A square with sides of length $1 \mathrm{~cm}$ is given. There are many different ways to cut the square into four rectangles. Let $S$ be the sum of the four rectangles' perimeters. Describe all possible values of $S$ with justification.
The answer is $60$ in all cases, we have $S>6$.

The maximum of $S=10$ is achieved by cutting the square into four $1 \times \frac{1}{4}$ rectangles with three parallel cuts, as in Squares 1 and 8.

All values in between can be achieved as well. Indeed, the total perimeters $S$ are written on top of each of the 14 cases. By shifting left or right the length 1 vertical segments in Squares 2,3,4, or by shifting up or down the length 1 horizontal segments in Squares 9,10,11, we can make the length $x$ vary from 0 to $1: 0<x<1$. Thus, for example, Square 4 can achieve any total perimeter $S=6+4 x$ between 6 and 10: $6<S<10$. In a similar way, Squares 5 and 12 can be drawn with lengths $x$ and $y$ varying between 0 and 1 , and so these squares can also achieve any perimeter $S=6+2 x+2 y$ between 6 and 10 .

Putting everything together, $S$ can be any number in the interval $(6,10]$; i.e. $6<S \leq 10$.