## Task 3 - 020833

Rainer, who is part of the school's football team, is organizing the football shoe cabinet. He knows that some shoes have been taken to the cobbler.

He notices that the shoes come in different sizes, namely 37, 38, 39, and 40.

Six pairs are neatly tied together, and these include shoes of each size. Most of these pairs are size 38.

Of the additional five right shoes, none are size 38, and most are size 39.

The additional eight left shoes are of each size, with the most being size 40 and the least being size 37.

a) How many football shoes are at the cobbler's?

b) What kind of football shoes are they?

Justify your answer!
There is at least one of each size; and more of size 38 than of any other.

If there were 2 of size 38, there would also have to be 2 of another size. There shouldn't be any other size at all. Therefore, the 38 must appear three times, and the others once among the pairs.

For the right shoes, we proceed analogously: there are only three sizes, but in total, there is one shoe less than among the pairs. So Rainer has 3 right 39s and one each of 37s and 40s in front of him.

For the left ones, we know that there is at least one 37 and that the others occur more frequently (i.e., at least twice). Since there are only 8 left shoes, the 40 is represented three times, the 39 and 38 twice each, and the 37 once.

When we look at which left and right shoes can still be paired, we find that 2 right 38s, one left 39, and 2 right 40s must be at the cobbler's.