Subdivide a rectangle into unequal columns

Solution 1:

Suppose when you subdivide the rectangle into $ n $ columns of decreasing sizes, the size proportions of the columns are $ n : (n-1) : \ldots : 2 : 1 $. In your examples, when $ n = 2 $, the size proportion is $ 2 : 1 $, and when $ n = 4 $, the size proportions are $ 4 : 3 : 2 : 1 $. The sum of the numbers from $ 1 $ to $ n $ is $ n(n+1)/2 $. So, if the length of the side of the rectangle, on which you are cutting the columns, is $ a $, you place the first cut at length $ n (2/(n(n+1))) a $, the second cut at $ (n + (n-1)) (2/(n(n+1))) a $, and so on. Hence, the $ j $th cut will be placed on

$$ \sum_{k=0}^{j-1} (n-k) \frac{2}{n(n+1)} a = \frac{2}{n(n+1)} \left( n j - \frac{j(j-1)}{2} \right) a . $$