Question

Click and drag the steps to their corresponding step numbers to show that if seven integers are selected from the first 10 positive integers, there must be at least two pairs of these integers with the sum 11.

Answer 1

You can first pair the numbers so that each pair sum 11: 1 and 10 sums 11, 2 and 9 sums 11, 3 and 8 sums 11, and so on. There are a total of 5 pairs of numbers with the sum being equal to 11.

You can only pick up to 5 numbers without picking any pair: in that case you should select one member of each pair and not the other.

If you select 6 numbers instead of 5 you will have to pick a pair, and then place the other 4 numbers as you wish. This way you can still select up to one pair by placing the 4 remaining numbers each one in a different pair.

However, if you want to place 7 instead of 6, you cant avoid to select the numbers without picking 2 pairs, because the remaining number you have after you have selected six of them in the way we described above (the only way to avoid selecting 2 pairs when you pick 6) will have to go to one of the four incomplete pairs, thus, you will have 2 pairs in your selection.