UVa 10970 - Big Chocolate

Summary
A chocolate bar is a $$M N$$ rectangle, composed of unit squares. Determine, how many straight-line cuts are necessary to split it into unit-square pieces. Each cut can only divide one connected piece of chocolate into two.

Explanation
Note the following facts:  In the beginning we have one piece of chocolate. At the end we want to have $$mn$$ pieces. Each cut divides one piece into two, thus the number of pieces increases by one. 

Thus we will need exactly $$mn-1$$ cuts regardless of the way we cut the chocolate.

A slower solution:

The constraints are so low that solutions using dynamic programming/memoization will be accepted. The idea is: for each piece chocolate not larger than $$m\times n$$ compute the smallest number of cuts needed. If $$f(x,y)$$ is the minimum number of cuts for a $$x\times y$$ chocolate bar, we get:


 * $$f(1,n) = n-1$$
 * $$f(m,n) = \min_{1 \le k \le m-1} [1 + f(k,n) + f(m-k,n)] (\mbox{for }m > 1)$$

Input
2 2 1 1 1 3 5 5 5 10 10 5

Output
3 0 2 24 49 49