Question 26
In the country of Oddland, there are stamps with values $1$ cent, $3$ cent, $5$ cent, etc., one type for each odd number. The rules of Oddland Postal Services stipulate the following: for any two distinct values, the number of stamps of the higher value on an envelope must never exceed the number of stamps of the lower value.
In the country of Squareland, on the other hand, there are stamps with values $1$ cent, $4$ cent, $9$ cent, etc., one type for each square number. Stamps can be combined in all possible ways in Squareland without additional rules.
Prove for every positive integer $n$: In Oddland and Squareland there are equally many ways to correctly place stamps of a total value of $n$ cent on an envelope. Rearranging the stamps on an envelope makes no difference.
In the country of Squareland, on the other hand, there are stamps with values $1$ cent, $4$ cent, $9$ cent, etc., one type for each square number. Stamps can be combined in all possible ways in Squareland without additional rules.
Prove for every positive integer $n$: In Oddland and Squareland there are equally many ways to correctly place stamps of a total value of $n$ cent on an envelope. Rearranging the stamps on an envelope makes no difference.