Where each number must go

The magic sum is 15. Next question: which numbers can sit at the CENTRE?

Try 9. If 9 is at the centre, then 8 must go in one of the other squares. Wherever it goes, 8 and 9 end up in the same row, column or diagonal, and then 8 + 9 + another number would have to be 15. But 8 + 9 is already 17. Impossible — so 9 cannot be at the centre.

Try 1. If 1 is at the centre, then 2 goes somewhere, and 2 + 1 + another number must be 15. That needs the other number to be 12, and we only have 1 to 9. Impossible — so 1 cannot be at the centre either.

Work through the rest the same way and one number survives. OBSERVATION 2: the number at the centre of a magic square filled with 1 to 9 must be 5.

Now, where do the smallest and largest numbers go? By Observation 2 they are somewhere on the boundary — either a corner position or a middle-of-an-edge position.

Can 1 be in a CORNER? A corner square lies on three lines: a row, a column and a diagonal. So there would have to be THREE different ways of adding 1 to two other numbers to make 15. We have 1 + 5 + 9 = 15 and 1 + 6 + 8 = 15. Is there a third? No. So a corner is impossible for 1, and by the same argument for 9.

OBSERVATION 3: the numbers 1 and 9 cannot occur in any corner, so they must occupy middle positions. Once you place 1 and 9, a whole row or column follows, and the rest falls out. Then generalise: write m for the centre number and express every other entry as how much more or less than m it is — and the same pattern builds a magic square with any centre, such as 25.