#Approach to problem For a section of consecutively occupied boxes, no cuts are obviously the cheapest way to go. I.e. if all boxes were occupied, the cheapest way would be to cut the plank only once, at the end. This is not the case though so we have to look at what happens with empty boxes.
Imagine that we step through the shed as an array in our mind, box by box. In order to step to the next box we must pay a certain cost given by the current box. The cost to leave an occupied box is that of 2m wood. The cost to leave an empty box is either that of 2m wood or the price of a cut. How do we decide what cost to choose here?
If we step through the boxes and find an unoccupied one, we have found a gap of size 1. If we find that the next one is empty as well, we have a gap of size 2. This gap size is important because it will give us a threshold where it will be cheaper to cut than to fill the gap with plank. If the gap is so wide that the price of cutting is lower than that of filling it with plank, we choose the former.
E.g. in the example case, a gap of size 2 is cheaper to cut than to fill so the lowest price is €40 (2 cuts and 20m of plank)
#Code In solution.js, you will find the code that performs the price minimisation. app.js is mostly jQuery that makes use of the solution but also allows to set custom prices and shed layouts.
#App Here is the app