Hearted Youtube comments on Mathologer (@Mathologer) channel.

  1. A fun proof for the 4x4 case: make the colors prime numbers 2, 3, 5 and 7. The product of all numbers in the grid is (2*3*5*7)^4. Then divide by the numbers in the 2x2 squares between the corners (forming a cross that overcounts the middle). This gives 1. Since we divided by the middle 2x2 square one time too much, we can multiply by the middle square again to compensate, which gives 2*3*5*7. The number we are left with is the product of the numbers in the 4x4 grid, when you take away the middle cross, i.e. the corners. Since we got 2*3*5*7 for that number, the corners must be 2, 3, 5 and 7.
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  2. One ring to rule them all
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  3. The manuscript in the end was really enlightening. It makes you wonder how much past (and possibly present) mathematicians were (/are) influenced in their thinking by the writing conventions of their time.
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  4. Ramanujan has no formal education,he taught himself and made himself a genius
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  5. Pascal's triangle is inaccurately named as it only has two sides!
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  6. I believe a person in the Swiss Patent Office published a number of papers in 1905 that garnered some attention and may have elbowed out other interesting papers.
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  7. I said the first pattern should continue with 31. I didn't expect you to add the "evenly spaced" criterion.
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  8. It's neat how this feels like a sort of "hacky" way of approaching calculus, if that makes sense. Colleges should teach this imo. Just from this 40 min video, I could see some of these concepts plugging into all sorts of annoying classwork problems we've done in calc and differentials
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  9. There is another sequence that goes 2, 4, 8, 20, infinity: the number of faces of a regular polyhedron made of triangles. The first one is a degenerate case with 2 triangles sharing all vertices, then tetrahedron, octahedron, icosahedron, and finally an infinite tiling when you attempt to meet 6 triangles at each vertex.
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  10. I really love how there's subtitles for every video since I'm still learning English Thanks for the great content
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  11. Great video <3
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  12. They should have called it the 'Barely Divergent Series'
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  13. "A conspicuously simple and universal pattern is more likely a feature of the observer's perspective than the universe being observed." ... seems a more profound lesson than anything one could wring from an obsession over Tesla circles. Thanks!
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  14. Im a math failure. I'm here for your t-shirts 😅
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  15. The power of 2 coins can make any amount in exactly one way! (binary representations :))
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  16. Earlier I wondered: "When is mathologer posting his next video?" Couldn't have gotten a better answer :)
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  17. Pro tip: if you are manipulating equations without regard for units, you are doing mathematics. If instead you DO consider units, you are probably doing physics, engineering, or some other useful thing ;-)
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  18. Thank God someone's finally talking about these things! I stumbled upon one of these Tesla 3 6 9 videos ages ago, and I knew that everything they said definitely wasn't magic or anything and probably had a simple explanation in number theory, but I could never put it into words myself. I'm so glad someone finally put together an understandable and informative response to those things.
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  19. The weirdest thing you showed is definitely the unusual optimal brick stacking pattern.
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  20. I am from Austria and we never learned that the number, which remains actually is the remainder (9:45). When I learned about modular arithmetic in math Olympiad, I guessed that fact to be true while doing an example. Not even my highly invested teacher was sure, whether the solution was right. Infuriating, that you do not learn these deeper truths about mathematics at school.
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  21. This is very much nit-picking, but clickbaity tiles like "Why was X lost/forgotten/ignored/missed for Y years?" does the channel a disservice. Like question in headlines, this makes it look like poor journalism (despite it being the opposite). Furthermore, the interesting thing that the video focusses on is the maths itself, not the sociology around it, which makes the title somewhat inaccurate. Again, not a major issue, but such great educational content deserves better than a gutter-level, click bait title - no matter what a YouTube brand consultant might say.
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  22. "Mathematicians, artists, wizards, and a lot of crazies." So... just crazies, then?
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  23. My dad was a property assessor, and had many well made straight and circular slide rules. My grandpa was a brick mason, but never used one. I showed him how they work, and he thought it was interesting. It was in the mid-70's. I then showed my grandpa my six-place logarithm table. It's an Austrian book, from 1924 by Dr. Ludwig Schrön. The logarithms are to six figures, and the column/row lookup goes to 5 (but you can use proportions between answers to get a sixth). I showed him how to turn multiplication into addition, and division into subtraction, and how to back-reference to find the answer. He was quite amazed by the whole process. I gave him my table for a while as he was quite fascinated and spent time using it on many different occasions.
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  24. The number of math-related shirts Burkard owns, is somewhere between 12 and Graham's Number. Burkard, do you ever throw out old shirts?
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  25. Don't let other YouTubers doing the same topics stop you. Please! I like to see different takes on the same topic from my favorite YouTubers.
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  26. I liked the color proof better, because the swivel proof seemed like it relied more on visuals, which could be deceiving.
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  27. The fact that amyone on earth can connect to the web and have their mind expanded in this manner, for free, is wonderful. I am very grateful for this and other maths content on Youtube. Thank you!
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  28. This channel and 3Blue1Brown have fantastic visuals that are very helpful. Both also have excellent, clear presentation. No other mathematics channel I've found come close to your skill in communication, which obviously involves a great deal of work to create all the graphis.
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  29. Hey, Mathologer is back after a short interruption of service. Nice to see you again!
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  30. Really cool that you mentioned Think Twice. He's such an underrated youtuber, you really, really should check him out!
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  31. I have to say (and I am hard to impress ) this is one of the best high level explanations I have seen. The amount of work involved in putting this to air both at the animation level and the level of explanation are truly top drawer. But as at least one other viewer has pointed out, don't burn yourself out on this stuff at the expense of your own research.
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  32. That image of Madhava seriously looks like you. I could not believe it wasn't an AI generated image of you from 15th century India.
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  33. My answer to the question on why people focus on calculus is because calculus is indeed a simple solution and thinking with calculus is way helpful for cases where the "neat" solutions don't work
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  34. Most memorable: The harmonic series misses all integers up to infinity
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  35. I never never ever thought that I might see as strong visual proof for this amazing formula as this guy's👏🏻👏🏻👏🏻 this is so 100% enough
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  36. Continued fractions are truly amazing and, for most people, mysterious mathematical objects. This is really a pity, because just as the natural base for logarithms is e rather than 10, and the natural measure of an angle is radians rather than degrees, the most natural representation of a real number, in a sense, is a continued fraction.
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  37. Another fun but elementary observation: the sines of the "nice" angles 0⁰,30⁰,45⁰,60⁰,90⁰ are √0/4, √1/4, √2/4, √3/4, √4/4. I could never remember what their values were until I noticed that!
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  38. Archimedes video is in the pipeline :)
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  39. Finally, a mathematical proof that devils and angels are equally evil
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  40. The most memorable part for me might be the idea of that gamma value: especially the super quick visual proof that it had to be less than one by sliding over all the blue regions to the left
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  41. I liked the binnary solution because not only is it "elegant", it also shows intuitively why the number always gets smalller.
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  42. I think it's easy to get distracted by the fact that there is a matching negative for every positive term in the sequence. A similar paradox makes it more intuitive what's wrong with rearranging terms. ∞ = 1 + 1 + 1 +... ∞ = (2 - 1) + (2 - 1) + (2 - 1) +... ∞ = (1 + 1 - 1) + (1 + 1 - 1) +... ∞ = 1 + 1 - 1 + 1 + 1 - 1 +... Then we can pull out positive and negative terms. 1 + 1 + 1... - 1 - 1 - 1... So every +1 is canceled by a - 1. You can even create a mapping from the nth positive 1 to the n*2 negative term, so every positive term has a negative to cancel it. This, to me, intuitively shows why you can't add infinite sums by rearranging terms. You need to look at how it grows as you add terms.
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  43. Math: '*exists* Euler: "First!"
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  44. How odd...
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  45. For me, the "no integers" part was the most memorable, but honestly the whole video was of great quality (as expected).
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  46. Apparently the names of the hyperbolic functions are pronounced differently in different places! I had no idea. Perhaps this is a US vs. UK thing? In my schooling in the US, I have never heard anything other than COSH for cosh(x) and SINCH (homophonous with cinch) for sinh(x). The other functions I've heard more variation. For tanh(x) some people say TANJ like 'tangerine' and some TANCH. The most difficult one by far to incorporate, I think, is csch(x) wherein you can try to say COSEECH or instead bail out to '1 over SINCH.'
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  47. Solution to the problem at 16:14 Start with this equation, which is true for every value of k: 5^k * 2^k = 10^k The digital root of any power of 10 is 1, so DR(5^k * 2^k) = 1 Using the multiplication rule you explained earlier, DR(DR(5^k) * DR(2^k)) = 1 In other words, DR(5^k) and DR(2^k) have to be multiplicative inverses of each other. Taking the “digital root” of an integer is equivalent to modding it by 9. (The only difference is that if DR(n) = 9, then n mod 9 = 0.) In mod-9 arithmetic, every number except for 0, 3, and 6 has a unique multiplicative inverse. Since the digital root of a power of 2 is never 3, 6, or 9, this means that DR(2^k) completely determines DR(5^k). As k increases, the value of DR(2^k) cycles as follows: 2 4 8 7 5 1 2 4 8 7 5 1 … Taking the multiplicative inverse of each number above gives the values of DR(5^k). 5 7 8 4 2 1 5 7 8 4 2 1 … So DR(2^k) and DR(5^k) cycle through the same values, but in reverse.
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  48. I liked your evil mathematician back story, with the teacher refusing to grade the "wrong" proof.
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  49. This is a great length of a Mathologer video, nothing wrong with this! Thanks
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  50. Most memorable: being invited to take a moment and post why it might be obvious that gamma is greater than 0.5 and then doing it. Hmm... why is it obvious that gamma is greater than 0.5? Well it didn't seem obvious... But imagine the blue bits were triangular; then there would be equal parts blue and white in the unit square on the left i.e. a gamma of 0.5. But the blue parts are convex, they each take up more than half of their rectangles and together take up more than half of the square.
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