Hi everyone,

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:

$\displaystyle \begin{matrix}a & b \\ c & d \end{matrix}$

Also assume you've been given the values for the total of each column, each row, and the two diagonals:

$\displaystyle a + b = R_1\\\indent c + d = R_2\\\indent a + c = C_1\\\indent b + d = C_2\\\indent a + d = D_1\\\indent b + d = D_2\\$

Pretty basic so far, right? Good. To find the value of

*a*:

$\displaystyle ((a + c) + (a + d) - (c + d)) = C_1 + D_1 - R_2\\\indent 2a + c + d - (c + d) = C_1 + D_1 - R_2\\\indent 2a = C_1 + D_1 - R_2\\\indent a = \frac{1}{2}(C_1 + D_1 - R_2)$

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

$\displaystyle \begin{matrix}a&b&c \\ d&e&f \\ g&h&i \end{matrix}$

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*...

$\displaystyle e = \frac{1}{3} (R_2 + D_1 + D_2 - C_1 - C_3)$

...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.

Cheers