First, take a look at the question that you would have done earlier in stage 1, had you been in this competition.

http://codersblock.net/ccc/2007/bowling/senior_stage1_2007.pdf


The stage 1 solution requires that you break it into easier subproblems with dynamic programming.

The stage 2 solution requires that you take add another dependency between these subproblems, to take into account what happens if we hit only (1, 2 ... w) additional pins.


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Stage 1 Solution Hint
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Let P(g, b) define the subproblem "What is the most number of points we can get if we are on our b'th ball, and only able to hit pins from g .. n?"

Then  P(g+1, b) gives us a smaller version of the same subproblem. 
Also, P(g+w, b+1) gives us the next subproblem if we decide to hit the current pins (g .. g+w).

The maximum of these subproblems is the best solution for P(g, b).

http://codersblock.net/ccc/2007/bowling/stage1_equation.gif

A further set of subproblems are required for the stage 2 solution.