Exercise 1. Mathieu takes a rectangular piece of paper and draws four distinct straight lines with a ruler, each going from one side to another side (none of these lines can be an edge of the rectangle). Then, he cuts the paper along each line drawn. How many pieces can he obtain? Sketch an example for each possible number of pieces, and explain why there are no other possibilities.
# Solution 

We will place the 4 lines one after the other. The first line necessarily divides the rectangle into two parts. The second can cross one of these two parts, or each of the two: for this, it must intersect the first. Similarly, the third line can intersect 0, 1, or 2 of the first two, thus crossing 1, 2, or 3 of the existing parts, creating 1, 2, or 3 new parts. And the fourth creates 1, 2, 3, or 4 new parts. In the end, after cutting, we will have $2+(1$ or 2$)+(1$ or 2 or 3$)+(1$ or 2 or 3 or 4$)$ pieces, which means all possibilities between 5 and 11. More precisely, the number of pieces is equal to $5+$ the number of intersection points (if intersection points belong to more than two lines, they must be counted multiple times, but we were not asked to consider this special case). The figure on the following page illustrates each of the cases.

![](https://cdn.mathpix.com/cropped/2024_06_04_c8c1ca304a882abd442fg-2.jpg?height=402&width=212&top_left_y=370&top_left_x=588)

5 pieces

![](https://cdn.mathpix.com/cropped/2024_06_04_c8c1ca304a882abd442fg-2.jpg?height=417&width=208&top_left_y=794&top_left_x=593)

![](https://cdn.mathpix.com/cropped/2024_06_04_c8c1ca304a882abd442fg-2.jpg?height=391&width=208&top_left_y=381&top_left_x=821)

6 pieces

![](https://cdn.mathpix.com/cropped/2024_06_04_c8c1ca304a882abd442fg-2.jpg?height=411&width=209&top_left_y=797&top_left_x=818)

10 pieces

![](https://cdn.mathpix.com/cropped/2024_06_04_c8c1ca304a882abd442fg-2.jpg?height=388&width=211&top_left_y=383&top_left_x=1048)

7 pieces

![](https://cdn.mathpix.com/cropped/2024_06_04_c8c1ca304a882abd442fg-2.jpg?height=399&width=197&top_left_y=797&top_left_x=1061)

11 pieces

![](https://cdn.mathpix.com/cropped/2024_06_04_c8c1ca304a882abd442fg-2.jpg?height=388&width=203&top_left_y=383&top_left_x=1286)

8 pieces