Discussion: This is an excellent example of an elementary application of the pigeonhole principle. The challenge here is to define the pigeonholes so that 2 of the 12 numbers must fall in the same pigeonhole and so that if they do then they satisfy the desired conclusion. This student first reduces to the case where all the numbers have different remainders when divided by 20 and then identifies the numbers with their remainders. Then the numbers are paired with their complement in 20, that is, 1 with 19, 2 with 18 and so on. 10 gets paired with itself. In this way one gets eleven boxes where the numbers in the box add to 20 and therefore satisfy the conclusion. Since there are 12 numbers some box (it therefore can't be the box with 10) gets w numbers.

Written response: Good use of the pigeonhole principle to prove this result.