I also want to point out that the exact solution really isn't necessary, since we are just trying to get a feel for the difference between real-world skill challenges, which will likely include a significant amount of variations (characters with different skill levels, one-time abilities, situational bonuses, etc). Dart-throwing seems quite a bit more flexible (with all due respect to the elegance of DClown's algorithm). But then, I was trained in physics, not math, so I've never been a stickler for exact solutions.