# Magic square math -- help the clueless

My puzzle-a-day calendar had a magic square today – a three by three grid in which every row, every column, and both main diagonals add up to the same value, which is to determined. They give you three numbers to start:

12 __ __
__ 10 __
7 __ __

I gave up quickly and peeked at the answer, which reads: “To start, since the left-to-right diagonal and the bottom row add up to the same amount” [okay, with ya so far] “we can work out the value of the bottom middle cell. It must be 12 + 10 - 7 = 15.” Uh … why? I mean, I see where the three numbers on the left side of the equation come from, but why is that what you do with those three numbers?

TIA.

The total for each row, column and diagonal (call it T) must match be 12 + 10 + X, where X is the value of the bottom right cell.

But since X is shared by both the diagonal and the bottom row, T = 12 + 10 + X = 7 + X + Y (the value of the center bottom cell).

So 7 + Y + X = 12 + 10 + X. Subtract X from both sides of the equation:

7 + Y = 12 + 10. Subtract 7 from both sides.
Y = 12 + 10 - 7 = 15

The top-left to bottom-right diagonal adds up to 12 + 10 + bottom-right.

The bottom row adds up to 7 + bottom-middle + bottom-right.

Since these are equal to each other, we have that 12 + 10 + bottom-right = 7 + bottom-middle + bottom-right.

If two things are equal, then doing the same thing to both of them keeps them equal. So let’s subtract 7 and subtract bottom-right from both sides. Doing this gives us:

12 + 10 -7 = bottom-middle

:smack:

Thanks!

Incidentally, you don’t even need to use that to solve the puzzle. The (IMHO) easiest thing to do is figure out what the value is that each column, row, and diagonal adds up to, and then you can immediately finish off the puzzle by filling in the missing square in each line where you already have two. And as for what the magic number is that everything adds up to,

It has to be three times the center square (thus, in this case, it’s 3 * 10 = 30). Why is that? Well, if you add the two diagonals and the middle column, you’ll be counting the squares in the top row and the bottom row once each, plus the middle square three times. Thus, 3 * the magic number = 2 * the magic number + 3 * the middle square, from which it follows that the magic number is 3 * the middle square, as I said.

Let’s call those two bottom unknown numbers A and B. We know that 12 + 10 + B is the magic value, since they’re all on a diagonal. And we also know that 7 + A + B is also the magic value, since they’re all in a row. So 12 + 10 + B = 7 + A + B. If we subtract B from both sides, we have 12 + 10 = 7 + A. And if we subtract 7 from both sides, we have 12 + 10 - 7 = A.