Dart Throw

In order to achieve the smallest possible number of games, Michel wants to score as many points as quickly as possible. After 8 perfect 15-point games, Michel will have 120 points (9 perfect games will put him at 135 points, which is too many) and only 14 remaining points.

Is it possible to score 14 points with three darts? No! The best score with three darts is 15 points and the second best score with three darts is 5 + 5 + 3 = 13. Since 14 points cannot be achieved in one game, Michel will need to play $\textbf{two more games}$ to achieve 14 points, which he can do as follows: $\underbrace{(3 + 2 + 2)}_{\text{first game}} + \underbrace{(3 + 2 + 2)}_{\text{second game}}= 14$ .

Therefore, the fewest number of games possible to reach 134 is $8 + 2 = 10.$

The fewest number of games possible to reach 134 is 8+2 = 10; n = 10

With 9 games scoring maximum points (15), he gets to 135. As there is no region that awards 4 points, so he can't lose only one point in any one of those games to get a 134 score. The least he can lose per game is 2 points. With 10 games, he needs to lose 16 points in order to score 134 instead of 150. 16 is divisible by 2 so he can score 134 by playing 10 games. He can do this by losing total 16 points in various possible combination such as scoring 5+5+3 in 8 of those games. Or scoring 3+2+2 in two of those games. Or he could score 2+3+0 in one game then 3+3+3 in another. Or (5+5+3)+(5+3+0)+(5+3+0). Etcetera.