra and abstract algebra are pretty different! In abstract algebra, the equations are similar but the letters stand for something very different from numbers.