Firstly, apologies if this is in the wrong section (or forum!). I had a look around and this seemed the likeliest place to put my question, though I'm not so much "pre-Algebra" as "severely post-Uni, recreational mathematician".
I have a few questions which come from years of playing matrix-based games like Sudoku and whatnot; bear with me as I try to word them properly. This is where I started:
Assume a 2 x 2 grid of unknown values as follows:
Also assume you've been given the values for the total of each column, each row, and the two diagonals:
Pretty basic so far, right? Good. To find the value of a:
Given a, we can now solve for b, c and d. I've tried this a few times on paper (and on my computer), and it seems to hold true. So far, so good.
So then I decided to take it up a level: a 3 x 3 grid
I won't write out all the equations; suffice it to say, you have values R1, R2, R3, C1, C2, C3, and the sums of the two diagonals, D1 and D2, to work with.
Using what little I remembered of my University mathematics classes, I tried to "solve" this grid with an 8 x 9 matrix and ended up, almost by accident, finding the value of e...
...which, again, has held true when I've tested it on paper and on my computer. Somewhere along the way I also managed to solve for a, but I can't seem to find that particular set of equations.
So, my questions are:
1. Is it possible to solve for *any* value in the grid? In the 3 x 3 example, having solved for both a and e, I could solve for i, but what of the others?
2. Are these solutions applicable to larger grids? I stopped at 3x3 (it gave me a headache). What about, for example, a regular Sudoku-sized 9x9 square?
3. Is there a formula (or a set of formulae) for solving these sorts of grids, or maybe a website or Wiki article on the topic?
Hopefully this makes sense to someone... thanks in advance for any help or information you might have.