Grids and their row sums

Here is a 3 × 3 grid filled by a simple rule: use the numbers from 1 to 9 without repeating any. Outside the grid are circled numbers. What do they represent?

They are the SUMS of the corresponding rows and columns. Once you see that, you can fill several such grids for yourself, and make up puzzles to challenge your classmates.

But some of these puzzles have no solution, and it is worth knowing why before you waste an afternoon. Consider a grid whose circles include 5 and 26.

What is the SMALLEST sum a row of three different numbers from 1 to 9 can have? It is 1 + 2 + 3 = 6. And the largest? 9 + 8 + 7 = 24.

So no circled number can be less than 6 or greater than 24. A grid asking for 5 is impossible, and so is one asking for 26 — no arrangement will ever work.

Kishor noticed something else: in every grid he solved, the sum of ALL the circled numbers was 90. Vidya noticed the sharper version: the three ROW sums, or the three COLUMN sums, always added to 45.

This is not a coincidence or a trick: it is maths, and we can say why. Why must that be? Because the three row sums together count every square of the grid exactly once — and the squares hold the numbers 1 to 9, each used once. So the row sums add to 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45. The same argument works for the columns, which is why all six circles come to 45 + 45 = 90.