[ Various tasks on cutting $]$

A sheet of paper is divided into cells with a side of 1. Vanya cut out a rectangle from it along the cells and found its area and perimeter. Tanya took the scissors from him and, saying "Watch, a trick!", cut out a square from the edge of the rectangle along the cells, threw the square away, and announced: "Now the perimeter of the remaining figure is the same as the area of the rectangle, and the area is the same as the perimeter was!" Vanya verified that Tanya was right.

a) What size square did Tanya cut out and throw away?

b) Provide an example of such a rectangle and such a square.

c) What were the dimensions of the rectangle Vanya cut out?

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a) The square could not share a common angle with the rectangle (see fig.), because then the perimeter would remain the same or decrease (convince yourself!), while the area would decrease. Therefore, the square adjoins only one of the sides of the rectangle (see fig.).
![](https://cdn.mathpix.com/cropped/2024_05_06_bca77ef9e87e9e47ba11g-11.jpg?height=374&width=1798&top_left_y=2288&top_left_x=40)

Let the side of the square be $x$. Then, by cutting out the square, Tanya reduced the area of the figure by $x^{2}$, while the perimeter increased by two sides of the square, i.e., by $2 x$. Thus,

Initial area - $x^{2}=$ initial perimeter, initial perimeter $+2 x=$ initial area.

Hence, $x^{2}=2 x$, from which $x=2$.

b) Let the sides of the rectangle be $m$ and $n$. Then, from the solution of part a), it follows that $m n=2 m+2 n+4$. Our task is to find all possible pairs of numbers $m$ and $n$ that satisfy this equation. We can reason as follows. Clearly, $m$ and $n$ exceed 2, otherwise it would be impossible to cut out a $2 * 2$ square. Let $m \geq n$. Then, for $m \geq n \geq 5, 2 m+2 n+4 \leq 4 m+4 < 5 m < m n$. Therefore, $2 < n < 5$. Substituting $n=3$, we find $m=10$, and for $n=4$ we get $m=6$. Thus, the rectangles could only be as shown in the figures - $3 * 10$ and $4 * 6$.
![](https://cdn.mathpix.com/cropped/2024_05_06_bca77ef9e87e9e47ba11g-12.jpg?height=436&width=1770&top_left_y=642&top_left_x=58)

We can reason differently. The equation $m n=2 m+2 n+4$ can be written as $(n-2)(m-2)=8$. Since $m$ and $n$ exceed 2, the problem reduces to finding the factorizations of the number 8 into two natural factors.

## Answer

a) $2 * 2 ;$ b) see fig. for part c) of the solution; c) $3 * 10$ or $4 * 6$.