Vojta is trying to store his 20 toys in boxes so that each box contains at least one toy and no two boxes contain the same number of toys.

a) Describe how he can store the toys in five boxes.

b) Can he store the toys in six boxes?

(J. Tkadlec)
To ensure that each box has at least one toy and no two boxes have the same number of toys, Vojta can start by placing one toy in one box, two in another, three in the third, and so on. If any toys are left over, he can place them in the last box.

a) Twenty toys can be stored in five boxes, for example, as follows: $1,2,3,4,5+5$.

b) For six boxes, at least $1+2+3+4+5+6=21$ toys are needed. Therefore, it is not possible to store 20 toys in six boxes using the specified method.

Evaluation. Award 3 points for each part of the task. A negative answer to question b) without providing a reason should be scored 1 point.