The Puzzle of 15: Magic Squares

Magic squares are grids of numbers where each row, column, and diagonal adds up to the same total. The classic version uses the numbers 1 through 9 — and there is only one total it can possibly add up to.

What is a Magic Square?

Imagine a square divided into smaller squares, like a tic-tac-toe board but bigger. Now, imagine filling each of those small squares with a number. If you can arrange the numbers so that every row, every column, and both diagonals add up to the same number, you've created a magic square.

The Puzzle of 15

The most famous magic square uses the numbers 1 through 9. The challenge is to arrange these numbers in a 3x3 grid so that each row, column, and diagonal adds up to 15. Here's one solution:

    8  1  6
    3  5  7
    4  9  2

Add the numbers in each row, column, and diagonal. Do they all add up to 15?

Why Does It Work?

The number 5 must go in the middle square. The pairs of numbers that add up to 10 (1 and 9, 2 and 8, 3 and 7, 4 and 6) have to be placed in opposite corners or edges, so they can be part of a line that adds up to 15 with the 5 in the middle. There are actually multiple solutions, but they are all rotations or reflections of the same basic arrangement.

Check yourself

That is why "multiple solutions" is a bit of a trick answer. Turn the square, or hold it up to a mirror, and every line is the same set of numbers as before — so it cannot help but still add to 15.

You can find more arrangements just by turning the square or flipping it in a mirror. But is there a genuinely different one somewhere, a square that is not just this one turned around?