Hearted Youtube comments on Mathologer (@Mathologer) channel.

  1. Challenges: 1) We can guarantee at least 5 on one side. There are two ways to split the pigeons: 3-2 and 4-1. The max you can hit for both of them is 3 and 4 respectively. So we have at least 3 pigeons on one side. Add in the two pigeons on the equator and we have 5. 2) let the mystery value equal x. by multiplying by 10^some integer until the digits line up again, and then subtracting, we have 10^whatever*x - x = the non-repeating part you're left with. Factor out x on the left and that 10^whatever - 1 becomes a bumch of 9s. Then divide and we have the non-repeating remainder over a bunch of 9s. If ever the remainder isn't an integer just multiply both the numerator and denominator by 10 to knock out the decimal. Example: 0.4318181818... x = 0.4318181818... 100x = 43.18181818.... 99x = 42.75 (all the 18s afterward cancel out) x = 42.75/99 x = 4275/9900 x = 19/44 (simplify) 3) let's pretend we color the "didn't shake hands" white. We leave the "shook hands" black. So all we have to do is prove a triangle exists somewhere, and it doesn't matter what color. Note that each dot will always have at least three lines of the same color connecting to it, for the same reason the pigeons in challenge 1 can be split into a side with at least 3 and a side with less. Look at a point, and follow three lines of the same color (I'll choose black) to three other points. Any line connecting two of these three points would either be black or white. If it were black, that would make a triangle with the two points and the starting point. If it weren't, then the three dots all have white lines between them, which makes another different triangle. 4) 315. And no, I didn't count manually. Here's how I found out: The 7 edges UB, UR, RB, DB, DL, LB, and UL, all commute in one big cycle of length 7. The remaining 5 edges do the same. The 5 corners URF, ULF, DRF, DLF, and DRB go back in their positions but in the wrong orientation. So it takes 3 cycles to get back to their original orientation. The same thing happens with the other 3 corners in the back, only this time they take 9 cycles because they permute around each other too as well as reorient themselves. All that's left is to compute the lcm of 7, 5, 3, and 9, which turns out to be 315. 5) 1, 2, 4, 8, 16, 32, and 64. Any collection just results in the binary representation of the number. Since every number can be written in binary in only one way, every single collection will add to a different number. 6) Queen of Hearts. The 9 of Hearts at the start signals the suit of the missing card is Hearts. The remaining three kings can be sorted as MBT (Diamond, Club, Spade), which is assigned 3. 3 spots after 9 is the Queen, and putting it all together we have Queen of Hearts. Edit: I forgot one, challenge 5 at chapter 6
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  2. the chemist version of the joke is very "basic"..pun intended.
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  3. Would be a very natural thing to do but sadly hardly anybody ever gets to know about this beautiful topic (until now of course :)
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  4. My father was a carpenter. He built various buildings, and the last one was a cottage where I helped. We checked that the corner was at right angle using the 3-4-5 measurement.
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  5. Never commented here before... Burkard, you seem like the coolest person! Loved every one of your videos that I have watched. I wish I had the internet when I was a kid. Learning math with you as a kid would have been so much simpler and so much more fun. Thanks for everything! You rock!
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  6. Thanks Burkard for putting together such a nice presentation and filling in so many connections. It has been a load of fun working with you and Henry on this.
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  7. 4:45 "On the right, what do we have. Oh, well, whatever that is, it should be infinity. HMMM" "On the left, we are dividing by zero. Double HMMMM" DEAD
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  8. Always found it insanely impressive when ancient mathematicians used normal language and text instead of our modern formalism to transfer ideas and results. Imagine how hard it must be to do mathematics that way.
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  9. I already had a python interpreter open, so I did Bernoulli's sum in about 20 seconds! 😝
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  10. Those "complete the sequence" questions are my pet peave. The thing is, any number can continue any sequence, and there will be a formula (a polynomial; actually, infinitely many polynomials) to produce the resulting new sequence. That type of question is routinely used in school tests and intelligence tests, but what it really tests for is a kind of learned bias toward small integers.
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  11. When the rubber band snaps, do you get complex numbers? Maybe surreal! Great stuff as usual
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  12. Regarding the tree puzzle at 15:50: The bottom square's area is 1. The Pythagorean theorem states the two squares attached it share the same total area, so they are each 1/2. The total area so far is then 2, with 1 contributed from the big square and 1/2 + 1/2 = 1 contributed from the small squares. The next level down, the tinier squares attached to one of the small squares has to add up to 1/2, so they are each 1/4. There are 4 of them, so the total area of these squares is 1, and the total area is now 3. You continue down the line, adding 1 to the total area for each iteration of the tree. There are 5 iterations, so the total area is 5.
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  13. 6:28 - The reason taxes come out first isn't some kind of greed on the part of government. It's because only paying the taxes first ensures that nobody will go bankrupt to their friends as a means of avoiding the payment of taxes to the government. If my assets are only just enough to cover my back taxes, I can call my friend and say "Do you remember that money I borrowed from you 20 years ago? We kinda forgot about it, but I think it's time to formalize it with a document because I'm about to go bankrupt". The friend comes over and I sign the note, which is for 99 times the amount of my assets. If taxes are not given priority, then when I go bankrupt the government gets its penny on the dollar in proportion to all debts, while my friend gets 99 cents on the dollar. When the dust settles, my friend simply gives me back that 99% of my assets. I end up losing 1% to back taxes instead of 100%. But if everyone knows that taxes come first, then everyone sees the fruitlessness of such schemes and they won't be attempted. Of course, I can run this same scheme on any NON-governmental entity to whom my debt equals my assets, and get 99% of everything back while the debt to that creditor is marked "paid in full". That creditor will go to court and challenge the validity of a promissory note signed one day before a bankruptcy-filing, but the government, by taking taxes first, doesn't have to go through that hassle.
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  14. Wow. 50 Minutes of Mathloger. It is like Christmas already!
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  15. Madhava, Aryabhatta, Brahmagupta, Baraha- the list of such geniuses goes on and on. This talk was so beautiful it brought me immense joy. A flowers fragrance benefits all - no problem. The problem is the same people not recognizing, others walking nearby not recognizing. It is good to see western and other intellectuals finally recognizing. We must recognize where it is due - like the agricultural genius from South America, Math and philosophy from India, material science from China.
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  16. So elegant. At 19:17, I understood where this proof is going, that is the happiest moment of your video when I understand where the proof is going 😃
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  17. 20:00 Number of tilings of the Arctic Circle: 2^(-1/12). Got it. 😏
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  18. Oh, that's what an empty lunch box means. I just thought my mom was a little absent-minded. But this makes so much more sense.
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  19. Am i really watching a 29 min video at 3am about the best way to tie your shoe laces ? YES.
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  20. Related to the coin rolling paradox: Despite there being 365.25 days in a year, the earth actually rotates 366.25 times on its own axis in a year.
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  21. I'll have to watch 3 times. It always takes me at least 3 times because I constantly want to pause, and go explore something that was said...😂
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  22. so this is a kind of duality between two different triangles, neat
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  23. One ring for each integer above 1. And one ring to rule them all.
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  24. What's better than an "AHA moment"? Over 40 minutes of endless "AHA moment"s, of course! Loved the video, make more, Burkard! (And preferably faster :D )
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  25. Here's the thing. I just got home and am extremely tired and definitely not ready to get my brain working on math, or anything demanding that is. So, I got hooked in the first few seconds and my brain started working again. Your skills, sir, are of another world.
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  26. -He says something about rings -I think it will be another Mathologer analogy or something -He is actually talking about damn rings from abstract Algebra -I get excited
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  27. As we’re all home-schooling at the moment, the school is sending challenge sheets out to all the kids. The final question on my daughter’s maths one was “how many different ways can you make £3.10 using coins”. She’s 7 years old... I just changed the question to “think of a few ways you could” instead, because even I didn’t know how to solve it. But now I do! I don’t think I’ll be teaching this to her yet though.
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  28. 24:22 For the 3,4,5 triangle the line connecting the incenter and the Feuerbach point is parallel to the shortest side. Thus, the parent triangle of the 3,4,5 triangle is the degenerate 0,1,1 triangle (and its clear why this construction cannot be taken further).
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  29. A few years ago I used to binge watch this channel. You might imagine my surprise when I started my science degree (undergraduate biochemistry where maths is thankfully a necessary subject to take) and walked into a maths lecture and was confronted by this wonderful person in the flesh. That was about 4 or 5 years ago. Keep up the saintly work you do!
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  30. There's a mistake at the end. In Brahmagupta's Formula the third bracket should be (A+C+D-B) not (A+B+D-B). But great video. That are some beautiful equations and you explained all of it really nicely with all these brilliant animations. Love it!
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  31. 22:21 cos 120 = -1? wtf?
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  32. The big flaw of the Talmud method is that it disincentivises big loans (no one wants to be the biggest creditor), and can be gamed by splitting up big loans into smaller ones. In the extreme case, it converges to proportional splitting.
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  33. The most memorable thing is how ugly the optimal leaning tower is
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  34. Archimedes' claw, seriously? What a shame not to name it Archimedes' pumpkin 🥲 Or Archimedes' Kabocha to be even more accurate.
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  35. Completely understood the math behind it, but I would never be able to come up with that binary analogy on the spot if you asked me to do the same.
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  36. Ramanujan may have been The Man Who Knew Infinity, but Mathologer is the Man Who Made Infinity Long Videos About Them :)
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  37. I'll be honest, I don't know why, but the second problem of integration, about there not being elementary antiderivatives of all elementary functions, just crushed me psychologically when I was learning calculus. I know there are ways around the problem, and I memorized them enough to do decently well in calc, but somehow the trial and error nature of it just lost some of the luster off something I previously quite enjoyed, and turned me off of pursuing any higher forms of calculus. To this day I can handle most high school math up to that point with only minimal references, but all the various methods of integration slipped away from me like water once the class was over.
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  38. Wow just the first 2 minutes of introduction is just amazing, what a great mind Ramanujan had, and beautifully explained professor!
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  39. Dear Mathologer, Seeing your video this morning has brightened my day so incredibly much. Your videos allow me to transcend my body(have pain) and live in a world of pure mathematics. Please never stop. - Your fan and student.
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  40. His shirt doesn't have a 27
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  41. “But of course close doesn’t win the game in carnivals or mathematics” Analysts: “Allow us to introduce ourselves” ???: “Amateurs” Analysts: “What did you say?!” Numerical analyst: “AMATEURS”
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  42. When I got 4th rank in Kerala for the Madhava Mathematics competition, I realised that I had the potential to be a mathematician. Thank you for letting more people know about him!
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  43. A Parker heptagon!
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  44. Twenty minutes later I felt the power of Matrices.
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  45. Wow, I've watched this video four times over the past 1+ year and each time I understand the cubic formula better, seriously! Especially since I learnt about complex numbers last year only. Amazing video, one of my favourites on all of YouTube!
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  46. It is good to see that mathloger is back online...
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  47. The most suprising part for me was the "terrible aim" the fact that odd/even is never an integer is so simple yet i would have never thought about it
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  48. It seems algebra, geometry, modular arithmetic, and fractals are all the same thing wearing different hats.
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  49. Speaking only for myself I appreciate when you explain the same problem in three or more ways. I feel like I gain a little bit of understanding with each iteration.
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  50. When you said he solved this instantly I couldn’t help but feel small
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