# Sum of perimeters of rectangles by $2$ lines in a square

A square with sides of length $$1$$ 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

Attemp: There are two ways to cut the square into four rectangles: with two perpendicular lines, or with three parallel lines.

The original perimeter, before cutting, is $$4$$. In each case, adding a cut that goes the full length (or width) of the square will add $$2$$ units to the total perimeter - one unit for the cut, and one unit as you separate the square (or rectangle) into two.

Thus, you can either have a total perimeter of $$6$$ or $$7$$.

But.... the lines need not go through the square completely. I do not know what else to do

• What about cutting one way then after removing one rectangle, cutting up the remaining rectangle in the perpendicular direction 2 times ? That would be another way. Jan 2 '20 at 18:12

I think the answer is $$6 $$6$$ can (almost) be obtained by cutting a thin slice of the square and then dividing it into three by lines of virtually zero length.

$$10$$ is obtained by using three full length parallel lines.