Mind-Bending Math: Riddles and Paradoxes

Course No. 1466
Professor David Kung, Ph.D.
St. Mary's College of Maryland
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4.6 out of 5
32 Reviews
81% of reviewers would recommend this product
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 32.
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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