# Mind-Bending Math: Riddles and Paradoxes

## 24 episodes in this series

Episode 1 Everything in This Lecture Is False
Plunge into the world of paradoxes and puzzles with a "strange loop," a self-contradictory problem from which there is no escape. Two examples: the liar's paradox and the barber's paradox.…
Episode 2 Elementary Math Isn't Elementary
Discover why all numbers are interesting and why 0.99999... is nothing less than the number 1. Learn that your intuition about breaking spaghetti noodles is probably wrong. Finally, see how…
Investigate a puzzle that defied some of the most brilliant minds in mathematics: the Monty Hall problem, named after the host of Let's Make a Deal! Hall would let contestants…
Episode 4 Strangeness in Statistics
While some statistics are deliberately misleading, others are the product of confused thinking due to Simpson's paradox and similar errors of statistical reasoning. See how this problem arises in sports,…
Episode 5 Zeno's Paradoxes of Motion
Tour a series of philosophical problems from 2,400 years ago: Zeno's paradoxes of motion, space, and time. Explore solutions using calculus and other techniques. Then look at the deeper philosophical…
Episode 6 Infinity Is Not a Number
The paradoxes associated with infinity are... infinite! Begin with strategies for fitting ever more visitors into a hotel that has an infinite number of rooms, but where every room is…
Episode 7 More Than One Infinity
Learn how Georg Cantor tamed infinity and astonished the mathematical world by showing that some infinite sets are larger than others. Then use a matching game inspired by dodge ball…
Episode 8 Cantor's Infinity of Infinities
Randomly pick a real number between 0 and 1. What is the probability that the number is a fraction, such as 1/4? Would you believe that the probability is zero?…
Episode 9 Impossible Sets
Delve into Bertrand Russell's profoundly simple paradox that undermined Cantor's theory of sets. Then follow the scramble to fix set theory and all of mathematics with a new set of…
Episode 10 Gödel Proves the Unprovable
Study the discovery that destroyed the dream of an axiomatic system that could prove all mathematical truths - Kurt Godel's demonstration that mathematical consistency is a mirage and that the…
Learn that determining the will of the voters can require a mathematician. Delve into paradoxical outcomes of elections at national, state, and even club levels. Study Kenneth Arrow's Nobel prize-winning…
Episode 12 Why No Distribution Is Fully Fair
See how the founders of the U.S. struggled with a mathematical problem rife with paradoxes: how to apportion representatives to Congress based on population. Consider the strange results possible with…
Episode 13 Games with Strange Loops
Leap into puzzles and mind-benders that teach you the rudiments of game theory. Divide loot with bloodthirsty pirates, ponder the two-envelope problem, learn about Newcomb's paradox, visit the island where…
Episode 14 Losing to Win, Strategizing to Survive
Continue your exploration of game theory by spotting the hidden strange loop in the unexpected exam paradox. Next, contemplate Parrando's paradox that two losing strategies can combine to make a…
Episode 15 Enigmas of Everyday Objects
Classical mechanics is full of paradoxical phenomena, which Professor Kung demonstrates using springs, a slinky, a spool, and oobleck (a non-Newtonian fluid). Learn some of the physical principles that make…
Episode 16 Surprises of the Small and Speedy
Investigate the paradoxes of near-light-speed travel according to Einstein's special theory of relativity. Separated twins age at different rates, dimensions contract, and other weirdness unfolds. Then venture into the quantum…
Episode 17 Bending Space and Time
Search for the solutions to classic geometric puzzles, including the vanishing leprechaun, in which 15 leprechauns become 14 before your eyes. Next, scratch your head over a missing square, try…
Episode 18 Filling the Gap between Dimensions
Enter another dimension - a fractional dimension! First, hone your understanding of dimensionality by solving the riddle of Gabriel's horn, which has finite volume but infinite surface area. Then venture…
Episode 19 Crazy Kinds of Connectedness
Visit the land of topology, where one shape morphs into another by stretching, pushing, pulling, and deforming - no cutting allowed. Start simply, with figures such as the Mobius strip…
Episode 20 Twisted Topological Universes
Consider the complexities of topological surfaces. For example, a Mobius strip is non-orientable, which means that left and right switch as you move around it. Go deeper into this and…
Episode 21 More with Less, Something for Nothing
Many puzzles are optimization problems in disguise. Discover that nature often reveals shortcuts to the solutions. See how light, bubbles, balloons, and other phenomena provide powerful hints to these conundrums.…
Episode 22 When Measurement Is Impossible
Prove that some sets can't be measured - a result that is crucial to understanding the Banach-Tarski paradox, the strangest theorem in all of mathematics, which is presented in Lecture…
Episode 23 Banach-Tarski's 1 = 1 + 1
The Banach-Tarski paradox shows that you can take a solid ball, split it into five pieces, reassemble three of them into a complete ball the same size as the original,…
Close the course by asking the big questions about puzzles and paradoxes: Why are we so obsessed with them? Why do we relish the mental dismay that comes from contemplating…

## Related videos

Secrets of Mental Math
One key to expanding your math potential--whether you're a CEO or a high school student--lies in the power to perform mental calculations. Solving basic math problems in your head offers lifelong benefits including a competitive edge at work, a more active and sharper mind, and improved standardized test scores. Discover…
Geometry - An Interactive Journey to Mastery
Inscribed over the entrance of Plato's Academy were the words, "Let no one ignorant of geometry enter my doors." To ancient scholars, geometry was the gateway to knowledge. Its core skills of logic and reasoning are essential to success in school, work, and many other aspects of life. Yet sometimes…
The Joy of Mathematics
Ready to exercise those brain cells? Humans have been having fun with mathematics for thousands of years. Along the way, they've discovered the amazing utility of this field--in science, engineering, finance, games of chance, and many other aspects of life. This course of 24 half-hour lectures celebrates the sheer joy…
The Power of Mathematical Visualization
World-renowned math educator Dr. James Tanton shows you how to think visually in mathematics, solving problems in arithmetic, algebra, geometry, probability, and other fields with the help of imaginative graphics that he designed. Also featured are his fun do-it-yourself projects using poker chips, marbles, paper, and other props, designed to…
Algebra I
Algebra I is an entirely new course designed to meet the concerns of both students and their parents. These 36 accessible lectures make the concepts of first-year algebra - including variables, order of operations, and functions-easy to grasp. For anyone wanting to learn algebra from the beginning, or for anyone…
Black Holes, Tides, and Curved Spacetime - Understanding Gravity Course
Gravity rules the universe. Without it, everything would dissolve into a gas of randomly interacting atoms. Yet gravity is one of the least understood forces in nature. Black Holes, Tides, and Curved Spacetime introduces you to key ideas in gravity research over the past 400 years. It's an awe-inspiring journey…
The Search for Exoplanets: What Astronomers Know
As recently as 1990, it seemed plausible that the solar system was a unique phenomenon in our galaxy. Thanks to advances in technology and clever new uses of existing data, now we know that planetary systems and possibly even a new Earth can be found throughout galaxies near and far.…
Chemistry, 2nd Edition Course
Many students struggle in high school chemistry. Even if they succeed in earning a good grade, they often still feel confused and unconfident. Why is this? And what can be done to help every student succeed in this vitally important course? Success in chemistry, according to veteran science teacher Professor…
Everyday Engineering - Understanding the Marvels of Daily Life
Look around you. Great masterpieces of engineering are everywhere. As soon as you get out of bed in the morning, you are putting centuries of scientific and mechanical ingenuity to use. So familiar are these structures, inventions, and services that they are all but invisible. Yet they are just as…
The Mathematics of Games and Puzzles - From Cards to Sudoku Series
Whether it's chess, poker, or Sudoku, most games have this in common: Everything you need to win is rooted in mathematics. Now, using nothing more than a simple grasp of math, you can discover optimal ways to win games and solve puzzles with the speed and accuracy of professional players.…
Mysteries of Modern Physics: Time Series
Time seems to be woven into the very fabric of the universe. But why? In 24 riveting half-hour episodes, Mysteries of Modern Physics: Time shows how a feature of the world that we all experience connects us to the instant of the formation of the universe--and possibly to a multiverse…
Physics and Our Universe: How It All Works Series
Physics is the fundamental science. It explains how the universe works! All you need to begin exploring physics is a grasp of high-school algebra. These lessons are intensively illustrated with diagrams, animations, graphs, and other engaging visual aids and introduce you Newtonian mechanics, oscillations and waves, thermodynamics, electricity and magnetism,…