Mind-Bending Math: Riddles and Paradoxes

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 a paradox?

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, and reassemble the other two into another complete ball, also the same size as the original. Professor Kung explains the mathematics behind this astonishing result.

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 23. Start by asking why mathematicians want to measure sets.

Many puzzles are optimization problems in disguise. Discover that nature often reveals shortcuts to the solutions.

Consider the complexities of topological surfaces. For example, a Möbius strip is non-orientable, which means that left and right switch as you move around it.

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 Möbius strip and torus.

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.

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 to connect an array of dots with the fewest lines, and test yourself with map challenges.

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.

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 everyday objects do strange things.

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 winning strategy.

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 everyone has blue eyes, and try your luck at prisoner's dilemma.

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 different methods and the origin of the approach used now. As with voting, discover that no perfect system exists.

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 the discovery that destroyed the dream of an axiomatic system that could prove all mathematical truths - Kurt Gödel's demonstration that mathematical consistency is a mirage and that the price for avoiding paradoxes is incompleteness. Outline Gödel's proof, seeing how it relates to the liar's paradox from Lecture 1.

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 axioms, designed to avoid all paradoxes and keep mathematics consistent - a goal that was partly met by the Zermelo-Fraenkel set theory.

Randomly pick a real number between 0 and 1. What is the probability that the number is a fraction, such as ¼?

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 to prove that the set of real numbers is infinitely larger than the set of natural numbers, which is also infinite.

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 already occupied. Also sample a selection of supertasks, which are exercises with an infinite number of steps that are completed in finite time.

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.

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, social science, and especially medicine, where it can lead to inappropriate treatments.

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 change their guess about the location of a hidden prize after revealing new information about where it was not.

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 averages - from mileage to the Dow Jones Industrial Average - can be deceptive.

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.