Math Circle Student Resources
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Below is a roundup of some of our favorite mathematical resources and examples of topical questions we cover during our Nashville Math Circle sessions.
Example Topics
- What is 0.9999999… ?
- Why does the Golden Ratio show up so much in nature?
- Is it really true that one can cut an apple into five pieces, rearrange them, and obtain two apples of the same size, doubling the volume of apples without creating any new material?
- How can one turn a sphere inside out?
- What is infinity? Are all “infinities” the same size? How can infinite hotels with all rooms full accommodate infinitely many more guests?
- If a local barber shaves everyone who does not shave themselves, and shaves only those people, then does that barber shave themselves?
- Do triangles always have angles which add up to 180 degrees, or are there shapes where triangles can have angles adding up to different numbers? Are there “exotic” geometry theories where all triangles are isosceles and every point inside a circle is the center of that circle?
- Three intelligent mice found a piece of cheese. How should they arrange dividing it so that each mouse would be sure to get at least 1/3 of the cheese?
Math Resources
- Art of Problem Solving – The ultimate resource for problem solving, information on competitions at a number of levels and other programs, and discussion forums about problem solving.
- Outside In and Moebius Transformations Revealed – These videos show you, in beautifully animated fashion, how sometimes going to a higher dimension makes complicated things simple, and how to visualize the very strange fact that it’s possible to turn a sphere “inside out.”
- Numberphile – A YouTube channel that regularly posts interesting new content explaining big math ideas in fun and broken-down ways.
- The Man Who Knew Infinity – This book tells the inspiring story of Srinivasa Ramanujan, who flunked out of college twice and with almost no formal mathematical training rediscovered much of modern mathematics. He was truly a man who loved numbers and formulas and wrote down so many shocking formulas that leading mathematicians of the day could not believe that he was real. In a short period of a few years after his chance “discovery” by a leading mathematician of the day, he went on to revolutionize entire fields of mathematics. Today, his formulas have been shown to contain secrets behind the theory of black holes, and they are being implemented into signal processing methods in cell phone computer chips. A truly breathtaking read and proof that mathematical ideas have the power to change the world, even from the humblest of origins. Also, this book was turned into a Hollywood movie, which is also highly recommended.
- The Pea and the Sun: A Mathematical Paradox – This brilliant book is one of the first exposures Dr. Rolen had to deep mathematics, and is a fun ride through various riddles, geometric puzzles, and paradoxes, all leading to a demonstration of the perplexing Banach-Tarski paradox. This paradox states that it is possible to cut, with veryvery precise scissors, a ball into just a few pieces, then to rearrange these pieces without stretching or adding new material, only using rotations and translations, into two balls of the same volume as the original! The title refers to the resulting fact that you can therefore cut a ball the size of a pea into finitely many pieces, rearrange, and turn it into a ball the size of the sun! In this book, you will be introduced in a very friendly and approachable way to different ideas about “infinity.” For example, you will see why some infinities are bigger than other infinities, learn how to fill a hotel with no vacancies with more guests without kicking any guests out, and see a complete explanation of the paradox as depicted by a big machine with various tubes and hoppers.
- The Number Devil – Originally written in German, but its English translation is also a classic. It follows the story of a boy who hates math as a number devil takes him on an adventure.
- Summer Mathematics Programs Consortium – U.S. based summer math programs for middle and high school students. Here is a list of camps and programs that are part of the consortium.