by Bill Graham

Answer to Darts Anyone? from Vol 1, No. 2

Write down possible scores in order (5, 7, 10, 12, 14, 15, 17, 19, 20, . . .). At this point, notice that you can always use any combination of the lower scores on the list to get a new score. You can use three sevens to get 21, and 10 and 12 to get 22 and so on. Make the list a bit longer: (. . . 21, 22, 24, 25, 26, 27, 28 . . . ).

You now have five scores in a row from 24 to 28. You can add five to those each of those scores to get the next five scores in sequence. From here on out, all scores are possible. Therefore, the largest impossible score is 23.

What happens if the dart board has different numbers on it?

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 is "b", then the largest impossible score is ab - (a + b).