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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  • Puzzles and solutions
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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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Mind-Bending Math: Riddles and Paradoxes is rated 4.5 out of 5 by 44.
Rated 5 out of 5 by from Excellent Professor Kung has made clear some difficult mathematic problems. Through school and college, there would be a reference to these questions but in my mind they were never answered. For example Zeno's paradox, and the dog running between the boy and slower girl all at different speeds. It was great to finally learn the answer about the dog as to where and what direction it was facing. The metapuzzles were also interesting and required expanded thinking to figure out those. The order of the presentations brought viewers to begin to get an idea of the Banach-Tarski theorem and partly understand what was going on with the solution, though some parts that were not explained would have been a stretch to try to understand the mathematics behind that part of the proof. Recently there was an article in Scientific American as to whether math was an abstract idea or real. For me it is like the wave-particle duality. It can be either depending on the point of view. Excellent, though will need to watch again. Love the tie to ideas in calculus.
Date published: 2020-11-20
Rated 5 out of 5 by from Fairly advanced course Creative choice of topics and proofs. Enthusiastic presentation.
Date published: 2020-09-21
Rated 4 out of 5 by from Worth watching I love the enthusiasm of Prof. Kung. He clearly put a lot of work into this course. I think it delivers, as promised, a thorough collection of riddles and paradoxes, and illustrates the complexity of our number systems and our world. I did not know about Banach-Tarski, so I really appreciated this topic. Overall, I think it was worth my time.
Date published: 2020-06-17
Rated 5 out of 5 by from The title piques your interest The material is an enjoyable treatment of historical as well as present-day paradoxes. The part on Goedel's work and advanced logic in general is worth the price of admission. I haven't completed the course as yet but have already gotten my money's worth.
Date published: 2020-04-27
Rated 4 out of 5 by from Counterintuitive Ideas Excellent explanation of complex concepts but diagrams coordinated with the explanation are not very good.
Date published: 2020-04-26
Rated 5 out of 5 by from Mind bending math concepts and paradoxes I'm an engineer and had engineering oriented math courses that emphasized getting it done and not much about possible difficulties; and I was good at that. In graduate school I took some "real" math courses and found out how inadequate my math education and knowledge was. In particular, I took a third year level undergraduate course in real function theory (Rudin text and Royden text and many others over time) that introduced many of the concepts and paradoxes that Prof Kung presents and discusses wonderfully. I struggled with them then but eventually, after many years and exposures, began to appreciate and understand them. These lectures reinforces and expands my understanding. I commend Prof Kung for wonderful lectures that I hope soon to complete. I want to point out that I have taken these lectures through through GreatCoursesPlus and waited to purchase the DVDs when the price came down and free shipping.
Date published: 2019-09-03
Rated 1 out of 5 by from Title is misleading. Very few riddles or puzzles I expected more in the way of riddles and puzzles and the first few sessions have been more of a math lecture
Date published: 2019-08-27
Rated 4 out of 5 by from Graphics (and replay) really helped I enjoyed Professor Kung's "Mind-bending Math: Riddles and Paradoxes" video course. I can't remember when I used the 15 second replay button so often. The graphics were a great help, but can only go so far in depicting concepts such as 4-dimensional topology. However, if the goal of the course is to improve critical thinking by enabling people to question long held perceptions and imagine the world in a new way, it is well worth the time and attention to watch. I'd already had a general familiarity with Riemannian and Lobachevskian geometries that deal with the negative and positive curvature of space as opposed to the linear universe of Euclid, but never visualized them in quite so complete a way as Professor Kung was able to present. Professor Kung is certainly a very gifted educator, talented speaker, and expert mathematician. I admit I may not have found some of the "riddles and paradoxes" quite as entertaining and / or mind-blowing as he did, but I certainly enjoyed and appreciated his enthusiasm which kept me engaged in topics that otherwise could have seemed far less interesting. Early in the course I approached each lecture with trepidation, often nervous that I'd be unable to follow it, get lost in some abstract quagmire, and end up confused, but by about lecture 8 I had confidence that wouldn't be the case. Some of the material is certainly challenging, and at times it was difficult to gauge which audience level it was designed for, going from simple intuitive concepts to more advanced calculus. But any confusion was rather quickly remedied by Professor Kung's further explanation, sense of humor, and amusing experiments. I regard all science as an attempt to understand the natural world, and numbers are one form of symbolic notation. But when concepts lead to conclusions that are mathematically provable while being physically impossible, I don't find myself all that amazed: although I appreciate the elegance of mathematical equations, I rather lose interest in what occasionally seems like a system that tends to become so self-referential it seems flawed in its inability to translate solutions into helping people or making the world better. The guidebook was helpful, only has a handful of typographical errors, and contains additional problems and solutions for each lecture (although for lecture 11, problem 1, part d, I got a different Borda count for C and D). A glossary may have been helpful: I sketched my own chart to clarify the relationships of natural numbers, whole numbers, integers, rational numbers, and irrational numbers in the real number universe, as these distinctions seemed crucial to grasping the concept of countable and uncountable sets with regard to the qualities of being finite or infinite.
Date published: 2019-02-20
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