You can throw as many darts at the board below.   Some total scores are impossible to obtain -- for example, all numbers less than 5, as well as 6, 8, and 9.

Extra Credit: Can you come up with an algorithm or rule to determine the largest impossible score for any two whole numbers on the dartboard?

Solution to the Problem:

Begin by writing down numbers which are possible:
5, 7, 10, 12, 14, 15, 17, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, ...
Once you get five scores in a row (like we did from 24 to 28), you can just add 5 to those scores to get the next five scores.   Then adding five to those scores gives you the next five scores.   From here on out, all scores are possible.   Hence 23 was the largest impossible score.

1. If both numbers are even, all odd numbers are impossible.
2. If one number is a multiple of the other, all numbers that are not multiples of the smaller number are impossible.
3. Otherwise, if one number is a and the other number is b, then the largest impossible number
     is     ab - a - b.