Making sums equal

In your book, Maths Mela (Chapter 4, We the Travellers — II), there are two groups of numbers. Your job is to swap (interchange) numbers between the two groups until both groups have the same sum, using as few swaps as you can.

The first two groups are 1, 2, 7, 9, which add up to 19, and 3, 4, 5, 9, which add up to 21. The book asks: what happens if we swap 2 and 5? The first group gives away 2 and gets 5, which is 3 more, so its sum goes up from 19 to 22. The second group gives away 5 and gets 2, so its sum goes down by 3, from 21 to 18. The sums are now further apart than before!

Here is the secret: a swap moves the difference of the two numbers from one sum to the other. The sums 19 and 21 are 2 apart, so they should meet in the middle, at 20. The bigger group must give away a number that is just 1 more than the number it gets back. Swap the 3 (from the group with 21) and the 2 (from the group with 19): the groups become 1, 3, 7, 9 = 20 and 2, 4, 5, 9 = 20.

Another pair of groups: 9, 11, 13, 14 adds up to 47, and 5, 7, 12, 15 adds up to 39. The sums are 8 apart, so each must move 4 to meet at 43. Swap the 9 with the 5, which is 4 less: 5 + 11 + 13 + 14 = 43 and 9 + 7 + 12 + 15 = 43.

Sometimes one swap is not enough. The groups 81, 82, 83, 84 (sum 330) and 77, 78, 79, 80 (sum 314) are 16 apart, so 8 must move. But the biggest difference we can make in one swap is 84 − 77 = 7. So we use two swaps: 81 with 77 moves 4, and 82 with 78 moves 4 more. Both sums become 322.