4. Divide a circular piece of paper into several smaller pieces of any size using straight lines. If you need to divide it into no fewer than 50 smaller pieces, what is the minimum number of straight lines required? Please explain.


4.【Solution】Let's draw the lines one by one. Drawing the first line divides the circular paper into 2 pieces. Drawing the second line, if it intersects the first line inside the circle, it will divide the circular paper into 4 pieces (increasing by 2 pieces), otherwise it can only divide it into 3 pieces. Similarly, drawing the third line, if it intersects the previous two lines inside the circle, and the intersection points are distinct (i.e., no three lines intersect at the same point), it will divide the circular paper into 7 pieces (increasing by 3 pieces), otherwise the number of pieces will be less than 7. The following figure shows various scenarios when drawing 3 lines.

From this, we can see that to divide the paper into as many pieces as possible, the new line should intersect all the existing lines inside the circle, and the intersection points should be distinct. In this case, the number of additional pieces equals the number of lines. The number of pieces divided by the lines is listed as follows:

Number of lines | Maximum number of pieces the paper can be divided into
--- | ---
$1 \quad 1+1$
$2 \quad 1+1+2$
$3 \quad 1+1+2+3$
$5 \quad 1+1+2+3+4$
$5 \quad 1+1+2+3+4+5$

It is not difficult to see that the number on the right side of each row in the table equals 1 plus the sum of all integers from 1 to the row number. Since $1+1+2+3+\cdots+10=56,1+1+2+3+\cdots+9=46$, it is clear that the number on the right side of the 9th row is less than 50, while the number on the right side of the 10th row exceeds 50.

Answer: At least 10 lines need to be drawn.