Problem Writing Party: June 6th to 26th

Problem Writing Party 8 was a great success, and we had so many comments that the page started loading slowly for some people. Thanks for being part of the conversation!

I'm working on creating the quizzes, and will update the following list ASAP. Some of you would also have received B-notifications that your problems were added to the quizzes over the course of the previous week. Keep it up! We value your submissions, and would love to feature more of them.

Here are the quizzes that the Brilliant community helped create:

31 New Brilliant Challenge Quizzes

Arithmetic Progressions: Level 1, Level 2, Level 3
GCD/LCM: Level 1, Level 2, Level 3, Level 4, Level 5
Conditional Probability: Level 2, Level 3, Level 4, Level 5
Euler's Theorem: Level 2, Level 3, Level 4, Level 5
Distribution into Bins: Level 2, Level 3, Level 4, Level 5
Limits of Functions: Level 1, Level 2, Level 3, Level 4
Chess Tactical: Level 2, Level 3, Level 4
Chess Abstract: Level 2, Level 3
Pattern Recognition:

Let's kick off our 9th Problem Writing Party!

How it Works

The party starts right now (June 6th, 2016) and will last for the next two weeks. Throughout the two weeks, we will be focusing on writing awesome problems for the topics listed in quizzes that need your help on the publish page. The topics are:

All levels:	Chess Tactical	Rectangular Grid Walk	Power Mean inequality	Roots of Unity
Low levels:	Functions	Classification of Triangles	Newton's Law of Gravity
High levels:	Triangle Centers	Markov Chains

To join, submit as many problems as you want to these listed topics. At the end of the party, Brilliant staff will be picking the best 5-10 problems for each topic. These problems will then be immortalized and formed into a challenge quiz. If we pick your problem, then you can brag to your friends because it will be displayed on Brilliant forever! Your problem has a better chance of being selected if you include a graphic (when appropriate) and a solution.

Happy writing and keep the party alive!

Use this note to

Ask questions about the party or brainstorming ideas from Brilliant staff.
Share links to great relevant problems.
Bounce your ideas off each other to help formulate the best problem you can.
If you're posting your problems, please keep it to one rooted comment (and I will be merging such comments from the same person). This helps us keep the page more orderly.
You can link to your own problems by using the markdown syntax of [text](url link).

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}

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

I had a really good one once but I forgot it. I hate that.

Whitney Clark - 5 years ago

Is it like a fabulous proof for which this comment is too small to contain? I hate that too.

Calvin Lin Staff - 5 years ago

:-) I have a bad memory for some things. Some of my best ideas come when I don't have a pen and paper on hand, it seems.

Hey guys, check out my new problems for the PWP. Hope you like it :) .

Roots of Unity - Let's get United.

Classical Inequalities - Classic ! , Classical ! .

Law of Gravity - Who are A and B ? .

Anish Harsha - 5 years ago

Ah yes, that's a really important property of roots if unity :)

Note that it's the "Power Mean inequalities" chapter instead of all classical inequalities. Yes, we're developing out this section and splitting up the chapter! Finally! So, the IMO question doesn't quite count, but the other one is great!

The phrasing of the gravitational question could be improved on. Right now, it sounds quite forced / unnatural. Can you tweak it slightly?

Ok, sir. I'll work on the question on Law of gravity. Anyways, thanks.

This is my submission for the Power Mean Inequality... And my first submission for the problem writing parties...

Manuel Kahayon - 5 years ago

This is an interesting geometric application of the Power Mean Inequality! It is well presented, and the diagram helps too. I am looking forward to more of your problems :)

Pranshu Gaba - 5 years ago

Here is a question on functions ( inverse ) . Log inverse

Prince Loomba - 5 years ago

The final answer format of $\ln(\log_a e)^{6}$ is very contrived. Can we simplify that further? It might be better to ask what the constant $a$ is, and provide options like $e^e, e^{ \frac{1}{e}, e^ 1 , e^2$ . Thoughts?

Sir I cannot change the options but you can. Surely it will be better then. But then the question can be solved by hit and trial. Thats why I did this. Anyways as you wish.

@Prince Loomba – I've edited the problem. Let me know what you think of it.

Calvin Lin Staff - 4 years, 11 months ago

@Calvin Lin – This is an excellent question that was fun to solve. However, I wonder if replacing the two choices that you can immediately eliminate because they are <1, violating the first assumption in the question, would partially address Prince Loomba's concern about trial and error. Perhaps $e^{e-1}$ and $e^{e^{-2}}$ can replace $e^{-1}$ and $e^{-e}$ ?

Aaron Miller Staff - 4 years, 11 months ago

Here is my question for Rectangular grid walk. I edited the previous one.

Arkodipto Dutta - 5 years ago

Thanks, the final answer is in a much simpler form now. It would be great if you could add a diagram to your problem; it would make the problem easier to visualize.

Ok, I will add one soon.

@Arkodipto Dutta – Thanks for the image! I've edited it to make it clearer, and to also show an example of a possible path.

Here is another rectangular grid walk program. It has to do with a random walk in four dimension, where time is the fourth dimension... Enjoy! :)

Geoff Pilling - 5 years ago

TARDISSSSSS! =D =D

Pi Han Goh - 4 years, 12 months ago

Hahahaha... Yup, an inside joke for Dr. Who fans! :) (Apropos for a time traveling problem!)

Geoff Pilling - 4 years, 11 months ago

Haha, I'm a Doctor Who fan!

Your question brings up a loophole in the Doctor Who universe, where events are "time locked". They could have just travelled to $T-1$ and then wait it out, or travel to a nearby city and then walk over :)

Here is mine for Roots of unity

Akeel Howell - 5 years ago

I think asking for the minimum number of sides was a good question if the point wasn't already of the form $( \cos \frac{2\pi}{n}, \sin \frac{2\pi}{n}$ . Giving the 3rd vertex of a 7-gon would have made it more interesting

A problem for Rectangular Grid Walk:

Heist of the Diamond

Tarmo Taipale - 4 years, 12 months ago

Woah! Very nice question! +1 for storytelling! Reshared

Thanks!

Thanks for inviting me to this party but I am currently facing crisis in my school that my free time is totally cut off. I will see to that I can add something good if I can!

Akhash Raja Raam - 4 years, 11 months ago

Same here, thats why I cant contribute nowadays :(

Ashish Menon - 4 years, 11 months ago

I would like to submit these problems for Newton's Law of Gravity: Entry 1, Entry 2, and Entry 3.

Oh, I really like question 1. That same thought occurred to me as I was reading the book and thinking about the environmental differences.

For question 2, I'm not quite sure that 5N of gravitational force between 2 stones that we're holding is a reasonable estimate. It might be better to just say "N newtons" and express the options in terms of that.

For question 3, one potential thought experiment is to ask what happens as $r \rightarrow 0$ . If the gravitational force goes to infinity, why am I allowed to touch something and release it?

Here are my entries for Triangle Centres :

A Former Brilliant Member - 5 years ago

Thanks! These are great problems that the community likes and will be in the list that I review. In fact, the first problem was already added to a challenge quiz :)

Oh! I didn't get a notification about it though. Thanks for selecting it. :)

@A Former Brilliant Member – Yea, that happened back in the day before we started sending out notifications lol. I'm so glad that we do it now.

Here is a first problem for Classification of Triangles

Hjalmar Orellana Soto - 5 years ago

Great. I've improved the phrasing of the problem. Old version

If $ABC$ it a triangle such that its orthocenter lies on its circumscribed circumference. Find the greatest internal angle of $ABC$ (in degrees).

into

If $ABC$ is a triangle such that the orthocenter lies on the circumcircle, then find the greatest internal angle of $ABC$ (in degrees).

Can you add a solution to it?

Here is one for rectangular grids: Avoiding the accident sites.

And a simple one on Newton's Law of Universal Gravitation: Rock.

Arjen Vreugdenhil - 5 years ago

Thanks! I combined your posts into 1, to keep this note cleaner.

The first problem could definitely benefit from an image, which will remove the need to define the streets and avenues.

On Markov Chain :

Christopher Boo - 5 years ago

Great! I've added both of these to the Markov Chains chapter. I really like Otto's problem, and the discussion. I was pleasantly surprised at the result.

Roots of unity!. will contribute to it

Prakhar Bindal - 5 years ago

Looking forward to it!

Sir like gravitation requires low level problems but i have some level 4-5 problems can i still contribute?

@Prakhar Bindal – Sure of course!

For Classification of Triangles: That's all?

For Roots of Unity: Unification

Jonas Katona - 5 years ago

The first question seems computational / tedious, as it's just memorization of an angle. It also doesn't help explain how/why we care about classifying triangles.

Similarly for the second question, which doesn't yet showcase the beauty of roots of unity. It would have been nicer if final form was $\cos \frac{pi}{n} \times e^{ i \theta }$ , which starts to get at the usefulness of the identities.

Here are some suggestions for writing a great problem. I look forward to seeing your improvements over time :)

Thank you for the advice! I will keep this in mind for next time.

Here is one on rectangular grid walk: Paths

My problem on Classification of Triangles: Get It Right

Problems on functions: Fitting Functions, Flipping Functions

Lee Care Gene - 5 years ago

Ah yes, that's pretty interesting. At first glance, it seems almost impossible to proceed and we have to resort to the old method of listing out paths to every single node. But as it turns out, there is a nice way to interpret the problem and play with it. Thanks for sharing! Could you add a solution to the problem?

The second problem is good too. Ideally, we want to avoid trigonometry in this basic chapter. I have moved the problem into trigonometry instead.

I've just posted a solution.

Not sure if it fits, but this is my question for Rectangular Grid Walk: Dark Room Hunt.
And for function: Fibonacci Reversed.

Worranat Pakornrat - 5 years ago

The first problem is really interesting! It doesn't fit under Rectangular grid walks, but I've placed it into the Grid puzzles quiz.

The second problem is pretty fun too. I see that it's been placed in the Functions chapter, and will review it later :)

Thanks. The first one is one of my favorite detective question. :)

Here is my first entry for classification of triangles section.

Here is my entry for triangle centers section.

Here is my third submission for triangle centers section.

Thanks. Here is that ${91}^°$ question.

Ashish Menon - 5 years ago

FYI I combined your posts into 1, to keep this note cleaner.

Great question about triangle classifications :) A follow up question could be "Which non-degenerate triangle cannot have $91^\circ$ as one of its angles.

I like the triangle centers question too. It's kept simple, and interesting to play around with.

The third problem seems like tedious computation, and I'm not excited to get started with it. You can also tell that from the low attempt rate of the community.

Overall, great improvement in the problems that you're posting, especially in comparison to what you started out from. Keep it up!

Than you, sir, for the invitation, I just posted my problems under the community page. I do not know how to add them here.

Hana Wehbi - 5 years ago

Well write the syntax:- [word you want to link](the url of the webpage) is my submission for xxx section.

Here is my question for Rectangle grid walk

Thank you! Can you add a solution to the problem? To me, it's more natural to ask for the total number of ways, in part because the $Z! ^3$ in the denominator seems out of place.

Not sure if this one counts as a Markov chain?

Very tangentially. It is more of linearity of expectation, then it has to do with understanding a Markov chain. Great question BTW.

Yeah, that's kinda what I figured... Still trying to think of a good Markov Chain one... Oh, and thanks! :)

Here is my question about Triangle Centers. :>

Nowras Otmen - 5 years ago

Thanks for making your first problem contribution to Brilliant! I look forward to seeing more :)

I have posted two more problems, both relating to classification of triangles. I hope that these two problems are more suitable than my last two:

Stubborn triangles

Believe in yourself, Tommy!

I am also planning to create some better problems for roots of unity, keeping in mind how Euler's formula can be used as a means of simplifying trigonometric functions.

Thank you! I enjoyed solving the problems :) They are easy to understand, and slightly tricky to solve.

I am looking forward to your problems in roots of unity!

Thank you for the compliment, Pranshu! :D

Jonas Katona - 4 years, 12 months ago

A set on Markov Chains :

Winnie the Poo!!!!!!!!!!

Akeel Howell - 4 years, 11 months ago

Here is another rectangular grid walk problem.

Here is mine for functions

abc xyz - 5 years ago

Well u learnt hoe to link!!! (+1)

Ya I refered to the latex help page :>

@Abc Xyz – Actually it is mentioned in the note too

How do you link?

Finn C - 5 years ago

@Finn C – Use the Syntax:- [Text you want to link](url of the page you want to link)

Here is my fourth submission for triangle centers section.

Here's my problem for Function( Sorry i was a little bit late ;) )

Jason Chrysoprase - 5 years ago

Can i post Similar Triangle Problem for Classification Triangle ?

Hmmmm, I don't think so. They are two different stuffs.

Pi Han Goh - 5 years ago

;(, that's mean i only have one posted problem

And if you are not busy, try to write solution in this problem. Yes i know the answer but i don't know how to prove it.

coupling a few 1008 roots!!

problem on roots of unity i just made !!! :)

Rohith M.Athreya - 5 years ago

Doesn't your expression of $\omega\omega^{2n} + \omega^2 \omega^{2n-1} + \cdots + \omega^{2n} \omega$ simplifies to just $(2n+1)\omega^{2n+1}$ ? Or am I interpreting your question wrongly

Interesting question nonetheless! +1

according to me, it adds upto $2n\omega^{2n+1}$ which is nothing but $2n\omega$ which can be expressed as

$2ne^{\frac{2\pi i}{2n}}$ which finally yields an answer of 2

but a moderator(perhaps) editted the question and the answer has been changed to 1 which i cant account for

Rohith M.Athreya - 4 years, 12 months ago

Here are my entries for the topics: Functions

Maps to itself... eventually and Use a telescope.

function problem

Margaret Zheng - 5 years ago

Cute question! I thought sinusoidal = periodic only. +1

An easy one on Classification of Triangles: Tri-Max.

Another Function Problem

Click Here

Here's the applied question for Classical Inequality: Rectangular Inequality.

Enjoy! ;)

Worranat Pakornrat - 4 years, 12 months ago

Two easy (I think) problems for functions:

Not many solutions? Part 1

Not many solutions? Part 2

This is my new question on Power-Mean Inequality: Yummy Jelly.

Should be pretty easy though. ;)

Triangle problem: It's a whole number!. Triangle centers: It's a Triangle!. Powers: Powers of 2. Roots of Unity: What value does the area approach?.

Akeel Howell - 4 years, 12 months ago

Short and simple problems. These problems test basic understanding of the concepts. Thanks for sharing these problems, Akeel :)

Pranshu Gaba - 4 years, 11 months ago

Here's one for Roots of Unity: Odd Powers Only.

Another one on Roots of Unity: Prime Power Roots.

Worranat Pakornrat - 4 years, 11 months ago

Welcome to my dazzling problem on Triangle Centers: Tangling Circles.

That's interesting (and colorful)!

Ahh i'm late, need to work fast ;)

There's still lots of time, don't worry.

And lots of parties too!

This is my question on Markov's Chain: Sun & Beaches.

Nice problem, Dr. Warm. I have added a diagram so it's easier to see what is happening.

Thanks. I was so busy with my work that I didn't have time to draw the diagram.

My entry for roots of unity

Joao Miguel Coelho - 5 years ago

Cool problem, Joao! It is simply stated, and clear and concise. Keep it up! :)

Here's a link to a problem I wrote for Newton's Law of Gravity: Splitting the Sun in Two

Aaron Miller Staff - 5 years ago

Here's another Gravity problem: A Gloopelhopper Problem

Interesting problems, Aaron! The scenarios are described in great detail, and the solutions are excellently written. Awesome!

Here's another problem I wrote for Newton's Law of Gravity: Voyage from the Sun

Thank you, Aaron! It is a lovely problem with a nice diagram.

I have a suggestion; the problem statement was long and slightly difficult to follow:

Express your answer as a ratio $\dfrac{F_2}{F_1}$ , where $F_1$ is the gravitational force applied by the (actual) Sun on the Earth at an orbital distance of $1 \text{AU}$ .

It can be simplified to:

Suppose the gravitational force applied by the Sun on the Earth at an orbital distance of $1 \text{ AU}$ is $F_1$ .

What is $\dfrac{F_2}{F_1}$ ?

By using shorter sentences, we can make the problem more easy to understand.

Hi Pranshu,

Thank you for your helpful comment! I totally agree with your rephrasing. I will change the problem to incorporate your suggestion.

Here is my rectangular grid walk question... Enjoy! :^)

Hope I'm not late to the party. Here's my entries for Power mean inequality::

Do I need to expand it all?

Maximum value is not as big as you think

Man you come up with the craziest questions, Pi! ;-)

Here's my entry for triangle centers

Final entries for this party::::

Triangle centers:
Centroid?
Some other center?

Grid walk::::
Sudoku
Tron
Limit

I have one more question to add to this party! I just created it today, for functions: Inseparable

Jonas Katona - 4 years, 11 months ago

Do it by algebra

Aditya Narayan Sharma - 5 years ago

Hm, after our discussion, I don't see what makes this problem interesting / special.

The problem is not in any of the chapters that I listed out. These chapters will get special attention over the next 2 weeks, and thus contributing into those chapters now would increase the likelihood that it makes it into the quizzes.

@Calvin Lin sir what about this? Will it work for the AP quiz: Just, AP

Rishabh Sood - 5 years ago

I did review the question as I was putting together the AP set, and it's interesting in it's own way. However, the community didn't find it interesting and wasn't excited to work on it, which is why I didn't place it into the Level 4 quiz.

Here are some guidelines to help you improve the quality of your problems, and I would love to feature them :)

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}`	$\boxed{123}$