So Alice and Bob decide to go bowling. They play one round. By the end of the game, Alice and Bob have each managed to knock down the exact same number of pins in total. However, Alice won, because some pins are worth more than others.
The question is, what is the largest possible difference in score between Alice and Bob? And can you prove that the number you have is in fact the largest possible?
In case you don't know, the scores in bowling work as follows:
- You play 10 frames; each frame you get 2 rolls.
- If you don't knock down all the pins between the 2 rolls, you get 1 point per pin.
- If you knock down all the pins on your second roll; you get a spare. For that frame, you get 10 points PLUS an additional point for each pin you knock down in your next roll (Meaning that those pins are worth 2 points each).
- If you knock down all the pins on your first roll; you get a strike. For that frame, you get 10 points PLUS an additional point for each pin you knock down in your next TWO rolls.
- If you get a spare on frame 10; you get a single roll for frame 11 to determine how much to add. Those pins only add to frame 10; they don't count themselves as well.
- If you get a strike on frame 10; you get 2 extra rolls. Again, pins you knock down on those rolls only add to your regular frames (10 or 9 & 10); they don't count for themselves as well.