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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

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    $\begingroup$ 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. $\endgroup$ Jan 2 '20 at 18:12
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I think the answer is $$6<S\le10.$$ $6$ 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.

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