In magic squares, all of the rows, columns and diagonals add up to the same number. Magic squares like this one, where we have negative numbers as well, work the same way as normal magic squares: all that you need to remember is that, with negative numbers, you need to subtract rather than add to get your total. So going along the rows, minus 3 minus 6 is minus 9, then minus 9 plus 3 is minus 6, so all of our other rows, columns and diagonals must total minus 6 as well. For this second row, 4 minus 2 is 2, then 2 minus 8 is minus 6. Here, minus 7 plus 2 is minus 5, and then minus 5 minus 1 is minus 6. Now going down the columns, minus 3 plus 4 minus 7 is minus 6, minus 6 minus 2 plus 2 is minus 6, and 3 minus 8 minus 1 is minus 6. We can also look across the diagonals, so minus 3 minus 2 minus 1 is minus 6, and minus 7 minus 2 plus 3 is minus 6.
So now that you understand how magic squares work, see if you can complete this magic square. We have a complete line in our diagonal, and because minus 1 plus 0 plus 1 is 0, we know that all of our other rows, columns and diagonals must total 0. So now, for this question, there are two different places we could start, because we have 2 numbers in this bottom row, but we also have 2 numbers in the middle column, so let’s start here. We have our missing number here, and if we add 0 and take away 6, we get 0, because remember, we’ve already worked our that all lines must total 0. So our number must be 6. Now, we have minus 1 plus 6 then plus or minus our missing number to make 0. Minus 1 plus 6 is 5, so to get 0 as our total, our missing number must be minus 5. If we take this column next, we have minus 5 plus or minus something plus 1 equals 0. Because minus 5 plus 1 is minus 4, we need to add 4 to make a total of 0, so our missing number is 4. Now, something plus 0 plus 4 is 0, so we have minus 4. Now to get our last number, we could use this row or this diagonal, but let’s use this column. We have minus 1 minus 4 plus or minus something is 0, so our missing number must be 5. So now we’ve completed our magic square, can you spot any patterns here? Well, notice that opposite squares are always the positive and the negative of the same number: so for this middle row, we have minus 4 and 4 either side of 0, for this middle column, we have 6 and minus 6, for this diagonal, we have 5 and minus 5, and for this diagonal, we have minus 1 and 1. Why do we get this pattern? It’s only because, for this magic square, all of our rows, columns and diagonals had to total 0, and because we have 0 in the middle, the only way to get the total of 0 is to have the positive and negative of a number. If we didn’t have 0 in the middle, or if we didn’t have line totals of 0, we wouldn’t get this pattern. But we would get a similar pattern.
Let’s have another look at the first magic square we looked at. Pause the video and see if you can see any patterns here. Let’s look at this middle row: 4 is 6 more than minus 2, and minus 8 is 6 less than minus 2. Or we could look at this column: minus 6 is 4 less than minus 2, and 2 is 4 more than minus 2. For this diagonal, minus 7 is 5 less, and 3 is 5 more, and for this diagonal, minus 3 is 1 less and minus 1 is 1 more.