The triangle puzzle

In your book, Maths Mela (Chapter 7), there is a number game on a triangle. A triangle has three corners, and there is one space in the middle of each side. That makes 6 spaces. Put the numbers 1, 2, 3, 4, 5 and 6 in them, each number once, so that the three numbers on every side add up to the same sum.

Here is a way to make 9. Put 1, 2 and 3 at the corners. Between 1 and 2 put 6: 1 + 6 + 2 = 9. Between 2 and 3 put 4: 2 + 4 + 3 = 9. Between 3 and 1 put 5: 3 + 5 + 1 = 9. Every side is 9!

A clever idea: each corner number is on TWO sides, but each middle number is on only one side. So small numbers at the corners give a small sum, and big numbers at the corners give a big sum. With 4, 5 and 6 at the corners we get 12: 4 + 3 + 5 = 12, 5 + 1 + 6 = 12 and 6 + 2 + 4 = 12. That is the biggest sum we can make.

With 1, 3 and 5 at the corners we can make 10, and with 2, 4 and 6 at the corners we can make 11. So the sums 9, 10, 11 and 12 can all be made. 13 cannot be made: even with the biggest numbers at the corners, the sum only reaches 12.