Abstract algebra and the knight. (as in chess, not the knave that the box suggests) - How do I study the patterns?

Question

(background removed -too long, note terminology may be wrong, I had to teach myself, armed with a 4AA powered chess computer)

When moving a knight in chess its colour changes the colour of the square on which the knight rests changes for each legitimate move. So in two moves it is on the same colour square as the one on which it started.

This means when scanning for what you can do with a knight you can discard everything of a different colour.

I learnt quickly that anything on a diagonal to the knight (same colour squares) is not reachable in two moves (as the above suggests), for example 2 across and 2 up from the knight cannot be reached in 2 moves.

I also noticed that most things that work with this rule can can be reached in two ways. For example the square of same colour as the knight can be targeted by two moves (2 in either direction, 1 up), although I usually looked further away from the knight than the square 2 in front.

When trying to find a path from a knight to a target, look at a square and then imagine a "poor attempt at a circle" around it, that's where it can go. So finding the place to put a knight from where it is to threaten that square is as simple as "where do these circles intersect?" which is 0 or 1 or 2 points. Unlike actual circles it is 1 point if the centre of one circle is directly vertically or horizontally in line with the intersection.

That's my method for quickly finding out how to get a knight from A to B, 3 steps is not much more complicated either!

Now there is algebra at play here (try drawing it, draw a big circle for squares reachable in 2 moves, and little crosses for 1 move, it's like a flower of sorts. Also that "poor circle" might be an iteration of a larger shape, because the boundary squares for what can be reached in 2 moves have boundaries parallel to that of the first step (granted there are only 8 directions), after 3 moves it becomes almost a filled in diamond shape.

Why is it to get to a square not reachable by the second step inside the bounding octogon of 2 moves, vertically or horizontally adjacent can be reached i 3, yet that unreachable diagonal takes 4? (Everything immediately diagonal to the starting square is reachable in 2 moves, 2 applications of that is 4, an application of moving 2 horizontally, followed by 2 vertically (reachable only in 2 steps) is also 4 - composition of something?))

There is algebra at play, but I have no idea how to write any of it, or say what is a group, there's obviously no unique inverse (as things are reachable via two paths) but there's something at play here!

The knight "function" fills the space it is in (composition again) but my experience with finite functions is limited to the "injective" and "surjective" types. What is the function here? Some things can be reached two ways!

I am constantly searching for examples of things I learn in abstract algebra (like an orbit that isn't something from modulo n) and combinatorics (the study of finite functions if you will) tells me, broadly, they exist.

Thanks to Abstract Algebra I can prove that there is no regular number of days I can do a task in (eg every other) without covering (eventually) every day of the week. This is because 7 is prime, and everything is co-prime to a prime (except multiples of the prime - eg doing something every 7 days) but is that it? Where might I see a centraliser!

Anyway there is something interesting happening and I may have bitten off more than I can chew but I'd love to know HOW to study this.