The topic turned somehow to scoring. In 301, a round of darts consists of three throws. Each throw can either score 0 (miss the board); any number from 1 to 20 (by landing it in the corresponding section of the board); 25 for an outer bullseye; or 50 for an inner bullseye. There are also Double and Triple rings on each of the numbered slivers that score from 2-40 or 3-60 by twos or threes, respectively. Therefore, the highest-scoring theoretically possible throw for a single round of darts is 180: 60 (triple-20) thrown three times.
"So, Bax," Rob asked me. "Is it possible to score any number from 0 to 180 in a single round of darts?"
I thought for a moment. "No. A trivial counterexample is 179. To get 180, you need three 3x20s. But the next highest single-dart score is a triple-19. The second-highest round you can score is therefore two 3x20s and a 3x19, which is 177. You can't score 178 or 179."
Rob looked thoughtful. "So, what's the lowest score that's impossible to get in a single round?"
The two of us looked at each other. Mental wheels started spinning.
Five minutes later, we were knee-deep in numbers, he'd grabbed a sheet of paper, and I was assuring Kady that, no, really, we were close, we'd have the answer in a few minutes, we could start heading home soon. (Like a true mathematician, I had told him, "Let's just find the answer first. That'll help us come up with the elegant method for deriving it." Indeed, that took the better part of an evening.)
Anyway, Kady only got home about five minutes late, and we've got our answer. Can you find it?
Extra bonus points for a proof showing why it's the smallest (and roaminrob, you're eligible here too -- I'd like to see you develop your idea from the phone message more, because it's different from the one I settled on).
n.b.: I'm also going to go ahead and unscreen the answers to the previous riddle now, for those of you who haven't gotten it yet. Answers here will be screened for, um, probably a few days, or until I post the next riddle, whichever comes first.)
Edit:, 8-1-2005: Answers unscreened.