Mind-Bending Math: Riddles and Paradoxes

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
Rated 5 out of 5 by from It’s fun to watch and fun doing the problems. It’s opening my mind about other things as well.
Date published: 2018-12-15
Rated 3 out of 5 by from Too abstract some interesting, but not useful stuff. For example, some other math courses have tricks that you can share with others. This course does not. I didn't finish it, not even close.
Date published: 2018-11-25
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
Rated 5 out of 5 by from it ends... that is the biggest flaw. In addition to other positive comments; I would add that the inclusion of topology only adds to the value of the course, challenging the mind of participant and allowing him to recognize the value of basic science. The issue of riddles in learning is gaining scientific community interest, as some of the recent publications can attest, for example: "Guide to Teaching Puzzle-based Learning" Springer 2014. I have a question to Professor Kung - Is his lecture included within the number PI, I know my birthday is -- when you search it within the billion known digits.
Date published: 2017-03-01
Rated 5 out of 5 by from Pure delight! Professor Kung's instructional style is creative, sometimes zany, sometimes serious and detailed. He makes great use of visual techniques - sometimes physical props, sometimes digital effects, and a regular Mr. Science-style clip of a related concept. His topics vary widely in complexity and difficulty. I'm nearly at the end of this package and am dreading running out! His style is so informal that it should appeal to people with curiosity who may have widely varying experience with the math concepts that he is treating us to.
Date published: 2017-02-28
Rated 4 out of 5 by from An average course. I was SO excited to get (and watch) this course, but I found it to be disappointing. That MAY be MY fault, because I expected it to be so much better. It's probably not a bad course, but I expected much more.
Date published: 2016-12-04
Rated 5 out of 5 by from Excellent presenter Professor Kung does a great job of presenting the material, explaining the background of the paradoxes and riddles, and then giving the participant extra thought problems to work on. In this course I found that, after listening to each lecture, reviewing the material in the course guidebook and then working on the chapter problems helped me to understand some pretty esoteric material.
Date published: 2016-12-04
Rated 5 out of 5 by from Another TGC homerun I was fortunate to get this great stores from my wife for Christmas. I loved it. Every course left me with something interesting to think about such as "what does it mean that there are multiple infinites?", "how can there be no set of math axioms to explain all maths?", and "how do fractional dimensions work? " The instructor is great and, if you like to challenge your conceptions, this series is outstanding.
Date published: 2016-10-09
Rated 4 out of 5 by from Well presented, interesting if math is your thing This review is actually a difficult one for me to write, so I'll get the good part out of the way first. Professor Kung is simply outstanding, one of the very best TGC instructors. I very much enjoyed his previous course on how mathematics and music relate and was looking forward to this course. His presentation style on this course was, if anything, even better than the previous course. He made extensive use of innovative video techniques and actual demonstrations in getting his points across. He has a very natural and pleasant manner and presentation style. Furthermore, of all the TCG courses that I have viewed, he is the only one that I can recall who at the end of the course mentions the contributions of others in putting together the course. I would give him more than 5 stars if I could. Now for the not-so-good part. In this course, Professor Kung presents a wide range of riddles and paradoxes from many different fields, ranging from very practical (and timely) discussions of systems of voting and allocating representation in a legislative body (i.e. US House of Representatives) to abstract mathematical concepts such as the existence of multiple infinities. By far the most incomprehensible to me was the next to last lecture, where he discusses in detail how a single body can be broken into two bodies each having the same size as the original (1 = 1 + 1). Frankly, this did not make any sense to me. I cannot understand how our intellectual existence can benefit by the "proof" of something that is physically impossible. Maybe it's my faulty intuition speaking, but I can't see the benefit of this. On the positive side, I did enjoy his coverage of several phenomena in physics that I was either unaware of or else had never seen explained so clearly. For instance, even though I worked in the field of applied hydrodynamics for almost 50 years, I had never heard of the phenomenon of "dead water", where a ship or craft's speed is reduced relative to normal conditions even though no waves are visible on the surface. This is caused by the existence of layers in the "dead water", due to temperature inversion or different water densities. The ship generates waves in the interface between the layers, which are not apparent to anyone on the surface but which cause the ship to slow down. I also had a few minor quibbles about the quality of the DVDs. I noticed a few brief instances of the audio dropping out and also some static in the audio. Not enough to return the DVDs, but unusual for the standards that TGC usually maintains. In short, I really am reluctant to criticize such a well presented course from an obviously brilliant instructor who is also a decent human being. However, I have to be honest that parts of this course were challenging. If you are really into math and puzzles you will probably be able to handle all this. Otherwise, prepare to have your intellect and common sense significantly challenged.
Date published: 2016-10-01
Rated 5 out of 5 by from Wow! I understand this stuff!! I never was good at math in school. But somehow, I was able to understand this material. Very well selected and presented. Further, the production values are excellent. The Teaching Company is really investing in improving their product. This is a good course for those of us who were never good at math, but want something to whet our appetites for more.
Date published: 2016-08-09
Rated 5 out of 5 by from I enjoyed the course I really enjoyed this course. He is able to take what I consider to be some challenging math concepts and can explain them very clearly. He has an easy-going lecture style and has a way of helping you understand the problem first (often in a concrete way) before explaining the underlying concepts. This is the second course by Dr. Kung that I have listened to, and I look forward to more.
Date published: 2016-07-25
Rated 5 out of 5 by from Fun and Deep The professor does a fabulous job of presenting significant math paradoxes. He is entertaining and fun, and I must say, far more intelligent than anyone you are likely to meet in real life. How someone with his intelligence can also be entertaining, personable, and fun is also a mind bending paradox. He delves into Godel's proof, so if you don't know what that is about, your mind will at least be bent, and you may not understand much of what he presents, though he presents if very well. It is mind bending difficult material. A most fascinating example: Suppose you are on a game show, and you are offered two envelopes to choose from. One has x dollars, and the other has 2x dollars. You choose an envelope, and the game show host asks if you would like to change your mind and choose the other envelope. Probability theory suggest that on average, you should choose the other envelope, because you will get a 1.25 x improvement in expected returns. But this can't be true. Apparently this is a current paradox, with many theories offered that attempt to resolve it, but is not yet solved. Amazing how many paradoxes exist that involve everyday life.
Date published: 2016-07-21
Rated 3 out of 5 by from A puzzling course, perhaps not for me While the professor speaks well and enthusiastically, and the course is well supported by graphics, the material in this course did not hold my interest. Maybe it was just the wrong course for me.
Date published: 2016-05-16
Rated 5 out of 5 by from Mind Bending Math I am a 74 year old retired physician who did an undergraduate Math major at the University of Michigan 55 years ago. Recreational Math, and Martin Gardner, have been my lifelong hobbies. So I was glad to see the Mind Bending Math course offering from The Great Courses., and I subscribed immediately. I have finished all 24 lectures, listening to many sessions more than once (or twice). You have exceeded my expectations in both the material chosen and Professor Kung's presentation and have refreshed my ways of thinking about Math and Life. Now I have to get to work on the problems you included in the course book! Thanks for your “Labor of Love” which is next on my list.
Date published: 2016-03-09
Rated 1 out of 5 by from Slow-paced, dull. In 2016, expectations on video quality and content are really high. There's nothing specifically wrong with the video quality, and perhaps my expectations were too high. After all this is marketed as a lecture; but what's the point of the video medium if it offers no more content than a printed book? I found the pace tediously slow. The "this sentence is false" paradox is presented in umpteen different forms without getting into discussion about its relevance. I'd got it the first time - the repetition was utterly boring and unnecessary, and lost me as an audience. Isn't repetition what pause and rewind is for? Perhaps I was too knowledgeable for this lecture - but my family also found this lecture completely un-inspiring and boring. My recommendation to anyone interested in this topic is to read Gödel Escher and Bach instead. Though I accept that the lecture series covers some additional topics too. The experience of this lecture means that I am unlikely to purchase any more "Great Courses" from this company.
Date published: 2016-02-07
Rated 4 out of 5 by from Mind Bending Math About half way through this. It's definitely though-provoking, and I really like the professor. Being a math geek, I like his take on things so far.
Date published: 2016-01-20
Rated 5 out of 5 by from Mind-Bending Math: Riddles and Paradoxes Professor Kung's presentation is very creative and his passion for the subject matter is evident. The course was fun to watch and I found that I was compelled to share his puzzles and paradoxes with friends, family and co-workers. He provides many references to additional sources for even more puzzles which I was also inspired to take advantage of. Not all of the material was new for me, but the presentation was always fresh and entertaining. All of the new material for me was presented in a way that was clear and easy to understand. The conundrums appearing in roughly the middle of each lecture provided a good mental break from the lecture format and provided additional clarity and insight to each topic.
Date published: 2015-11-09
Rated 5 out of 5 by from For The Intellectually Curious... I can remember clearly, even as a kid in high school and eventually throughout college, I was (and always have been) profoundly attracted to the concepts that exist on the fringe of current human understanding. To this day I feel that there is value in pondering the deep questions even though I know that We will most likely not have "the answers" in my lifetime. To me, the journey is nearly always as interesting and rewarding as the destination turns out to be. For all of the reasons listed above -- and many more -- I strongly encourage you to consider picking up a copy of this lecture series. This course does the best job I've seen of wading in the waters of the curious, deeply esoteric and universally beautiful concepts that lurk in mathematics, though usually revealed in sufficient detail only to those with years of training in the rigors of mathematical thinking. The other reviewers have done a good job describing content so I largely defer to them for this. I just want to add that this is easily the best coverage of the concept of countable vs. uncountable infinities that I've seen outside of university. There was one item I wanted to address from a previous review; a comment made by a fellow reviewer below mentions that this course would be best suited for fans of puzzle creators like Martin Gardner. I wanted to comment that I personally do not think that this course is only for people who enjoy mathematical puzzles a la Gardner -- indeed, I'm not a big fan of Gardner myself -- but that doesn't prevent me from putting this course in my top 5 TTC/TGC courses to date. What I suspect may be true (though I can't say for sure) is that someone who is already a fan of Gardner's work would most likely also enjoy this lecture series as well. I thought it possible that the Gardner comment may give the wrong impression of this course. While some topics may skew a bit 'Gardnery', the material overall felt to me more like a conversation between an amateur mathematician and an amateur philosopher than it did a math-puzzle session. Also I should add that I greatly enjoyed the lecturer for (and his presentation of) this course -- he clearly enjoys the topics and for me that came across quite clearly and definitely made an impression on me. I got the video download edition of this lecture series so I can see how something might be lacking in a purely audio context (more in terms of connecting with the speaker and seeing his facial expressions telegraph his joy and sense of playfulness and humor -- maybe the word is pleasure? -- than in terms of any graphics or visualizations that you might miss (though seeing some of the proofs did help me substantially)). I think this course is for anyone who is deeply curious about ideas like infinity, self-reference or our faulty mathematical intuition. Anyone who wishes they went a little deeper into the ideas of mathematics without having the time for all that is usually prerequisite for understanding these topics. I wish this course had been around when I was in college.
Date published: 2015-10-04
Rated 5 out of 5 by from A terrific course I thought this was a excellent course. One of the best parts was on self reference, one would have thought that mathematics could not talk about itself but Kurt Godel found a way to do it and Professor Kung explains in a very clear and concise way how he did it. Even better Godel got mathematics to say "I can not be proven" and he explains how Godel did that too and the revolutionary implications that come from that fact. Professor Kung also corrected some misconceptions about the nature of infinity, particularly uncountable infinity, that I've had for a long time. I highly recommend this Great Course .
Date published: 2015-09-10
Rated 5 out of 5 by from This course will teach you how to think! This course is so much more than a collection of puzzles and brain teasers. Professor Kung uses these items to show his viewers how to correct poor intuition and to develop sound reasoning. The breadth of topics in the course is quite vast. It starts off with a few basic, well-known paradoxes. Then, as the viewer gets more comfortable with their ability to reason correctly, Professor Kung shows how this “unnatural" way of thinking has led to ground-breaking results in statistics, political science, economics, physics, and abstract mathematics, just to name a few... Our minds do an incredible job of processing massive amounts of information, as well as making decisions based on that information. But in order to do this effectively, sometimes we take a few shortcuts. We make generalizations by comparing new information with old knowledge that seems "similar.” Most of the time this strategy is quite effective, but sometimes it can get us into trouble. “Mind-Bending Math” sheds light on this dilemma, and it teaches us not to dismiss certain ideas just because they “feel wrong.” Perhaps more importantly, it teaches us not to accept faulty reasoning just because it “feels right." I have a degree in mathematics, so I feel qualified to say that the content of this series is accessible to virtually anyone. Professor Kung builds everything from the ground up. If there is something you need to know, he explains it to you. Keep in mind that this is a survey course. So those looking to come out the other end having mastered Einstein’s theory of relativity should curb their expectations. But for those of us (myself included) who like to dig a little deeper, Professor Kung offers his personal favorites for advanced works in each topic. I have to say that these recommendations are probably my favorite aspect of the entire course. It keeps everything moving along swiftly, but it also gives you a direction to go in if you find a topic that you want to study more in-depth.
Date published: 2015-08-24
Rated 2 out of 5 by from For Math Puzzle Fans Only If you love math puzzles (i.e., Martin Gardner is one of your favorite people), you may well appreciate this course. If not, I suggest you pass it by. The topics are certainly fascinating - everything from Zeno's paradox and Cantor's infinities through Gödel and fractals to topology and Banach-Tarski. The problem, for me, is the way they are presented. The level of difficulty ranges widely, from just plain obvious at the start to an assumption that we are familiar with the basics of calculus by the end. (I very much appreciate TGC's occasional offerings of more challenging math and science courses! But the level of a course should be internally consistent.) And, with the exception of several topics towards the end, the discussions are superficial and left me wishing for more completeness and depth. This may well be appropriate for a course covering many unrelated riddles and paradoxes, but I did not find it satisfying. (FWIW, I am a non-mathematician who loves learning about math.) Most importantly, I found Professor Kung difficult to listen to. He is clearly knowledgeable and enthusiastic about his subject. But he speaks in a hesitant, choppy style, and I often found his explanations unclear or incomplete. At the time I am writing this there are no reviews posted, and I regret - if this is the first - that it is negative. Again, if you are a math puzzle fan, do consider the course; you may love it! But I cannot recommend it to anyone else.
Date published: 2015-08-19
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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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