What is the largest number of strikes you can get in bowling, yet still lose to someone who gets no strikes, and only that same number of spares?
That is, the loser gets X strikes. Winner gets no strikes, but gets X spares.
If the bowler gets a strike in frame ten, he gets to roll the bowl two additional times, solely for determining his score in frame 10.
Similarly, if the bowler gets a spare in frame 10, he gets to roll his ball one additional time, soley for determining his score in frame 10.
So, with 9 strikes, a bowler gets a score of 120, if his strikes are in frames 1,2,4,5,7,8, 10,11 and 12.
With 10 strikes, he gets a score of 180 if his strikes are in frames 1,2,3,5,6,7,9,10,11,12.
And with 10 spares, the other fellow can get 190, with a spare in every frame and a 9 in frame 11.
And since it is impossible to get 11 spares in a game, the answer is 10.