Most interesting problem

Hey Brilliantinians, here I'm posting the most beautiful problem I've ever seen. The idea is that everyone can share the most interesting/beautiful/awesome problem he/she has ever faced to. The problem needs not to come from Brilliant. Solutions to the problems are also accepted in the comments

Here's mine. Let $p\geq 3$ be a prime number. We divide every side of a triangle in $p$ equal parts and every point of division is connected to the opposite vertex. Compute the maximum number of pairwise disjoint parts in which the triangle is divided in funciton of $p$ .

Enjoy!!!

#Combinatorics #NumberTheory #Awesome #Interesting

Easy Math Editor

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.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

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

Math	Appears 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}`	$\boxed{123}$

Comments

Well, if $p$ is an odd prime, then the maximum such number must be $3{ p }^{ 2 }-3p+1$

so that for $p={3,5,7,11,13,19,...}$ , we have ${19,61,127,331,469,817,...}$

We first note that the triangle is divided into ${ p }^{ 2 }$ disjoint parts by the first $2$ sets of rays from $2$ vertices. Then each ray from the last vertex intersects those rays $2(p-1)$ times, thereby increasing the number of disjoint parts by $2(p-1)+1$ , and there are $p-1$ such rays from that last vertex. Hence, the total is

${ p }^{ 2 }+(p-1)(2(p-1)+1)=3{ p }^{ 2 }-3p+1$

This derivation depends on the fact that no point is shared by $3$ such rays, hence the primes only.

This graphic shows the case where $p=5$ , so one can see how this is done. Affine transforms of this triangle leaves the number of parts unaffected.

Michael Mendrin - 5 years, 12 months ago

Math	Appears 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}`	$\boxed{123}$