We divided the plane into 16 parts with five lines. Among the parts,

a) at least

b) at most how many triangles are there?
First, we will show that five lines precisely divide the plane into 16 parts if and only if no two of them are parallel and no three of them pass through the same point (in the following, we will call such lines in general position).

Let \( s_{n} \) denote the number of regions into which \( n \) lines in general position divide the plane. It is obvious that \( s_{1}=2 \) and \( s_{2}=4 \). We will show that \( s_{n}=s_{n-1}+n \). Since the lines are in general position, a fixed line is divided into exactly \( n \) segments by the other \( n-1 \) lines (Figure 1). Each of these \( n \) segments splits one of the regions determined by the other \( n-1 \) lines into two. Therefore, the number of regions increases by \( n \).

From our reasoning, it also follows that \( s_{n} \) is independent of the arrangement of the lines; \( n \) lines in general position always divide the plane into the same number of regions.

Using our formula, we get \( s_{5}=s_{4}+5=s_{3}+4+5=s_{2}+3+4+5=16 \). If the lines are not in general position, then the number of regions is less than 16 (because there is a line that is not divided into five parts by the other four, but into fewer parts).

We need to determine how many triangles can be formed among the regions created by five lines in general position. Four lines in general position always form one quadrilateral, two triangles sharing one side with it, and 8 regions extending to infinity (Figure 2). By considering the essentially different positions of a fifth line (as shown in Figure 3), it can create 3, 4, or 5 triangles. (The details are left to the reader.) Therefore, the number of triangles is at least 3.

We will show that the maximum number of triangles is 5. If there were at least 6 triangles, then together they would have at least 18 sides. Since no three lines pass through the same point, two triangles cannot share a side, so at least 18 finite segments would be associated with the 6 triangles. One line, as we saw in Figure 1, is divided into three finite segments and two rays by the other four lines, so the number of finite segments is \( 5 \cdot 3 = 15 \). Therefore, there cannot be 6 (or more) triangles among the regions.

Thus, the answer to the question is: among the regions, there are at least three and at most five triangles.

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