Mind-Bending Math: Riddles and Paradoxes

Plunging into the world of paradoxes and puzzles with a "strange loop"; it is a self-contradictory problem from which there is no escape; the liar's paradox and the barber's paradox; "proving" that 1+1=1; the Island of Knights and Knaves.

Why all numbers are interesting; why 0.99999... is nothing less than the number 1; learning that one's intuition about breaking spaghetti noodles is probably wrong; how averages, from mileage to the Dow Jones Industrial Average, can be deceptive.

Investigating the Monty Hall problem, a puzzle named after the host of "Let's Make a Deal"; Hall lets contestants change their guess about the location of a hidden prize after revealing new information about where it is not.

Some statistics are the product of confused thinking due to Simpson's paradox and similar errors of statistical reasoning; how this problem arises in sports, social science and especially medicine, where it can lead to inappropriate treatments.

Touring Zeno's paradoxes of motion, space and time; exploring solutions using calculus and other techniques; looking at the deeper philosophical implications, which gain new relevance through the discoveries of modern physics.

Explore several paradoxes involving infinity, then study supertasks: exercises with an infinite number of steps that are completed in finite time.

Stretch your mind and attempt to solve the timeless riddles and paradoxes that have confounded the greatest philosophical, mathematical, and scientific minds in history.

Randomly picking a real number between 0 and 1; considering the probability that the number is a fraction or zero; probing this and other facts about infinite sets; the discovery that makes Cantor exclaim, "I see it, but I don't believe it!".

Delving into Bertrand Russell's paradox that undermines Cantor's theory of sets; following 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.

Studying Kurt Gödel's demonstration that mathematical consistency is a mirage; Gödel shows that the price for avoiding paradoxes is incompleteness; outlining Gödel's proof; seeing how it relates to the liar's paradox.

Learning that determining the will of the voters can require a mathematician; the paradoxical outcomes of elections at various levels; Kenneth Arrow's Nobel prize-winning impossibility theorem; assessing the U.S. Electoral College system.

How the founders of the U.S. struggle with a mathematical problem; apportioning representatives to Congress based on population; the results possible with different methods; the origin of the current approach; learning that no perfect system exists.

Leaping into puzzles that teach the rudiments of game theory; dividing loot with pirates; pondering the two-envelope problem; learning about Newcomb's paradox; visiting the island where everyone has blue eyes; tackling the prisoner's dilemma.

Spotting the hidden strange loop in the unexpected exam paradox; contemplating Parrondo's paradox that two losing strategies can combine to make a winning strategy; trying challenging hat games; using the axiom of choice from set theory.

Classical mechanics is full of paradoxical phenomena; Professor Kung shows this using springs, a slinky, a spool and oobleck; the physical principles that make everyday objects do strange things; how to float a cruise ship in a gallon of water.

Investigating the paradoxes of near-light-speed travel; separated twins age at different rates; dimensions contract; venturing into the quantum realm; exploring the nature of light; the true meaning of the Heisenberg uncertainty principle.

Searching for the solutions to classic geometric puzzles such as the vanishing leprechaun; scratching one's head over a missing square; connecting an array of dots with the fewest lines; map challenges; how to ride a bicycle with square wheels.

Entering a fractional dimension; honing one's understanding of dimensionality by solving the riddle of Gabriel's horn; venturing into the fractal world of Sierpinski's triangle and the Menger sponge, both of which have different dimensions.

Visiting the land of topology, where one shape morphs into another; starting with figures such as the Möbius strip; the Alexander horned sphere and the Klein bottle; studying the minimum number of colors needed to distinguish their different regions.

Here, consider the complexities of topological surfaces, including the iconic and non-orientable Möbius strip.

Many puzzles are optimization problems in disguise; nature often reveals shortcuts to the solutions; how light, bubbles, balloons and other phenomena provide hints to these conundrums; closing with the surprising answer to the Kakeya needle problem.

Proving that some sets cannot be measured; this is crucial to understanding the Banach-Tarski paradox, the strangest theorem in mathematics; asking why mathematicians want to measure sets; learning how to construct a non-measurable set.

The Banach-Tarski paradox shows that one can take a solid ball, split it into five pieces and reassemble them into two complete balls the same size as the original; Professor Kung explains the mathematics behind this result.

Asking the big questions about puzzles and paradoxes; why people are obsessed with them; the mental dismay that comes from contemplating a paradox; why one expends so much effort trying to solve conundrums; there is a method to this madness.