# The Challenge Problem

Suppose you want to tile the plane (a flat floor that goes on and on, forever, in every direction) with flat tiles, and you want all of the tiles to be identical, but you also do not want the tiling to be periodic (i.e. if you shift your whole tiling straight, in any direction, for any (> 0) distance, you do not want the tiles to "line up" perfectly with the tiles of your non-shifted tiling). Is it possible (assuming you do not want overlapping or gaps anywhere in your tiling, as well) . . . . ?

## Yes, no or mathematically undecidable - prove your answer.

### Background

Click here for an introduction to aperiodic tilings - an article by David Austin, Grand Valley State University, from the American Mathematical Society's Feature Column.

### Research Tips

Click here for some tips on doing mathematics research.

# About the Archimedean Challenge

Try to solve the challenge problem! What will you discover? Each month during the challenge, you may submit any related (even distantly related) results (original or not) - your own work ("from scratch" or "on the shoulders of giants") - to earn honors (and math-themed prizes).

• Submissions are not required to participate.
• You may participate individually or as a group.
• You need not solve the challenge problem to earn an honor (or a prize)!
• All submissions will be kept confidential.
• You may request comments from your submission's reviewer(s).
• We strongly recommend that you have a research mentor, but it is not required.
• You must be 13 - 18-years-old to participate.
• If during the challenge, the challenge problem is solved, the challenge problem will shift to whether there is a novel alternate method of proof.

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, . . . . ).

The Archimedean Challenge is an incredibly fun, exciting and challenging, mathematics research, friendly competition for youth! The challenge will be held each year, beginning in the fall - a different problem posed every time (usually). The Archimedean Challenge is held concurrently with the Pythagorean Challenge, its "little brother/sister" competition for under 13-year-olds.

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

Your odyssey will begin with trying to solve the challenge problem - a famous, long-standing, unsolved problem in mathematics. A research mathematician commonly has such a question in the back of his/her mind inspiring and guiding his/her mind-tingling quest . . . .

### Judging Criteria

• Content
• Exposition

Submissions will be judged 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, . . . . ).

#### Judges' Panel

Chief Judge, David Gay, Director, Euclid Lab

Juliette Benitez, Deputy Director, Euclid Lab

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### Honors

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

Every month, 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 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

• Participating individuals and groups (including their members) will be given the challenge poster.
• Monthly prizes (e.g. math-themed books, DVD's, software, games, science kits, posters, art, oddities, curiosities, etc.) will be given to participants randomly chosen from among the laureates.
• Each special/exceptional distinction laureate will be given a math sculpture/trophy (after the challenge).
• Each exceptional distinction laureate will be invited to become a member of Euclid Lab (participate in our internship program)!
• + A Klein bottle (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 13 - 18-years-old to participate.

### A sponsor is strongly recommended.

We strongly recommend that you have a sponsor. A sponsor may be any mentor (>= 13-years-old) who is your senior (mathematically) and has an interest in mathematics and mathematics research. You may add sponsors over the course of the challenge.

### You may participate individually or as a group.

#### Individuals

Each individual must register.

#### Groups

Each group must register. A group's membership may increase over the course of the challenge. A group's prizes will be mailed to it's mailing address.

### How to Register

#### When to Register

You must be registered to make a submission. Otherwise, you may register anytime between now and the end of the challenge. Early registration is strongly recommended!

#### Registration Information

##### For Groups

Your (group's) name and mailing address, a listing of your sponsors (if any), plus your members' names.

#### You may submit/update 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.

#### You may pay your registration fee online or by mail.

##### Online

PayPal securely processes our online payments; you do not need a PayPal account.

 Registrant's Name
##### By mail
Include the registrant's name and mailing address. Make your check/money order out to Euclid Lab. Mail it to our main office.

### How to Make a Submission

• Submissions are not required to participate.
• Your submission need not solve the challenge problem!
• All submissions will be kept confidential.
• You may request comments from your submission's reviewer(s).

#### Submission Cover Letter Information

##### For Individuals

Your name, mailing address, and whether you are requesting comments from your submission's reviewer(s).

##### For Groups

Your (group's) name, mailing address, and whether you are requesting comments from your submission's reviewer(s).

#### You may submit your results (+ cover letter) online or by mail.

You may submit a copy of a (hand-writen is acceptable - but it must be legible) document presenting your results for any month(s) of the challenge. If you would like to submit it online, email This e-mail address is being protected from spambots. You need JavaScript enabled to view it prior to the respective deadline (to be given an FTP account, etc.). Otherwise, mail it (+ your cover letter) to our main office. It must be uploaded or postmarked by the respective submission deadline.

## Archimedean Challenge #2 TBA December 2012!

What will the next challenge be . . . . ?