It is possible to arrange 3 lines in the plane to create 7 regions. You can arrange 3 circles to create 8 regions. In each case, this is the best you can do. What if you had more lines or circles?

The Challenge:

1. What is the maximum number of regions you can create with n lines?
2. What is the maximum number of regions you can create with n circles?
3. What is the maximum number of regions you can create using a combination of n lines and circles?

If arranged optimally, with <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> lines, the maximum number of regions that can be created is <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\frac{n(n+1)}{2} + 1</annotation></semantics></math>; with <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> circles, the maximum number of regions is <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>−</mo><mi>n</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n^2 - n + 2</annotation></semantics></math>; and when combining <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> lines and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> circles, the maximum number of regions is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>L</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac><mo>+</mo><mn>1</mn><mo>+</mo><msup><mi>C</mi><mn>2</mn></msup><mo>−</mo><mi>C</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\frac{L(L+1)}{2} + 1 + C^2 - C + 2</annotation></semantics></math>.

Source: Among Us

Posted by asher on 2025-03-20 01:43:40