Maximum pieces that can be cut from a Circle using 6 straight lines | Puzzles for Interview - Interview Preparation PDF Download

Problem Statement

You are provided with a circle and six straight lines. The challenge is to determine the maximum number of pieces into which you can cut the circle using these six straight lines. Each line can be drawn anywhere within or intersecting the circle, and the goal is to maximize the number of individual pieces created by the intersections of the lines.

Solution

To solve this problem, we must find a pattern or formula that can be applied for any given number of straight lines. Let's start by drawing the lines one by one and counting the maximum pieces after each line is drawn.

1. With the first straight line, we can only cut the circle into 2 pieces.
2. With the second straight line, we can cut the circle into a maximum of 4 pieces if the second line intersects the first line within the circle.
3. With the third straight line, we can cut the circle into a maximum of 7 pieces if it intersects the previous two lines within the circle.
4. With the fourth straight line, we can cut the circle into a maximum of 11 pieces if it intersects the previous three lines within the circle.
5. With the fifth straight line, we can cut the circle into a maximum of 16 pieces if it intersects the previous four lines within the circle.
6. With the sixth straight line, we can cut the circle into a maximum of 22 pieces if it intersects the previous five lines within the circle.

By observing the pattern, we can see that the maximum number of pieces increases as follows:

2, 4, 7, 11, 16, 22

The difference between the consecutive terms in the sequence above is:

2, 3, 4, 5, 6

We can notice that the difference between the consecutive terms increases by 1 each time. Therefore, to find the maximum number of pieces for any given number of straight lines n, we can use the formula:

P(n) = P(n-1) + n

In our case, n = 6, and we have already found that P(6) = 22.

Conclusion

Using the formula derived from our observations, we can conclude that a circle can be cut into a maximum of 22 pieces using 6 straight lines. This can be achieved by ensuring that each line intersects all the previous lines within the circle.