In 501 double out darts, it is possible to reach the final score of 501 with only 9 darts. The dartboard has fields 1 - 20, and also double and triple fields and the semi-bull (25) and bullseye (50). The last dart needs to hit a double field, i.e. either of:
[2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 50]
I already know all possible 9-dart-finish combinations (you can look them up on the internet). I have noticed that you need a minimum of three 60s to obtain a 9-darts finish. Now I wonder: is there a way to prove this? Or is there at least a way to see that you need to hit a 60 at least once in order to obtain a 9-darts finish? Because in 301 darts you can obtain a 6-darts finish without ever hitting a 60.
Would anyone care to show an example calculation?