Choosing $7$ numbers from $[1,2,...,11]$ will give us $2$ that have sum $12$. I tried:
There are only $5$ pairings possible:
Suppose I pick $6$, and then not to be a pair I will need to draw exactly 1 number from each of the above pairs. Suppose I picked, $7,8,9,10,11$. I will use up $6$ numbers. Thus any number that is left belongs to some pair.
How can I put this into mathematical terms and prove it?