Clark and Harry are siblings. As they had been rivals since their early childhood, their father decided that both should concentrate on a different sport when they were thirteen. That way, they would not have to compete for success. Now both are twenty years old and excel in different fields: Clark plays chess while Harry participates in dart-tournaments.
Having won a series of three tournaments in a row, Harry started teasing Clark about not having as much success. Clark retorted that chess was less luck-based and thus more difficult. That offended Harry and led him to the reply that in order to play darts optimally, a lot of combinatorics are necessary. Clark returned an icy smile and the comment that memorizing all different late-games could hardly be called “combinatorics”.
This is how it came to the wager. Harry bets that he can find all possible late-games for generalized dart-boards where memorized late-games do not help him. When Clark showed him a list of possible dartboards, Harry had to admit that he probably bit off more than he can chew. As his friend, you have to help him!
A dart-board consists of different areas. Each area has an assigned score for hitting it. Each area also has a double- and a triple-field that are worth twice and three times the score of the area. The only exception is the area for the highest score: It has only a double- and no triple-field! Given the values of the different areas you have to find the number of possible scores that can be obtained with a given number of darts.