Constructing magic squares, and where they came from

0:00A magic square is a grid where every line adds up to the same number. Every row. Every column. And both diagonals, corner to corner. Those count too, and they are the easiest thing to forget. Here is a grid where the rows all make fifteen, and the columns all make fifteen. It looks finished. It is not magic. One diagonal makes twelve. The other makes twenty-four. And that is not a rare accident. Of the seventy-two grids whose rows and columns are all level, only eight are magic.

0:38The other sixty-four fail on the diagonals alone. So: eight lines, one total. And that total has a name. The magic sum. So how do you actually build one? The obvious answer is to try things. Put one to nine in, look at it, shuffle them, look again. There are three hundred and sixty-two thousand, eight hundred and eighty ways to fill that grid. If you checked one every second, without ever stopping, it would take you more than four days.

1:14That is not a method. That is a hope. So we are going to do something else entirely. Every step from here cuts the search down, and by the end there will be nothing left to search. Not one square found. One square forced. Start with something we already have. The numbers one to nine come to forty-five. And the three rows of the grid, between them, hold all nine of those numbers.

1:43So the three row totals come to forty-five as well. But in a magic square, the three row totals are all the same number. Three equal numbers, adding to forty-five. Forty-five divided by three. Fifteen. So the magic sum is not something anybody chooses. Before a single digit has been written down, it is fifteen. Next question. What goes in the middle? Try a nine there, and watch what breaks.

2:16The middle cell shares a line with every single other cell on the board. That is the fact doing all the work here. So wherever the eight ends up, it is on a line with that nine. Nine and eight already make seventeen, and the line is only allowed to make fifteen. The third number on it would have to be minus two. There is no minus two. Now try a one in the middle instead. The two is somewhere, and it is on a line with it.

2:47One and two make three, so the third would have to be twelve. There is no twelve either. Keep going, and the same thing happens to almost every candidate. A number can only sit in the middle if every one of the other eight can find a partner to make fifteen with it. I tried all nine. Exactly one survives. Five. And here is the same conclusion in one line, if you would rather have it short.

3:17The four lines through the middle pair the other eight numbers off, and each pair has to total fifteen take away the middle. Those eight numbers add up to forty-five take away the middle. Set the two against each other, and only one number works. Five. So the middle cell was never a choice either. But why is the middle so special? Count the lines running through each kind of cell.

3:45Through the middle: a row, a column, and both diagonals. Four. Through a corner: a row, a column, and one diagonal. Three. Through a cell partway along an edge: a row and a column. Two, and no diagonal at all. Four, three, two. That is the whole shape of this puzzle. A number in a busy cell has to satisfy four demands at once. A number in a quiet cell has to satisfy two.

4:17So the awkward numbers — the ones with the fewest ways of working — belong where the fewest demands are made of them. Take the one. Where can it go? Suppose it sits in a corner. Three lines run through a corner. Each of those lines has the one on it, so each needs two more numbers making fourteen between them. Three lines. Three different pairs, all adding to fourteen.

4:45Let us see what is on offer. Five and nine. Six and eight. And that is the lot. There is no third pair. So the one cannot be in a corner. Do the same for the nine and you need three pairs making six: one and five, two and four, and nothing after that. Both get pushed onto the edges. And it is never about being big or small — five has four such pairs, and this argument says nothing about five.

5:17We now know enough to build the thing. Five in the middle. One somewhere along an edge. Put the one on the left edge, in the middle row. The five is already next to it. One and five make six, so the last cell in that row has to be nine. And there it is. One, five, nine, straight across the middle. Everything else now falls out. The two corners on the left are in a column with that one, so they have to make fourteen between them.

5:48Six and eight, and there is no other pair that does it. The diagonals decide which way round. Two ways to finish, and they are mirror images. In fact there is only one magic square here at all — it turns up in eight guises: four turns and their reflections. Now something worth being careful about. Here is the square. Add one to every single entry. Is it still magic? Yes, it is.

6:19But look at the magic sum. It was fifteen. It is now eighteen. Not sixteen. Eighteen. Because every line has three cells in it, and you have just added one to all three of them. Add one, the sum goes up by three. Add ten, the sum goes up by thirty. Now double every entry instead. Still magic, and the sum doubles to thirty. But the numbers are no longer consecutive — so a magic square never needed consecutive numbers at all.

6:56Which brings us to the best idea in all of this. Forget the actual numbers. Call the middle one m, and write every other cell as an amount above or below it. In our square the middle is five, so the nine becomes m plus four, and the one becomes m minus four. Do that to all nine cells, and the offsets run from minus four up to plus four, with nothing in the middle.

7:24Now add up any line at all. The offsets cancel out. They come to nothing. And what is left is three m. Three times the middle number, on every line, every time. So you can build one to order. Want the middle to be twenty-five? The magic sum is seventy-five, and you are finished. Want a magic sum of sixty? Divide by three. The middle is twenty, and the nine numbers run from sixteen to twenty-four.

7:56One more square, and this one is not made of one to nine. It is four by four, and it is carved into the stone of a temple at Khajuraho, in central India. It was cut in the tenth century. It has been sitting on that wall for about a thousand years. Every row makes thirty-four. Every column makes thirty-four. Both diagonals, thirty-four. But this one goes a good deal further than it had to.

8:26Take the four numbers in any quarter of it. Thirty-four. The four in the very middle. Thirty-four. The four corners. Also thirty-four. Altogether eighty-six different groups of four cells in it come to thirty-four, out of one thousand eight hundred and twenty you could pick. Whoever cut that was not guessing. And the three by three we built is older still. It appears in China more than two thousand years ago, in a story about a flood and a turtle with dots on its shell.

9:03Two, seven, six. Nine, five, one. Four, three, eight. Check it. Fifteen every way you read it. It is our square, turned round. You will also find these on painted yantras, where nine panels each carry their own small square, and every panel has a different magic sum. Fifteen. Eighteen. Twenty-one. And so on, up the panels. Every one of those is the same square with a number added to every cell. One square, nine times over, exactly as the offsets promised.

9:40Which leaves one last question. If adding shifts the magic sum, what if you shift it all the way to nothing? Keep the offsets, set the middle to zero, and every line adds up to nothing at all.