If there are n rectangles, each with the size 1×2, 2×3, 3×4, 4×5 ... n(n+1), what would be the smallest rectangle in which we could fit all n of these rectangles without any of them being overlapped?
For instance, if n=2, the answer will be 2x4 - simply stick the 1x2 rectangle and 2x3 rectangle together. if n=3, the answer will be 4x5, sticking the previous 2x4 rectangle with the 3x4 rectangle. if n=4, the answer would be either 4x10 or 5x8, depending on which way we decide to stick the rectangles. if n=5, the answer would be 5x16, sticking the 5x6 and 5x8 rectangle together.
However, starting from n=6, there's no way(in my knowledge) to orientate these n(n+1) rectangles in a certain rectangle without leaving some blank space - and the smallest rectangle to do so is 6x19, with the 6 rectangles filling in the 6x19 rectangle with only 2 blank spaces.
Here, I was intrigued to find out if there would be a way to generalize this - to find the smallest rectangle that could fit n rectangles - 1×2, 2×3, 3×4, 4×5 ... n(n+1) - whichever n is given. How would I be able to do this?
(Just as a side note, the original maker of this problem listed it in "factorization." Maybe that could be used in a smart way to solve this, but I don't see how yet)