by Bill Graham

You can throw as many darts as you want at the board shown to the right (a circle with only two scoring rings, five and seven).

What scores could you get? You could get a five, or a seven, or a 10, or 12 and so on. What scores are impossible to get? You couldn't get a score of one, or two, or three, or four, or six, or eight, etc. Figure out all you can about the scores that are impossible. Is there a largest impossible score?

Suppose the numbers on the dart board were different. Then what? Can you come up with an algorithm or rule to determine the largest impossible score for any two numbers on the dart board?