Mind-Bending Math: Riddles and Paradoxes

Rated 5 out of 5 by from Loved this course for its mental stimulation Wonderful professor! The course content was very interesting and engaging.
Date published: 2020-09-22
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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Mind-Bending Math: Riddles and Paradoxes
Course Trailer
Everything in This Lecture Is False
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 seems 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!...

33 min
Elementary Math Isn't Elementary
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 gas mileage to the Dow Jones Industrial Average-can be deceptive....

28 min
Probability Paradoxes
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. Also consider how subtle changes of wording-"my elder child is a girl" vs. "one of my children is a girl"-change...

31 min
Strangeness in Statistics
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 even medical students and doctors can be misled....

31 min
Zeno's Paradoxes of Motion
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....

30 min
Infinity Is Not a Number
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....

29 min
More Than One Infinity
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....

31 min
Cantor's Infinity of Infinities
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 ¼? 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!"...

33 min
Impossible Sets
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....

29 min
Godel Proves the Unprovable
10: Godel 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....

30 min
Voting Paradoxes
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....

30 min
Why No Distribution Is Fully Fair
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....

29 min
Games with Strange Loops
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....

32 min
Losing to Win, Strategizing to Survive
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....

29 min
Enigmas of Everyday Objects
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): how to float a cruise ship in a gallon of water....

30 min
Surprises of the Small and Speedy
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....

33 min
Bending Space and Time
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....

30 min
Filling the Gap between Dimensions
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....

32 min
Crazy Kinds of Connectedness
19: Crazy Kinds of Connectedness

Visit the land of topology, where one shape morphs into another by stretching, pushing, pulling, and deforming-no cutting or gluing allowed. Start simply, with figures such as the Möbius strip and the torus. Then get truly strange with the Klein bottle and the Alexander horned sphere. Study the minimum number of colors needed to distinguish their different spaces....

31 min
Twisted Topological Universes
20: Twisted Topological Universes

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. Watch as Professor Kung plays catch with himself in a 3-torus, and twists his way through a quarter-turn manifold!...

31 min
More with Less, Something for Nothing
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....

29 min
When Measurement Is Impossible
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....

34 min
Banach-Tarski's 1 = 1 + 1
23: Banach-Tarski's 1 = 1 + 1

The Banach-Tarski paradox shows that you can take a solid ball, split it into six pieces, reassemble three of them into a complete ball the same size as the original, and reassemble the other three into another complete ball, also the same size as the original. Professor Kung explains the mathematics behind this astonishing result....

33 min
The Paradox of Paradoxes
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!...

32 min
David Kung

I've loved both math and music since I was a kid. I was thrilled to discover the many connections between these two passions of mine. Sharing that excitement with Great Courses customers has been incredibly gratifying.


University of Wisconsin


St. Mary’s College of Maryland

About David Kung

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 undergraduate and graduate degrees in mathematics, he performed with the Madison Symphony Orchestra. Professor Kung's academic work focuses on mathematics education. Deeply concerned with providing equal opportunities for all math students, he has led efforts to establish Emerging Scholars Programs at institutions across the country. His numerous teaching awards include the Homer L. Dodge Award for Excellence in Teaching by Junior Faculty, given by St. Mary's College, and the John M. Smith Teaching Award, given by the Maryland-District of Columbia-Virginia Section of the Mathematical Association of America. Professor Kung's innovative classes, including Mathematics for Social Justice and Math, Music, and the Mind, have helped establish St. Mary's as one of the preeminent liberal arts programs in mathematics. In addition to his academic pursuits, Professor Kung continues to be an active musician, playing chamber music with students and serving as the concertmaster of his community orchestra.

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