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Mind-Bending Math: Riddles and Paradoxes

Want to stretch your brain in surprising ways? If you like to think, solve riddles, and exercise your brain, this captivating course examining logic-based brain teasers is for you.
Mind-Bending Math: Riddles and Paradoxes is rated 4.4 out of 5 by 55.
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Rated 5 out of 5 by from Makes math fun! The Professor is EXCELLENT and makes math fun! Really enjoyed the puzzles, etc. Please make more courses like this.
Date published: 2024-04-16
Rated 5 out of 5 by from A review and a thought of finite infinity. This course is an excellent brain teaser, thought igniter, maybe you will start seeing math with an amplified vision, maybe nut just math, even life! Please allow me also to share the following regarding infinity, ---- can you imagine an infinity that dies, an infinity that has an age! It goes infinetely on and on, and doesn't end, because it is infinite, but only for a certain amount of time. So it is only infinite as long is it is alive! There is also an infinity that is longer/more infinite than other infinities. if 2 inifinities are born togethor, and they keep moving with the same speed, and die at the same time, they are equal. If one is faster than it is longer. If one lives longer, and it is speed is higher, it is even more long and so on. Some infinities change form, the infinities keep going infinites until they can't in their current form, so they continue in a different form. like an infinite that can only exist inside another infinity, when infinity vessel end, then the inside infinity would exist infinitly without a form, It will live again once it is given another form. So we have different types of infinities, of all sorts as diverse as life it self. If you a vehicle to travel to the end of the univers, you will never reach to the edge, because it keeps expanding infinitely, but expanding where?! it is the universe itself, is there a space that is not part of it to expand itslef into it? so is the universe's infinity wraped inside another that is bigger and room for it is expansion? Does that wrapping infinity has end this expansion will Stop once it reaches a certain age, or are everlasting wrapping infinites, each keep expanding inside of one another infinitely?! isn't the nature of numbers is the same as nature of life, don't they match?! the life line, and the number line for example, isn't the number line an infinite representation of life+ and life- from zero life? the infinity of life and death, where everything one of us is within! This infinity I believe is created, and the creator is affected by it. We are all limited to it, the finite infinity! Hey those where just some thoughts pass, as thoughts infinitely keep passing by as long as there are living minds to cross!
Date published: 2023-01-28
Rated 5 out of 5 by from Fascinating This is one of the very best food-for-thought courses in The Teaching Company’s repertoire. Especially insightful were Lectures #8 on Georg Cantor’s findings, #11 on voting, #12 on apportionment, and #18 to #20 on geometry and topology. Lectures #22 on measure and #23 on the Banach-Tarski paradox are ‘heavy’ and we will have to repeat our viewing to fully comprehend. Professor Kung does a marvelous job of presenting this intriguing material. Time well spent. HWF & ISF, Mesa AZ.
Date published: 2022-12-27
Rated 4 out of 5 by from Excellent instructor / Complex concepts This course is truly mind-bending, full of experiments that are fun to observe and puzzles and contradictions that are anything but obvious. Professor Kung is a fine teacher and demonstrates great joy in presenting bizarre, often contradictory, and complex mathematical proofs and concepts. His lessons are well thought out and well illustrated with props, graphs, and animations to help you visualize the subject. Though I enjoy math, puzzle solving, and logical thinking, I found the latter parts of this course concerning topology and measurement difficult to understand. They relied on advanced math that was beyond my level of study. Though interesting, I couldn't quite wrap my brain around those concepts. No doubt the more mathematical theory you have under your belt, the greater your appreciation of the course will be.
Date published: 2022-07-04
Rated 2 out of 5 by from Too complicated As a scientist I have a reasonable understanding of math. While a few of the lectures were interesting, most were way to complicated for me and the information went way over my head.
Date published: 2022-06-17
Rated 4 out of 5 by from Good start, looking forward to more. I’m just 3 lessons in but so far this course has definitely kept my interest. Really makes you think about how easy it is to manipulate “facts”, twist numbers and how you can shape statistics.
Date published: 2021-09-23
Rated 4 out of 5 by from Entertaining if not always easy to follow I recognized a lot of these puzzles and concepts from other courses but he presented them in an entertaining way. Some of the later puzzles were hard to follow the concepts and math but I got the general idea.
Date published: 2021-03-15
Rated 1 out of 5 by from Stuck with it through two lectures Stuck with it through two lectures, not sure where this course was going.
Date published: 2021-02-05
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Discover the timeless riddles and paradoxes that have confounded the greatest philosophical, mathematical, and scientific minds in history. Stretching your mind to try to solve a puzzle, even when the answer eludes you, can help sharpen your mind and focus-and it's an intellectual thrill! Mind-Bending Math will have you contemplating everything from the enthralling paradox of paradoxes to the mysteries of infinity.


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.


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 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.

By This Professor

How Music and Mathematics Relate
Mind-Bending Math: Riddles and Paradoxes
Mind-Bending Math: Riddles and Paradoxes


Everything in This Lecture Is False

01: 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

02: 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

03: 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

04: 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

05: 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

06: 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

07: 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

08: 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

09: 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