# Talk:Pigeonhole Principle

## Contents

# Response to Checklist

## Messages to the Future

- Made two suggestions for future editors.

## References and footnotes

- Original sources of "borrowed" images are marked if you click them.
- Direct quotes are cited.
- References are listed with links at the bottom of the page.

## Good writing

### Context

- This topic “Pigeonhole principle” is my personal favorite! The main image is appealing. "More Mathematical Explanation" is comprehensive. I gave a lot of examples for readers to know this principle better. I believe those examples make this page a lot more interesting than other pages.

### Quality of prose and page structuring

- The beginning paragraph defines pigeonhole principle and provides another definition, name.
- Each section is related to the main topic.
- Real world applications and subsections in the mathematical explanations are listed from easy to hard, from fundamental to expanding. I hide some long proofs and just show the statements, in case people don't want to know the proof but just the examples themselves. The heaviest math is in “Why interesting.” Well, they are not heavy math but hard to understand.

### Integration of images and text

- Every image is referred in the context. In every image, the denotations are noted and readers know what each symbol means. Some images are just for better structure; they are not really illustrative (i.e. birthday cake, socks, hair, cards).
- Readers are clear about which picture they should look at while viewing this page. Sometimes they can get inspired by the picture and know how to prove the problem.
- There are no large chunks of words.

### Connections to other mathematical topics

- There is a link to another mathematical topic outside of Math Images.

### Examples, Calculations, Applications, Proofs

- The equations, calculations, and examples are clear to readers.
- Every statement or property has its proof.
- I wrote a summary and some guiding text to help people know how to solve this sort of problems using pigeonhole principle. (especially the “how to construct pigeonholes” section)

### Mathematical Accuracy and precision of language

- I try to make everything as clear as possible. Hopefully readers with any level of math will understand it.
- I try to make everything error free. Corrections and suggestions are appreciated.
- The definition of every mathematical term, theorem or rule readers might not know is either explained in the body text or via a mouse-over, or linked to another page.

### Layout

- Texts are short, not very long, and broken up by images or broken in paragraphs.
- Mathematical terms are boldfaced.
- Hide and see is appropriately used.
- No awkward white chunks.
- No weird computer codes.

### Thank You

# Comments from Chris 7.20.11

This is both a fun and interesting page. You've done an excellent job with this. My comments are mostly minor ones.

Thank you very much Chris!! Very helpful comments! Phoebe 7/20

Near the end of the Basic Description, you make an excellent statement that I think belongs instead in the intro: "The pigeonhole principle (PP) itself may seem simple but it is a powerful tool in mathematics." I'd make it the strong conclusion to your intro section.

- I like your suggestion! You want me to move only this sentence or the whole paragraph this sentence is in?

Basic Description

~~Layout issue: It's clear on my browser that the last two paragraphs begin "The pigeonhole principle is also called the Dirichlet.." and "The PP itself may seem..." It's hard to tell. however, where the other paragraph(s) end(s) and begin.~~

- I see. Fixed it.

- Paragraph 1, Sentence 2 (P1S2) Is the principle really famous?

- I think so... I've learnt it in middle school.

~~P1S3 Add "that" between "principle is" and "for any."~~

- Fixed it.

~~Near the end of that long paragraph (if it is only one paragraph), there must be~~*at least*two pigeons in a pigeonhole.

- Fixed it.

Interesting Applications

~~Birthdays: P1S1: Remove "the" from between "including" and "leap."~~

- Fixed it.

~~Hair: P1S2: Make "hair" into "hairs a human normally has."~~

- Fixed it.

~~Chairs: P2S1: Instead of "(see the two figures below)", add a new sentence that talks about how the first two examples show two ways of partitioning the chairs without two empty chairs next to each other.~~

- Fixed it. Like your suggestion!

~~Strangers or Friends: really well done!~~

- Thanks!

More Mathematical Examples

- 3. This example is very hard to follow. At the very least, give a concrete example to relate all the variables with subscripts to.

- Good to know. Add a more concrete example.

Summary
~~*P1, Subsection 3. You didn't let the limit be 50 or "want to have" the limit be 50; the limit ends up being 50 because only 50 distinct groups can be made. ~~

- Fixed it.

How to Find the Bounds

~~P1S1 Change the tense to past perfect; "we have only been told" and "whether we have reached"~~~~P1S3 Either make the question are full sentence or say "This leads to the interesting question of how to find the bounds."~~

- Fixed them.

- Examples #1: Do you want a direct link to another website? (Ramsey Numbers) I don't think that's the convention for Math Images. I'll check tonight at Swarthmore; maybe you can also investigate this.

- I'm sorry if I did anything inappropriate. I don't really want to link to another website either. It's just that I don't have time to make a page for Ramsey Number, but I put it as a future direction for other people. How about I create a helper page and link PP to Ramsey Number? I'll add more things to it if I have time and if I'm not able to do that, future editors could investigate it as well. What do you think?

# Richard Comments 7/20

- Phoebe, the page looks great! I know you've done a ton of editing and fixing already, so take these comments however you'd like. Richard 7/2

- After reading this a second or third time, I'm realizing more what I was having trouble saying last time. In the basic description, you outline these two great ways to think about the pigeonhole principle: (1)a very intuitive way of thinking in terms of pigeons fitting into pigeonholes (2)a more "mathy" way with average and maximum value. You do a great job of explaining why they are the same in the basic description, but I think you are selling yourself short on the math side of things especially with many of the examples that you use. I guess what I'm trying to say is that there's more math to this than your examples show. For example, you could use math to describe the suit of cards example as well as describing it in terms of pigeons and holes. you could say that the avg value=5/4 so the max must be at least that, and since we can't pick a fraction of a card, there must be at least two cards of the same suit. I think it could be helpful to describe each example using both ways (1) and (2) like you did in the basic description.
- This also provides the potential for a MME that is more than just example. I think you can more thoroughly describe avg value and max value pictorially and with words. By defining average, you can show mathematically that there must be a number in the non-empty finite set that is greater than the average.

- I like your idea. Good point, but I might need more time to make those changes. For some easy examples, I didn't apply the max and avg value because they are understandable and pretty obvious. Maybe I'll just add those description to some harder ones.
- Alright, I add the max and avg thing to each example. They look better now! Phoebe

- After reading this a second or third time, I'm realizing more what I was having trouble saying last time. In the basic description, you outline these two great ways to think about the pigeonhole principle: (1)a very intuitive way of thinking in terms of pigeons fitting into pigeonholes (2)a more "mathy" way with average and maximum value. You do a great job of explaining why they are the same in the basic description, but I think you are selling yourself short on the math side of things especially with many of the examples that you use. I guess what I'm trying to say is that there's more math to this than your examples show. For example, you could use math to describe the suit of cards example as well as describing it in terms of pigeons and holes. you could say that the avg value=5/4 so the max must be at least that, and since we can't pick a fraction of a card, there must be at least two cards of the same suit. I think it could be helpful to describe each example using both ways (1) and (2) like you did in the basic description.

- For Interesting applications number 6 I'd reword the main statement to be something like "In a room with six people in it, there will always be either 3 people who know each other or 3 people who don't know each other.

- In the same example, I'd get rid of "This seems to be a chaotic problem but we need to find order in this mess." and "How should we prove that it is true all the time?" You can start the section off instead with something like "We can use a diagram to help us show that this statement is true. Note that these..."

- Fixed it.

- you sort of just say that by the pigeonhole principle, at least three lines must be the same color. I think it would be useful for you to show how you got that in terms of avg and max value here.

- I'm a little confused still by MME example 3.

- And you used the wring less than symbol I think...try this one
- Fixed it. I think I made some changes after you viewed it. Other people thought that this part is confusing as well, so I added more guiding texts to this problem. Thank you for letting me know.

- For How to Construct example 2, I think you could maybe put this one in terms of pigeons and pigeonholes as well and put the factorial form of 15 choose 2 like you have for the example 11 choose 4.

- Fixed it.

- In your summary section, you mention limit several times and I'm not exactly sure what you're talking about.

*Think about how many possible ways you can find in the problem; this is your restriction and also the pigeonholes you are trying to find.*Limit means the restriction. Fixed it.

- Same with bounds. Maybe define
**limits**and**bounds**in terms of the principle.

- Fixed it. Add a mouseover

- This is a really cool page, Phoebe. I really like its roots in application and that you use some pretty cool examples to teach the topic in the process.

- Glad to know. Thank you for spending time on my page!

Richard 7/20