# BdMO combinatorics question

What is the minimum number of lines needed to separate $20$ points on a plane?No 3 points lie on a straight line.

I have been struggling over this question for some time.It seems that the minimum number of lines corresponds with the maximum number of separations of a plane using some lines.Using 1 line,we divide plane into 2 regions.Using 2,we can make 4 separations.Using 3,we can make 7.There is a pattern here.Using this we can,with some easy calculation,figure out that it takes 6 lines to separate 22 points and 5 lines to separate 16.Therefore,the answer is 6.

The above holds true because using the (n+1)th line ,we can intersect n lines.We can obtain the recurrence $a_n=a_{n-1}+n$ with $n>0$ and $a_0=1$.Now I have the following questions:

1)Am I correct?

2)How can I,without solving the recurrence,figure out the formula for my sequence? I believe the last question can be resolved by any google search.So I am more interested in my first question and wondering if there is a better way to solve this.

Note by Rahul Saha
6 years, 7 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

The 20 points are fixed or you can move the points as you want?

- 6 years, 7 months ago

As the question doesn't put any restrictions,I think the points aren't fixed.

- 6 years, 7 months ago