Question

With a straight cut you can slice a circular piece of paper into two pieces. A second cut that crosses the first cut will produce a maximum of four pieces. What is the maximum number of pieces that you can get with a total of four straight cuts?

Hint:

This is a well-known sequence. Remembering the formula for the maximum number of regions from n cuts, \( L(n) = \frac{n^2 + n + 2}{2} \), can provide a direct and quick answer, saving time in an exam.

Answer 1

Correct Answer: 11

Step 1: Understanding the Question:
This is a classic mathematical puzzle about dividing a plane (or in this case, a circle) into the maximum number of regions using a given number of straight lines. This is also known as the "Lazy Caterer's Sequence".
Step 2: Key Formula or Approach:
To achieve the maximum number of pieces, each new cut (line) must intersect all the previous cuts at distinct points, and no three cuts should intersect at the same point.
Let L(n) be the maximum number of pieces from 'n' cuts. The pattern is:
L(n) = L(n-1) + n
The general formula is:
\[ L(n) = \frac{n^2 + n + 2}{2} \] Step 3: Detailed Explanation:
Let's find the answer by following the pattern step-by-step.
- 1st cut: A single cut divides the circle into 2 pieces.
L(1) = 2.
- 2nd cut: The second cut must cross the first cut. It passes through 2 existing regions, dividing them and adding 2 new pieces.
L(2) = L(1) + 2 = 2 + 2 = 4 pieces.
- 3rd cut: The third cut must cross the previous two cuts at different points. It passes through 3 existing regions, adding 3 new pieces.
L(3) = L(2) + 3 = 4 + 3 = 7 pieces.
- 4th cut: The fourth cut must cross all three previous cuts at different points. It will pass through 4 existing regions, adding 4 new pieces.
L(4) = L(3) + 4 = 7 + 4 = 11 pieces.
Using the formula for n=4 also gives:
\[ L(4) = \frac{4^2 + 4 + 2}{2} = \frac{16 + 4 + 2}{2} = \frac{22}{2} = 11 \] Step 4: Final Answer:
The maximum number of pieces that can be obtained with four straight cuts is 11.