Magic squares: what the magic sum must be

A square grid of numbers is called a magic square if each row, each column AND each diagonal all add up to the same number. That number is called the magic sum.

Could you make one by simply filling a grid with numbers at random until it works? It would be a long afternoon. There are a very large number of ways of filling a 3 × 3 grid using the numbers 1 to 9 without repetition — exactly 3,62,880 of them.

(Surprisingly, that count can be found without listing them all, which you will see in later years.) The point for now is that randomness is not a method. We should proceed SYSTEMATICALLY, by asking questions.

Here is the first question. What can the magic sum be? Can it be any number at all?

Focus only on the row sums for the moment. We already know that in a 3 × 3 grid filled with 1 to 9, the three row sums always total 45 — because together they use every number once.

In a magic square, the three row sums are all EQUAL to each other. Three equal numbers adding to 45 must each be 45 ÷ 3.

So each row sum is 15. OBSERVATION 1: in a magic square made using the numbers 1 to 9, the magic sum MUST be 15. Not 'usually is' — must be. One question asked well has removed 3,62,879 possibilities' worth of guessing.