Hearted Youtube comments on Mathologer (@Mathologer) channel.

  1. Honey, wake up, new Mathologer video just dropped!!
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  2. Why can't you start the pattern a^3+b^3=c^3+d^3 or a^3+b^3+c^3=d^3+e^3 or similar?
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  3. Just open Youtube to take a break from Algorithms exercises and sees this(
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  4. Finally, one of my favorite theorems! I want to leave you an exercise about this theorem that made me appreciate it even more: Let p be a prime such that p is either equal to 2 or 3 modulo 5. Prove that the sequence n!+n^p-n+5 has at most a finite number of squares
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  5. Always love your videos, a great sunday afternoon
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  6. The video so nice I liked it twice.
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  7. 24:30 For a while, I struggled with figuring out, why the ”>”-sign can’t become a ”<”-sign, straight away, as the quadrilateral becomes flat. But, I’ve finally got it: The transition between 3D and 2D is continuous; and, as such, the difference (Aa+Bb)-Cc must either stay positive (corresponding to the ”Strictly greater, than” -case), or, on its way to negative (corresponding to ”Aa+Bb < Cc”), it must pass through 0 (Aa+Bb = Cc), as you indicated, in your video on the ”Turning the wobbly table” -theorem (in a slightly different context). But; because, in a continuous transition, there is only 1 instant, where the quadrilateral is flat (and, therefore, the ”strictly greater than” -inequality doesn’t necessarily hold), the difference is always either positive or 0; and therefore, for all quadrilaterals, Aa+Bb >/= Cc. Very satisfying, I must say. 😌
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  8. Who else started thinking about how you can create a dictionary of 52C5 cards such that you can give 4 cards to someone and they can predict the last card. Of course, the last thing you would have to do is fit that dictionary in marty's head. Sorry marty!!!!!!!!!!
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  9. First! :D
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  10. The necklace made me think of the jewel-division problem from an earlier video. I'm now imagining a band of burglar-mathematicians trying to divide a necklace into equal-value pythagorean triples.
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  11. Thank you sir for this explanation
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  12. Seems like an E6B paper flight computer, to me....
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  13. Big fan of Heron's formula here, so I may go on a bit. Here in the US, New York State, under the "Regents" math curriculum, in the late 1960's, we did learn Heron's formula. We were not required to memorize the derivation, but it was in the textbook. Decades later, I decided to test my algebraic chops, and try to derive it with the only clue I remember from the book, that it relied on factoring the difference of two squares. If you drop an altitude on one of the sides, you can solve for the altitude and one of the unknown segments on the base simultaneously, using the Pythagorean Theorem. Mathologer takes a shortcut by using the law of cosines, but my method is how you get the law of cosines as well (and you can get rid of the fraction in the Mathologer's version with some convenient cancellations). But once I was there, and I got the formula knowing what I was looking for, I thought of four justifications for "discovering" Heron's formula instead of calling it a day having a formula for the area in terms of the sides: 1) it lacks symmetry. There is nothing special about any side, other than that you chose one to drop an altitude on; 2) it's pretty nasty to calculate from. Many of us have probably calculated nastier ones, but we can do better; 3) it is badly scaled. You end up raising numbers to the fourth power, which usually results in something large, then subtracting them, leading to truncation or round-off errors if you don't keep a lot of decimal places (I realize Heron wasn't thinking about floating point calculations. Heck, he didn't even have a decimal system) 4) it's not obvious from the original formula, at least to me, that your area isn't going to turn out to be the square root of a negative number. If you expand the trinomial and collect terms, you do get something symmetrical, but all the other objections remain. That might tell you that, since expanding didn't work out very well, maybe the opposite--factoring--is the solution. In addition to factoring the difference of two squares, twice, at one point you have to collect some terms and recognize it as the square of a binomial. It's all quite pretty. But it's 100%, algebra, none of the geometric insight of this video. With the final formula, in addition to being much prettier and easier to calculate from (if the need arises which, truthfully, it rarely does) you can see at once that for a legitimate triangle, no one side being longer than the sum of the other two (or equivalently, no side being longer than the semi-perimeter) you'll never get a negative number and moreover, if a "triangle"s one side is exactly equal to the sum of the other two it has zero area, as expected.
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  14. IT'S MATHOLOGER TIME!!! runs around the room crazily
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  15. In truth, that is exactly the only rule that i use to solve any classic Tower of Hanoy puzzle. I only have to care about the "sub-stack" of disks where the smallest one is at the time and if it is even or odd. In between i just do the moves that are unique, avoiding any "repetitions".
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  16. only in exactly one case, the equilateral, the triangle is the same as it's folded counterpart.
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  17. I think the answer to the problem on the board is 1 and 364/365. I was adding 100, 121, and 144 when I realized that makes 365, so the problem is just 1 + (169 + 196)/365 Edit: Apparently it's 2. I guess I forgot to completely add a number.
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  18. I really really love this type of video, please continue Master Class videos :D
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  19. That panel at the end...a reminder how easily momentous things can be overlooked, by anyone not understanding what they're holding.
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  20. Glad to hear it! Much love to you and yours!
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  21. Coin paradox blew my mind. Great video, great visual and explanations.
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  22. I have always wondered who could watch this and give it a thumbs down?
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  23. 2019 is the last 4 digit on the second line of the christmas tree stem thing on your clothes
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  24. I must have Stockholm Syndrome. I didn't mind being kidnapped at all. :)
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  25. Loved this Video! Always wondered whether there was any research on the addition of odd numbers to get squares. I knew how to get the cubes, but was pleasantly surprised to know that the pattern continues! Made my day, (the week actually). Loved the fact that some of the approaches in the video share a big similarity to the ones in your power sum video, grids and various others (And pascal's triangle!). Also, how you can use geometry to not only visualize, but also solve what is, essentially, a number theory and algebra problem. Gently makes you appreciate how interconnected everything in mathematics is. I hope you can keep making these videos, and that they reach an even larger audience, so everyone can fully enjoy and learn advanced mathematics. PS : Is your geometric proof the first one? PPS : I think, the output sequence might be n!*(n-1)! ? if a =1 and b, c, ... =2?, Since then each term would become n + 2(n(n-1)/2) = n^2.
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  26. I was supposed to study english for an exam tomorrow, but this is just way too fascinating
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  27. Thank you, for your videos. "fresh" ideas and content and ton of work behind them.
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  28. Wow, I love this mix of algebra, calculus and clever manipulations ♥️ For me the most beautiful part of the derivation was 1/1/2/3/4/...=sqrt(pi/2) Thank you for your work 👌
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  29. fixed 👍
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  30. you kept using the wrong infinite sum from 17:13 to the end
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  31. Please make your vids in dark mode
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  32. How are we able to draw equilateral triangle on a paper when its simply an infinite grid with small spacing
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  33. As always, I really appreciated the history bits that one comes to expect from this awesome channel :) But I especially liked the closing bit about ideal mathematical objects vs reality. Seems to me that often, a lot of otherwise great mathematical content tend to get a bit lost in idealism, and lose sight of that difference. It's a small detail, but it makes a big difference. Thank you for all the work that goes into these videos!
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  34. Beat me to it
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  35. Burkard, many thanks for your very informative videos on mathematics. I am a retired professional engineer and it is a pleasure to look back on the areas of mathematics I had to learn for my engineering degree, and you make it so much more interesting and enjoyable as a subject. Kind Regards Tony
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  36. I've always been intrigued by this paradox, calculus-involved approach or not. The visual aesthetics of your videos are really sleek as well. Thank you Mathologer for posting such interesting content in such approachable and easy-to-grasp ways :)
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  37. 여기서 한국 드라마를 보게 될 줄은 꿈에도 몰랐네요. Never expected to see K-drama in this channel!!
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  38. If there is 33*33 tiles The number of sum must be ((33*33) * (33*33 + 1) / 2) (gaussian sum) Suppose every row must sum to the same number. That's mean ((33*33) * (33*33 + 1) / 2) should be 33*n Each row then must be ((33*33) * (33*33 + 1) / 2) / 33 or 33 * (33*33 + 1) / 2 which is 17985 Generally (n * (n^2 + 1)) / 2
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  39. I now have a panic attack whenever I see a du/dx written somewhere. Goddamn integrals
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  40. My favorite channel!
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  41. just watched mathvengers , so this is like a sequel to your post credit scene
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  42. Galileo was one of the first to notice the "part equals the whole" weirdness with infinities. I find it interesting to contemplate how this then plays out in another puzzle he wrestled with - the shape of cables or ropes under gravity. (He got it wrong- he guessed that they were parabolic.)
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  43. Before people knew the analytic solution to the shape of hanging cables under gravity, the architects would build models upside down, with hanging strings and fabric, and then study the shape to specify how to build arches and domes right-side-up. They recognized that since a string hangs with a purely axial load, and so should an arch be built in a shape so it also carries a purely compressive axial load.
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  44. first time I got confused on these videos, heh warning about baby calculus wasn't strong enough to include baby differential equations as well I wasn't ready
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  45. Yes!! I was thinking of linoleum floor patterns myself.😊👍
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  46. 15:08 please make a complicated video about 4d stuff :] might not be for everyone but i´m sure many people will appreciate it!
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  47. I created a Geogebra application on Petr-Douglas-Neumann myself a year ago (see at Geogebra ... /classic/sdftbe2y). It has a slider to switch from triangle (n=3) to decagons (n=10). And there is a play button to run through the 47 steps. (If you choose a smaller n, then note that some of the 47 steps are hidden because they are meant for the larger n's.) I choose a different but equivalent approach to Petr, Douglas and Neumann. I prefer using regular n-gons instead of the ears. Imho, this is (mathematically) more elegant. Instead of connecting the tips of the ears, you have to connect the centers of the added regular n-gons. And instead of using ears with different angles each round, I add the regular n-gons where each round has a different distribution of vertices that get to each side of the edge. One round has all (n-2) vertices on one side, another round has (n-3) vertices on one side and 1 vertex on the other side, etc. You have to see the example in the app to understand what I mean.
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  48. A solution like this was probably inconceivable before expressions like "liquidate" and "financial liquidity" came into common use.
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  49. That visual proof that e^a = cosh(a) + sinh(a) is really satisfying! I've always wondered how the exponential related to hyperbolas, i.e. why we call them hyperbolic trig functions. I'd seen the construction in terms of hyperbolas, and I'd seen the representation in terms of the exponential, but whenever I tried to find an explanation how these two were connected, all I got was, "That's just the definition." (Read: I don't know, and I've never thought about it.)
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  50. Another fun fact you missed about the Letters. „Æ“ is the sound you make when you hear about the paradox for the first time and „O“ is the sound you make after someone explained it
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