The Archimedean Challenge
#1 Does a planar, aperiodic monotile exist?
The Challenge Problem
Suppose you want to tile the plane. And you want to use only one size and one shape of tile. But you also do not want the tiling to be periodic, have gaps or have overlaps. Such a tiling is called a planar, aperiodic monotiling. And, a tile which can only tile aperiodically (and never periodically) is called a planar, aperiodic monotile.
Problem: Does a planar, aperiodic monotile exist?
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 Archimedean Challenge #1 (#2, etc.)
A Mathematics Research "Odyssey"
The Archimedean Challenge is an incredibly fun, exciting and challenging, friendly competition for all! The challenge is ongoing, and until the problem is solved, it will be repeated each academic term.
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.
Definition: Results are anything you might write down in your research journal (e.g. examples, conjectures, ideas, guesses, hopes, mistakes, intuition, arguments, proofs, halfproofs, discoveries, . . . .).
If during the challenge, the challenge problem is solved (!), the challenge will go on until the next academic term, and participants may consider whether there is a novel, alternate method of proof and other related questions . . . .
Submission Review
Criteria
Submissions will be reviewed, in order of receipt, 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, halfproofs, justify your assertions, explain your ideas logically and clearly, . . . . ).
Reviewers are researchers; some may review anonymously.
 David Gay, Director, Euclid Lab
 Juliette Benitez, Associate Director, Euclid Lab
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Honors
Honorable Mention and Special Distinction Honors
Honorable Mention and Special Distinction will be bestowed upon selected participants.
The Archimedes Award
The Archimedes Award will be bestowed upon the first participant(s) to submit a solution of 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).
Laureate Notification
Laureates will be notified as soon as possible after they are determined.
Prizes
Dependent upon funding:

Each laureate will be awarded the challenge poster!

Each special distinction laureate will be awarded the challenge trophy!

+ the challenge Pi (a whole number multiple of $3.14) if you solve it!
Euclid Lab is a nonprofit, publicly supported 501(c)(3). Funds for the Archimedean Challenge may vary per academic term.
You may participate individually or as a group.
An individual may become a group over the course of the challenge. A group's membership may change over the course of the challenge.
Having a research mentor is recommended, but not required.
We recommend that you have a mentor. A mentor can be anyone 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.
A submission is not required to participate.
And, you don't need to solve the challenge problem to earn an honor! You may make multiple submissions. All submissions will be kept confidential.
How to Make a Submission
Submit your results online, anytime during the course of the challenge.
Send:
 Your name/Your group's name
 Your mentor(s) name(s), if any
 (Optional) Your institutional affiliation(s), if any
 Your results
To:
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