# The Challenge Problem

What is the minimum number of colors neccessary to color the plane such that whenever any two of its points are a unit distance apart, they are different colors? This number is called the chromatic number of the plane. This question is called the Hadwiger-Nelson Problem.

## Try to solve the challenge problem!What will you discover?

Submit any related (even distantly related) results (original or not) - but they must be your own work ("from scratch" or "on the shoulders of giants") - to earn an honor:

• You must be <= 19-years-old to participate.
• You don't need to make a submission to participate.
• You may participate individually or as a group.
• You don't need to solve the challenge problem to earn an honor!
• All submissions will be kept confidential.
• We recommend that you have a research mentor; but it is not required.
• If during the challenge, the challenge problem is solved, the challenge will go on! Research focus can shift to whether there is a novel alternate method of proof and to other related questions . . . .

Definition: Results are anything you might write down in your research journal (e.g. examples, conjectures, ideas, guesses, hopes, mistakes, intuition, arguments, proofs, half-proofs, discoveries, . . . . ).

## A 4-Month-Long Research "Odyssey"

The Archimedean Challenge is an incredibly funexciting and challengingmathematics research, friendly competition for youth! The challenge is held throughout the year - a different problem posed every time (usually).

### Reviewing Criteria

• Content
• Exposition

Submissions will be reviewed based on content (e.g. questions you explored, conjectures you made, methods you tried, mistakes you made, discoveries you found, . . . . ) and exposition (e.g. give proofs, half-proofs, justify your assertions, explain your ideas logically and clearly, . . . . ).

### Reviewers are research mathematicians; some may review anonymously.

• David Gay, Director, Euclid Lab
• Juliette Benitez, Deputy Director, Euclid Lab
• +

### Honors

#### Honorable Mention, Special Distinction and Exceptional Distinction Honors

Honorable mention, special distinction and exceptional distinction will be bestowed upon selected participants.

#### The Archimedes Award

The Archimedes Award will be bestowed, specifically, upon the first registered participant(s) to solve the challenge problem (i.e. to earn the Archimedes Award your submitted solution must be 1.) your own original work, and 2.) generally accepted by the mathematical community).

### Prizes

• Each participant will be given the challenge poster.
• Additional prizes (e.g. math-themed books, DVD's, software, games, science kits, posters, art, oddities, curiosities, etc.) may be given to participants randomly chosen from among the laureates.
• Each special/exceptional distinction laureate will be given the challenge sculpture/trophy.
• + the challenge mystery prize (the Archimedes Award) if you solve it!

Laureates will be notified, by mail, as soon as possible after they are determined.

### Eligibility

You must be <= 19-years-old to participate.

### Having a research mentor is recommended; but it is not required.

We recommend that you have a mentor. A mentor can be anyone >= 13-years-old who is your senior (mathematically) and has an interest in mathematics and mathematics research. Your mentor(s) may change over the course of the challenge.

### How to Register

#### When to Register

Register anytime during the course of the challenge.

#### Submit your registration information online or by mail.

Email it to This e-mail address is being protected from spambots. You need JavaScript enabled to view it or mail it to our main office. NB, you must be >=13-years-old to email it.

#### Pay online or by mail.

##### For Online Payment:

PayPal securely processes our online payments; you do not need a PayPal account. NB, you must be >= 13-years-old to pay online.

##### For Payment by Mail:
Include the registrant's name and address. Make your check/money order out to Euclid Lab. Mail it to our main office.

### How to Update Your Registration Information

Update you registration information anytime during the course of the challenge.

#### Submit your registration information update online or by mail.

Email it to This e-mail address is being protected from spambots. You need JavaScript enabled to view it or mail it to our main office. NB, you must be >=13-years-old to email it.

### Submissions

• A submission is not required to participate.
• Your submission does not need to solve the challenge problem!
• All submissions will be kept confidential.
• You may make multiple submissions.

### How to Make a Submission

#### Submit your results (+ your submission information) online or by mail.

Email your submission to This e-mail address is being protected from spambots. You need JavaScript enabled to view it or mail it to our main office. NB, you must be >= 13-years-old to email it. It must be emailmarked or postmarked by the last day of the challenge.

# Archimedean Challenge #3 TBA!

What will the next challenge be . . . . ?

# Congratulations laureates!

University Park Campus School Cougar Enigmas