Mind-Bending Math: Riddles and Paradoxes

Course No. 1466
Professor David Kung, Ph.D.
St. Mary's College of Maryland
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Course No. 1466
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Course Overview

Great math riddles and paradoxes have a long and illustrious history, serving as both tests and games for intellectual thinkers across the globe. Passed through the halls of academia and examined in-depth by scholars, students, and amateurs alike, these riddles and paradoxes have brought frustration and joy to those seeking intellectual challenges.

In addition, it’s well known that brain exercises are as fundamental to staying sharp as body exercises are to staying fit. Stretching your mind to try to solve a good puzzle, even when the answer eludes you, can help improve your ability to focus.

Now, in the 24 lectures of Mind-Bending Math: Riddles and Paradoxes, you’ll explore the ageless riddles that have plagued even our greatest thinkers in history—confounding the philosophical, mathematical, and scientific minds grappling to solve them. You’ll learn how to break down, examine, and solve these famous quandaries.

From Ancient Greek philosophers to noodling through an unusual enigma involving spaghetti, you’ll cover a wide range of amazing—and in some cases history-changing—conundrums, such as:

  • Zeno’s astonishing argument that motion itself is impossible
  • The compelling conundrum of infinity, which didn’t garner a resolution until the 1900s
  • Gödel’s strange loop, ascertaining no axiom system would work to prove mathematical theorems
  • The Banach–Tarski paradox, proving that one can cut up a ball and reassemble the pieces into two balls, each the same size as the original
  • More mind-bending math games that have endured through time, including the Liar’s Paradox, the Prisoner’s Problem, and the Monty Hall Problem

When it comes to delving into topics such as bending space and time, and topological universes, you need a knowledgeable and captivating instructor, which you get in abundance with Professor of Mathematics David Kung. He infuses each lesson with fun tangents, stories, and real-life riddles, making this one of the most intriguing and entertaining math courses available.

This mesmerizing course will have you contemplating everything from the enthralling paradox of paradoxes to the potential pitfalls when it comes to buying apples—using basic logic and math principles as the fundamental connector to solve exciting, mind-bending mysteries.

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24 lectures
 |  Average 30 minutes each
  • 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. Then "prove" that 1+1=1, and visit the Island of Knights and Knaves, where only the logically minded survive! x
  • 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 averages - from mileage to the Dow Jones Industrial Average - can be deceptive. x
  • 3
    Probability Paradoxes
    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. x
  • 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, social science, and especially medicine, where it can lead to inappropriate treatments. x
  • 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 implications, which have gained new relevance through the discoveries of modern physics. x
  • 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 already occupied. Also sample a selection of supertasks, which are exercises with an infinite number of steps that are completed in finite time. x
  • 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 to prove that the set of real numbers is infinitely larger than the set of natural numbers, which is also infinite. x
  • 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? Probe this and other mind-bending facts about infinite sets, including the discovery that made Cantor exclaim, "I see it, but I don't believe it!" x
  • 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 axioms, designed to avoid all paradoxes and keep mathematics consistent - a goal that was partly met by the Zermelo-Fraenkel set theory. x
  • 10
    Gödel Proves the Unprovable
    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. x
  • 11
    Voting Paradoxes
    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 impossibility theorem, and assess the U.S. Electoral College system, which is especially prone to counterintuitive results. x
  • 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 different methods and the origin of the approach used now. As with voting, discover that no perfect system exists. x
  • 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 everyone has blue eyes, and try your luck at prisoner's dilemma. x
  • 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 winning strategy. Finally, try increasingly more challenging hat games, using the axiom of choice from set theory to perform a miracle. x
  • 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 everyday objects do strange things. Also discussed (but not demonstrated) is how to float a cruise ship in a gallon of water. x
  • 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 realm to explore the curious nature of light and the true meaning of the Heisenberg uncertainty principle. x
  • 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 to connect an array of dots with the fewest lines, and test yourself with map challenges. Also learn how to ride a bicycle with square wheels. x
  • 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 into the fractal world of Sierpinski's triangle, which has 1.58 dimensions, and the Menger sponge, which has 2.73 dimensions. x
  • 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 and torus. Then get truly strange with the Alexander horned sphere and Klein bottle. Study the minimum number of colors needed to distinguish their different regions. x
  • 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 other paradoxes, and learn how to determine the shape of the planet on which you live; after all, it could be a cube or a torus! x
  • 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. Close with the surprising answer to the Kakeya needle problem to determine the space required to turn a needle completely around. x
  • 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 23. Start by asking why mathematicians want to measure sets. Then learn how to construct a non-measurable set. x
  • 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, 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. x
  • 24
    The Paradox of 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? Why do we expend so much effort trying to solve conundrums and riddles? Professor Kung shows that there's method to this madness! x

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Your professor

David Kung

About Your Professor

David Kung, Ph.D.
St. Mary's College of Maryland
Dr. David Kung is Professor of Mathematics at St. Mary's College of Maryland. He earned his B.A. in Mathematics and Physics and his Ph.D. in Mathematics from the University of Wisconsin, Madison. Professor Kung's musical education began at an early age with violin lessons. As he progressed, he studied with one of the pioneers of the Suzuki method and attended the prestigious Interlochen music camp. While completing his...
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Reviews

Mind-Bending Math: Riddles and Paradoxes is rated 4.6 out of 5 by 31.
Rated 5 out of 5 by from Very entertaining This could easily be described as a mind-expanding course, given by an enthusiastic and knowledgeable teacher. It was great fun, and I look forward to going through it all again, not least for its entertainment value
Date published: 2018-11-01
Rated 5 out of 5 by from Puzzled growth explained, as civilized advances Mind- Bending Math details much of the frontiers of current mathmetical exploration. The incredibly insightful exploration is done via paradoxes and "conondrums", the intellectual itches that needed to be insightfully presented and scratched. It is postulated that perhaps that's how we grow as a civilization. Presenting the new stuff along with the issues that prompted exploration is incredibly beneficial. I have not seen a better overview of the state of current findings and problems, with the best follow-up references; and I've been looking. I view this offering as a great sequel to "Zero to Infinity: History of Numbers" work by Berger from The Great Courses, which presents a great history of mathematics up to the mid 1900's. Kung presents the bleeding edge beyond, with just enough history to help substantiate the issues, calling in Zeno, Euclid, Pythagoras, especially Euler, and others. What should be addressed is that if one has never heard of (at least some of) Euclid, Zeno, Archimedes, the Pythagorean group, Fibonacci, Euler, Hilbert, Cantor, Godel, Russell, Einstein, Mandelbrot, Arrow, and so many others; and if the ideas of calculus, probability and statistics, various infinities, fractal dimensions, relativity, Mobius strips, Klein bottles, voting and democratic representation paradoxes, seem just that bridge too far; please consider that perhaps you should spend time and money getting that education first, because this material is way heavy. Don't be fooled by the simplistic "Mind-Bending Math" title, which has led some other reviewers to think that this should be soothing pablum for the mathematically challenged. Kung's material often works very much like Richard Gardners', in that it seems like it should be an easy read, then all the sudden you have to read it twice or more, then have no clue what to think, then "a-ha!". A question to be addressed is whether computer science academics and practitioners like me, will benefit. I have always found it fun to take the time to try and understand the history and currents in mathematical thought. Having been a part of both mathematics and computer science departments, my observation is that both strongly benefit from interaction. One recommendation to Prof Kung is that the paradoxes and problems of automata, algorithms, concurrency, AI, and the ambiguities and insecurities specific to programming languages await. I am especially impressed by a subliminal intellectual message that Kung's offering leaves. I was once advised by an acting department head, as an undergraduate, that looking into computer languages and algorithms was not a good idea, because the job offerings were non- existant (yes, decades ago), and that any new challenges beyond what mankind currently knows was minimal. The counter impression that Kung, and Burger, leave behind is that intellectually we are at the dawn of intelligence, perpetually. Yes, indeed!
Date published: 2018-10-24
Rated 5 out of 5 by from Interesting span of topics! I thoroughly enjoyed each presentation. Dr. David Krug is pleasant, engaging, very knowledgable and speaks directly to the student [camera.]
Date published: 2018-09-13
Rated 5 out of 5 by from Paradoxes Dr. Kung I was stunned by Dr. Kung's presentation. I have math courses by Drs. Starbird, Benjamin, and Burger. I like all of them. I know have Dr. Kung. I would definitely by another one of his. I did also buy his music and math course.
Date published: 2018-04-08
Rated 5 out of 5 by from Highlights of an Undergraduate Math Degree! This was a terrific course covering many of the interesting topics covered in an undergraduate math major focusing on logic, measure theory and topology. Professor Kung was lively, and did a really good job of explaining the material without the need for lots of heavy machinery. (Often in math courses you make work on technical stuff for a few weeks just to be able to state and prove some of these interesting results. ) He also did quite a few interesting physics experiments (and a few physics paradoxes) that challenged our intuition. I can't recommend this class enough! P.S. I am a mathematician by training but have not done any math for over 20 years, so the material came back to me quickly. This class does require some level of math sophistication, but not a lot of actual background is required. Just be smart and curious. P.P.S. I wish he had covered some of the results in Galois theory. I think he deserved a 36 lecture course.
Date published: 2017-10-03
Rated 5 out of 5 by from I wish I had Professor Kung in graduate school. Each lecture gives an understandable overview of subjects and disciplines that I believe every student perusing degrees in Mathematics should have. For everyone else, the lectures create a delightful insight into the wonderful world of Mathematics.
Date published: 2017-08-24
Rated 5 out of 5 by from A very fun course I just finished all the lectures of this course and thoroughly enjoyed it. I also enjoyed the lectures in the course How Music and Math Relate, by the same professor.
Date published: 2017-07-10
Rated 5 out of 5 by from Good Analogies Make Complex Math Understandable Professor David Kung makes complex mathematics understandable using a variety of analogies and examples. I wish that I had this video when I was majoring in math. This video should be made available to all high school and college students, regardless of major, as math is essential to understanding all subjects. What is great about The Great Courses formats is that you can study them when you have time, that you can redo classes, that you can take classes in any sequence, that classes are available offline and online, that video/audio/text are available, and that the courses on sale prices are affordable. I have courses downloaded on my Windows PC and Android phone and tablet for convenience of use at places and times of my choosing.
Date published: 2017-05-30
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