Number of Regions in Plane Defined by Given Number of Lines/Examples/6
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Example of Number of Regions in Plane Defined by Given Number of Lines
With $6$ lines, the plane can be divided into a maximum of $22$ regions:
Proof
From Number of Regions in Plane Defined by Given Number of Lines, the maximum number $L_n$ of regions in the plane that can be defined by $n$ straight lines in the plane is:
- $L_n = \dfrac {n \paren {n + 1} } 2 + 1$
Here $n = 6$, so:
\(\ds L_6\) | \(=\) | \(\ds \dfrac {6 \paren {6 + 1} } 2 + 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {6 \times 7} 2 + 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 22\) |
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $22$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $22$