Péter took one of each of the stamps worth $1, 2, 3, \ldots, 37$ forints from his stamp collection. He would like to group these stamps so that the sum of the face values of the stamps in each group is the same. In how many different ways can he do this?
The sum of the face values of all the stamps is $S=\frac{37 \cdot(1+37)}{2}=19 \cdot 37=703$.

This sum is the product of two prime numbers, which means that Peter can put all the stamps in 1 group, in which case the sum of the face values is 703. Clearly, it is not possible to put the 37 stamps into 703 groups, and it is also not possible to put them into 37 groups due to the distinct face values.

The other possibility is to put the stamps into 19 groups such that the sum of the face values in each group is 37. In this case, the 37 forint stamp can only stand alone, the 36 forint stamp can only be paired with the 1 forint stamp to make a sum of 37. Then, the 35 forint stamp can only be paired with the smallest remaining value, the 2 forint stamp, so the 34 forint stamp must be paired with the 3 forint stamp, and so on. Therefore, the only valid grouping is:

$\{37\}, \quad\{36,1\}, \quad\{35,2\}, \quad\{34,3\}, \ldots, \quad\{19,18\}$.

In total, the desired grouping can be done in two ways.