# How to get squares of different sizes from an A4 sheet

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One of the ideals of origami requires folding the original square without preliminary drawings and measurements. With this folding, a problem arises if it is necessary to fold a sheet of paper, for example, three times. How to do this without resorting to a pencil? This article addresses the issue of dividing a square sheet of paper into two, three, four,. ten equal parts.

Dividing a sheet of paper into two parts is not difficult, because it is realized simply by folding the basic form of the book. Let's move on to the more complex problem of dividing a square sheet into three parts.

This task is not so simple. To solve it, we need the Hague theorem. Add the angle of the square to the middle of the opposite side. In this case, the intersection point of the other side opposite to this corner and the side adjacent to it divides the side in a one to two relationship. Thus, using only the folds, we found a third of the side of the square. The next task is to divide the side of the square into four equal parts. To do this, it is enough to divide them in half, and then, each of the halves again in half. This is exactly what happens when we fold the basic shape of the door. As you might guess, dividing a square into five parts by folding is a much more difficult task. Its solution is shown in the figure. Try to prove for yourself that in this way we really divide the square into exactly five parts.

In order to divide the side of the square into six parts, it is enough for us to divide it into three parts, as shown earlier. And then, divide each part in half. You may notice that the division of the square by the number of parts, which is a prime number, causes particular difficulties. We proceed to dividing the parties into seven identical parts. To do this, first divide the square into five equal parts, and then, do the action shown in the picture.

Dividing the square into eight equal parts is quite simple. To do this, it is enough to divide it into four equal parts, and then divide each of them in half. Two methods can be proposed for dividing into nine equal parts. The first one is to divide first into three equal parts, and then repeat the division into three for a small square. However, this method is bad in that when applied in practice, it will be difficult to maintain sufficient accuracy, since errors made at different stages add up.

Another method is more original and based on the development of the Hag theorem, which was proposed by Koji and Mitsue Fushimi. Note that this method was used to divide the square into seven and nine parts. Maybe it is applicable for dividing it into nine parts? And finally, dividing into ten parts is a sequential division into five parts, and then halving each of the five parts.

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