How did Ramanujan solve the STRAND puzzle?
ฝัง
- เผยแพร่เมื่อ 20 พ.ค. 2024
- Today's video is about making sense of an infinite fraction that pops up in an anecdote about the mathematical genius Srinivasa Ramanujan.
00:00 Intro
04:31 Chapter 1: Getting a feel for the puzzle
08:27 Chapter 2: Algebra autopilot
12:37 Chapter 3: Infinite fraction
17:51 Chapter 4: Root 2
21:19 Chapter 5: Euclidean algorithm
30:15 Chapter 6: The best of the best: 17/12
36:34 Chapter 7: Outramanujing Ramanujan
This was supposed to be a short video but in the end turned out to be quite a tricky to sort out. Anyway, as it sometimes happens, I got carried away and now the video really covers a lot of ground : Pell equations, visualising continued fractions by dissecting rectangles into squares, the relationship between continued fractions and the Euclidean algorithm, the irrationality of root 2. Overall quite a few things that you won't find anywhere else :)
The way I tell the anecdote in this video is based on the following account by Ramanujan's friend Prasanta Mahalanobis: Current Science, Vol. 9 (3), pp. 74-75.
"On another occasion, I went to his room to have lunch with him. The First World War had started some time ago. I had in my hand a copy of the monthly Strand Magazine which at that time used to publish a number of puzzles to be solved by the readers. Ramanujan was stirring something in a pan over the fire for our lunch. I was sitting near the table, turning over the pages of the Strand Magazine. I got interested in a problem involving a relation between two numbers. I have forgotten the details but I remember the type of the problem. Two British officers had been billeted in Paris in two different houses in a long street; the two numbers of these houses were related in a special way; the problem was to find out the two numbers. It was not at all difficult; I got the solution in a few minutes by trial and error. In a joking way, I told Ramanujan, 'Now here is a problem for you'. He said, 'What problem, tell me', and went on stirring the pan. I read out the question from the Strand Magazine. He promptly answered 'Please take down the solution' and dictated a continued fraction. The first term was the solution which I had obtained. Each successive term represented successive solutions for the same type of relation between two numbers, as the number of houses in the street would increase indefinitely. I was amazed and I asked him how he got the solution in a flash. He said, 'Immediately I heard the problem it was clear that the solution should obviously be a continued fraction; I then thought, which continued fraction? And the answer came to my mind. It was just as simple as this.' "
There is a complete digital archive of The Strand magazine. You can find the page with the puzzle here: tinyurl.com/y2lnb8xf (page 790)
If you read the puzzle in the Strand you'll find that the problem is actually phrased somewhat differently to what Mahalanobis remembers and Mahalanobis also does not spell out the infinite fraction that Ramanujan came up with. And if you do the math(s) some of the other things he says also don't quite sound right. What I am presenting in this video is my best guess for what really happened.
In particular, the continued fraction that I am talking about in video is probably the most natural candidate for Ramanujan's infinite fraction, but others have argued that it could have been a different continued fraction (which I don't buy :) You can find these other infinite fractions here: 'Ramanujan's Continued Fraction for a Puzzle" by Poo-Sung Park tinyurl.com/yyfdscgr and here 'On Ramanujan, continued fractions and an interesting street number' by John Butcher tinyurl.com/yy6nv2yg
Solution to the red cross puzzle from Dudeney's book "Amusements in Mathematics" p. 168 :) imgur.com/a/bBuLOZN
Another interesting way to systematically search for solutions to the Strand puzzle is this: The equation we want to solve is 2 x^2=y^2+y. You can rewrite this as x^2 = y(y+1)/2. The formula on the right is just the formula for 1+2+3+...+y. So just keep adding 1+2+3+... and at every step check whether the number you get is a square ... :)
Other short formulas: 1) Expanding (1+√ 2)^n gives a number a+b√2. Then a/b is the nth partial fraction. 2) Play with powers of the matrix {{2, 1}, {1, 0}}
Some number Easter eggs are hiding on this slide • How did Ramanujan solv... :)
Link to the unlisted Marching Squares video: • Root 2 and the deadly ...
Here is a version of the t-shirt I am wearing: tinyurl.com/y5vgo7zb This one is about that other famous Ramanujan anecdote: tinyurl.com/y626c86x actually features prominently in another one of my videos.
The music in this video is by Chris Haugen, Fresh Fallen Snow (playing in the video) and Morning Mandolin (for the credits) and Nate Blaze 'Tis the season, all from the free TH-cam music library
Enjoy!
Burkard
14.9.2021: Thank you very much Michael Didenko for your Russian subtitles.
Ramanujan was such a great mathematician that on 22nd Dec.,as his birthday is celebrated as national mathematics day in India.
It's also near the average date of the Winter Solstice, which logically (to me) should fall on the *first* day of the year, so we should reset the calendar to make that happen. After all, if a Pope in 1582 can do that why not some actual scientists??
Imagine if the calendar consisted of 4 identical quarters of 30, 30 and 31 days each, adding up to 364 days every year plus one extra to get to 365 and another on Leap Years - how easy would that be! Weeks would start on Monday as most of the world already agrees, and the last one or two extras would be inserted between that last Sunday and the first Monday of the following year so that *every year* would look the same as well as being much more culturally neutral than now.
I would think that the scientific and business worlds would love this kind of standardization and predictability even if doesn't appeal to traditionalists.
@@deborahkeesee7412 huh u actually have a point there
@@deborahkeesee7412 French tried this type of the calendar after the revolution but it did not catch. Too much weight on church.
@@deborahkeesee7412 I don't understand why we can't have a 360 day year and just let the stars and seasons slide around.
@@tinfoilhomer909 then there is no point in having a year, why can't a day be equal to 20 hours?
Ramanujan is the classic kid that doesn't listen in class, forgets to take notes, does no homework but then FREAKIN' ACES the test because he found his own way of doing things... He will never cease to amaze me!
Sir Ramanujan was very polite and disciplined. He respected elders and the ethics of a place(school, office, neighbour....)
@@user-cv1jb9xv2p Of course, I meant it as a metaphor, didn't mean to disrespect him. Have a nice day!
I misinterpreted it. The times are wierd now. I staying much on social media, I think that's why it happened.
Stay home, stay safe, eat healthy and do riddles.
I think that's Einstein? Or not.. He just didn't ace the tests.
Ramanujan was just exceptionally good at math, but bad at everything else. His teachers and community recognized it, and had great expectations..
He did *more* homework (his tutors gave him books and materials), took copious notes on his own, etc. So in a way, it's kinda the reverse of our modern day expectations of a brilliant mind.
@Robert Slackware tell me about it, story of my life!
Truly the man who knew infinity.
Everyone: maths is boring :(
Mathsloger : let me take care of it. ;)
Btw your videos are very interesting and full of knowledge...... Love from india 🇮🇳❤❤
>ramanujan answered instantly
>takes 40 minute video to explain how
This dude was insane
Instantly means 5 - 10min
The question wasn't that hard though
Ramanujan solved the first Strand-type puzzle. Very impressive
was looking for this comment
Only if Ramanujan lived longer we would have had mathematicians who would have had their PhDs with him and how much more he would have inspired the next generation. His intuition in mathematics is Insane its God-like.
A large part of his earlier life was to personally rediscover the maths of 2000 years already done by previous generation due to his poor schooling till he arrived at present time .
The beauty of mathematics lies in the way how seemingly unrelated threads interweave to create the fabric of utmost mathematical elegance. And Mathologer
.......you do a great job untangling those threads and making us see and appreciate the beautiful connections lying underneath. Thank you so much.
Yup. The most fascinating results are those that connect seemingly unconnected fields, like Taniyama-Shimura.
Ramanujan was a freaking force. What a beast!
42:40 "To be continued"
I see what you did there.
Let's hope not, 'cause at the depicted progression fifth video from this one would be just under 33 seconds long.
42:37 "Until next time remember, it's okay to be a little crazy"
@@gabor6259 Hey 👋 ma buddy from Science Asylum.
th-cam.com/video/4YGqHJP50h4/w-d-xo.html
@@gabor6259 Nick Lucid
When you said he solved this instantly I couldn’t help but feel small
Don’t feel bad at being small to a titan like Ramanujan.
@@idjles it's true! We are not geniuses but that does not mean our lives are not meaningful! Very insightful there amigo!!!
Cause we are small!! It's all good!
One day I hope to answer your “did you see it?” with an equally enthusiastic “Yes!” 💕🐝
Sometimes I almost see it but not until he says "Did you see it?".
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I saw that coming
Cathi shaner best of luck
Ramanujan was and is great.
You forgot "alwæs will be"
Pratik Sonavane my! What antiquated spellynge!!!
Well, not 'is'. Because he passed away already. RIP.
@@SomeRandomGtaDude-zl3us A big part of Ramanujan's character was his independent and unique approach to mathematical thinking and proofs. There is no guarantee he would have been nurtured into a better mathematician if he had been made to memorise the tricks of the field like an average student. He might have lost his knack of finding clever and tricky insights out of thin air. Also it is a disservice to Gauss.
@@SomeRandomGtaDude-zl3us he would never have shine today like how he is remember today. Education would have wasted his time and would have train him in particular direction. And not have found numerous ways of finding the answer
Have you ever wondered why his t-shirt says TAXI 1729?
The number 1729 is known as the Hardy-Ramanujan number after a famous visit by Hardy to see Ramanujan at a hospital.
This number is the number of the taxi Hardy used to visit him, and Ramanujan looked at the number of the taxi and said 'very interesting'.
The great mathematician Hardy did not understand what Ramanujan was talking about and asked.
Ramanujan, who kept his mind busy with only numbers, said that 1729 is the smallest number, which is the sum of the cubes of two positive numbers in two different forms.
Indeed!
Knew this one, but a slightly different story...
@@AmarDamani Please tell it!
@@AmarDamani Yes, there is also a slightly different version of this story.
In Hardy's words:
I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one and that I hoped it was not an unfavorable omen. "No", he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."
Is this the story you're talking about?
The first time I heard the story, I *immediately* blurted out, in reply: "it's also the difference of the squares of two triangular numbers" ... the triangular numbers being (1, 2, 6, 10, ⋯) = (1·2/2, 2·3/2, 3·4/2, 4·5/2, ⋯) ... and in case you weren't paying attention, two of the solutions to the problem for house numbers are 12·17, 29·41.
At the time, I *was* going to say in reply that it was the difference of the squares of two triangular numbers in *two* ways, but stopped short, because the other one is off by one.
This is perhaps the best math video I've seen. Clever, well-explained, and elegant. Keep up the great work. Stay safe amid the Covid.
Rfrfrry3q TV r
Ramanujan was a genius
Can anybody ever say anything again without referring to Covid? I AM safe, and have not felt unsafe a single second ever since this hubris started! So stop it already, will ya? Thanks!
Disagree. There are many math videos out there much better than this.
@@velvetpaws999 Well said.
If I could only have had 30 seconds in Ramanujan's brain
Greetings from Melbourne. For a change I am posting this video at a reasonable time, 8:51 a.m. on a Sunday morning. We are still in lockdown around here, but things appear to be improving: 63 new cases.
There is a very interesting footnote to what I am talking about today contained in the description of this video. Check it out :)
Glad to hear it! Much love to you and yours!
Saturday night in Rockport Texas.
Reasonable time here too.
Love your vid
Mathologer Are in Melbourne. ?? From a Melbournian
@@windrush104 Yes, I teach maths at Monash :)
"...The Euclidean Algorithem, an ancient mathematical superweapon"
perfect description!
Having it demonstrated from 23:36 onwards in the video along with the accompanying soundtrack - a moment of poetic beauty ...
An impossible problem
Thank you from the heart. You have such kindness, generosity, and humor in your lectures. I trained to be a mathematician but realized that I didn't have any real talent, so I became an Engineer ( all three). And tutored math in my free time.
Splendid 21st Century math honoring the great Ramanujan. He would have loved this digital age!!
I feel like Ramanujan was an alien that was sent to Earth to accelerate our knowledge in mathematics, and once he taught everything he could to the human race, he left Earth to teach another underdeveloped civilization.
If you have not read it yet there is this very nice biography of Ramanujan by Roger Kanigel (I found the video pretty much unwatchable :)
You should read up about Indian science and Maths.
1) How Fibonacci was introduced to Indian mathematics
2)How maharishi kannada postulated (kinda) the atomic theory
3) How Schopenhauer had declared, “In the whole world there is no study so beneficial and so elevating as that of the Upanishads. It has been the solace of my life. It will be the solace of my death.”
4) How schrödinger named his dog Atman after getting inspired by Hindu texts..I've got endless stuff to write!
The dude killed himself. Not every intelligent move! 🤯🤯🤯
Well, you said Ramanujan was an Alien! In hinduism, we can say he was avatar of god who came here to teach the mankind! He was a really very brilliant great mathematician and it must been great for the other fellow mathematician who contemporary of Ramanujan. Well, we know that Ramanujan was highly self taught but there are many more examples of scientist that have appeared in the History that are self taught genius! The real ability of Ramanujan that made him brilliant and compared him will god or Alien that was his brilliant ability to play with mathematics!😁
What are you talking about? Ramanujan didn't kill himself.
I was watching one of your videos about infinite fractions and... WOW, NEW VIDEO, THAAAANKS
That reminds me that I should really add some cards linking to those videos :)
I paused and worked out as much of the problem as I could on paper. Then I unpaused and he covered everything I did in 5 seconds. :~(
Your infinite fractions/sum videos have been absolutely amazing. Please don't ever stop making videos, they are super clear and entertaining.
I love your Ramanujan inspired TAXI 1729 T-shirt
Check out this wiki page en.wikipedia.org/wiki/Taxicab_number :)
It was in 'the man who knew infinity'
I am so happy to have access to such great content without any charge. I love mathematics so much and this satiates my curiosity! Looking forward to more of your amazing work ❤️
I love the geometric intuition you continue to provide in your videos. I can't wait to see what you have next with this series.
this is a great presentation. easy to understand and breaks down seemingly mysterious mathematical intuition. thank you!
This video is really something special, the level of insight, beauty, deep concepts and even mathematics history is staggering. Thank you so so much!
Your videos are really calming to the mind. Pleasant music during algebra autopilot and then fascinating math explained in a natural way!
He might not be your typical name, which you could give and everyone would recognize it, such as its with Newton. Despite that however, Ramanujan is a genius, who sadly didn't live for long and would probably be one of the most important mathematicians, should he have lived and published more papers, perhaps even be an advisor to some future mathematicians :)
The notebooks he left is an area of research to this day.
I read Hofstadter’s “Gödel Escher Bach” at age 13, that was the first time I remember him being mentioned.
@@rjwh67220 I ended up becoming a mathematician, so its influence was profound. :)
No, he *was* a genius. He is not alive.
"Not be your typical name"? What do you really mean?
Mathologer and on Ramanujan , absolutely amazing. Highly appreciated work sir.
No words could explain the infinite joy I get while going on this journey with your way of telling this story. Thanks 🙏
29:49
The width of the white rectangle is sqrt(2) - 1 .
Its height is 1 - (sqrt(2) - 1) = 2 - sqrt(2) = sqrt(2) * (sqrt(2) - 1) .
This height divided by the width is:
sqrt(2) * (sqrt(2) - 1)/(sqrt(2) - 1) = sqrt(2) .
Thank you!
@@sanferrera You're welcome.
WTFhappenedWITHyou factor out the √2 from the left side you get the right side
@@WTFhappenedWITHyou 2 = sqrt(2) * sqrt(2), therefore with factorising sqrt(2) *sqrt(2) - sqrt(2) = sqrt(2) * (sqrt(2) - 1)
Mathologer, Just a note to thank you for all your videos, this is great work, so inspiring. Just the right level of compromise between rigour and popularisation to deal with such amazing topics! Thanks! Sebastien
Your way of explaining things is just lovely: pleasant, clear, open to anyone who's curious and knows basic algebra. Great work once again!
I have almost no grasp of basic algebra. I watch these videos in complete aww of the innate problem solving potential of human beings. I feel like learning math is akin to finally being able to leave Plato’s allegory cave, in that math seems like a key to understanding the entire world around us.
Well, pretty sure that the more of this kind of video you watch the more you'll understand :)
Are you interested to study together?
It's sad that Ramanujan did not achieve the Lucasian or Plumian Chair ( Although he would have to of graduated from Cambridge) It would have been nice to see his name on that list of Luminaries.
Sometimes Incredible men are taken before they can make those contributions that would leapfrog our Society ahead.
Possibly because as a group we are not worthy of what they could gift us with.
*had to have
Why on earth do you say that? They were human just like the rest of us..Who says we can't do the same as he? I could never admit I'm not as gifted and smart as Ramanujan and couldn't make as great contributions..why elevate one person unduly?
@@leif1075 Well I can tell that you are neither gifted or smart from your single comment. But obviously pride and ignorance are your forte.
I discovered this only today (October 20th) and thoroughly enjoyed it. Really looking forward to watching some more like this. Many thanks for producing it.
Burkard - having struggled with math all my life, you bring clarity and fresh air to an otherwise somewhat rarefied endeavor. Your efforts are most appreciated. Thank you kind sir!
Your videos are amazing and very amusing! I don’t think I understand all that you present but I enjoy them a lot! These videos are like a brain “oil change” for me. Used to enjoy math when I was in school a century ago. I have gotten rusty now but thanks to videos like yours I can enjoy math again! 👍👌
Always quality content 💪💪 one of the best channels on YT 💕
Absolutely love your presentation style: challenging, yet fun, engaging, and clear explanations!
Your videos are so fascinating! Looking forward to watching your next video!
I feel like a similar version of this problem is in NCERT Ex. 5.4 (Optional) Class X Mathematics. Who else have tried that problem using AP?
Maybe share this problem with the rest of us ?
@@Mathologer The houses of a row are numbered consecutively from 1 to 49. Show that there is value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x.
Ya. I too thought the same...
@@nishadthakur If you already know there are 49 houses, this problem becomes much easier. You can just write and solve a single equation for x, though you do need to know or find a way to sum 1 + 2 + ... + k for any whole number k.
Yeah that's a hot question for the board examinations of the CBSE here in India 😅
If Sir Ramanujan was alive for more 50 years mathmetics could prosper more ,specially number theory.
He’d have solved Riemann and Fermat rather quickly I presume.
Even more identities? 😨
Fantastic. This is the first time I have watched this video and I understand it! ! !
Congratulations Mathologer, I look forward to the two hour extension you promised? ? ?
my apologies for asking an unnecessary question yesterday. I was watching this wonderful lesson on continuous fractions using my cell phone and was unable to navigate to your full explanation which included the credits and titles for the background music. Thanks for helping us readers get really excited and interested in furthering our math education well beyond what we learned in high school.
*_/w magic_*
:)
Yayy two of my favourite maths TH-camrs
@@Mathologer (:
@@shantanunene4389 hehe :3
@@PapaFlammy69 Papa flammy spotted .
Algebra gets very interesting when it's described with geometry. I love it that way and probably Euclid's approach to the problem was derived from geometry as well.
Wonderful video highlighting the genius of Ramanujan and the power of continued fractions.
There’s another nice way to quickly see that the solutions to the Strand puzzle are related to sqrt(2) :) Since the sum of all the house numbers (1 + 2 + 3 + ... + y) is a triangular number, it will approximate the area of a triangle with base = height = y. Then, wherever house x is, it’ll be around however far along the base of the triangle you need to go to cut the area of the triangle into equal areas, and this length is x ≈ y/sqrt(2) :D
To see why this is the case, notice that if you make a vertical cut from some point along the base of the triangle, you will split it into a left area and a right area. The area on the left will just be a scaled down version of the original triangle. Since the areas are split equally, the area of this triangle is half the area of the original triangle, and the way to achieve this is to scale the original down by a factor of sqrt(2) ^^
I really like how you brought this all down to the pictorial version of squares. Then to stop it from going on forever, adjusted it slightly... which in the end allowed the process to work out very well. I am not a math scientist like you, but use it every day in programming, bookkeeping, estimating, and formula... for automation and business. I was able to follow along, and what you did... made perfect sense in the end.
On another note: when I was 17 years old, I took the number 7 to the power of 277, and calculated this by hand. The resulting length of paper ended up from floor to ceiling... or maybe more than 12 pages or so. What became very interesting, is somewhere down the pages... the answers, or the next calculated figure was a pattern. I could write out directly, as it developed a pattern that went right on down. So I could simply just write the number. Why did I go this... I don't know, but it was fun!!! Maybe I'm weird?
Wow. That is a show. An interview.
Me when I see a mathologer video:
"My *excitement* is immeasurable and my day is *made* ."
13:36 - Numerator is equal to 2x the denominator of the previous term plus the numerator of the previous term, denominator is equal to the numerator minus the denominator of the previous term. Maybe not the simplest rule but it’s the first one I saw, by looking at the sequence of partial fractions.
I appreciate the little challenges included in these videos. Not many math TH-cam channels include them. Most of the time I don’t go for them, but whenever I do and find the solution, it’s rewarding 🙂
Another slightly cleaner way to phrase this same pattern is: The denominator is equal to the sum of the previous term's denominator and numerator. The numerator is equal to the sum of the previous term's denominator and the current term's denominator.
Dude, I teach this stuff at the college level and all I can say is you are the most incredible math educator I have ever seen. Hands down. I had heard somewhere (can't remember where) that Gauss used to consider intuition about proofs to be sort of like the ugly scaffolding around a large structures in restoration. Assuming this is true, it explains so much about why math is feared. What you are doing is the antithesis of that perspective and you are totally nailing it. Thank you.
Yu don't refer t this man as"Dude!!"
Red Cross solution:
Align the centre of one of the smaller crosses to the centre the big cross, matching their orientation. Rotate the smaller cross until its vertices touch the edges of the big cross. The 4 segments produced form the 2nd smaller cross.
That's it :)
@@Mathologer I am confused. This solution assumes that the smaller crosses are of the right size. But we don't know the size yet to begin with. So while rotating the smaller cross, what if it keeps freely rotating without touching the bigger cross?
@@therealsachin the area of the large cross should be twice the area of the small crosses, therefore the ratio of the lengths has to be 1 : √2. So the diagonal of a unit square in the small crosses equals the length of the edges of the squares in the large cross.
@@MarkVersteegh I will pretend to understand that.
Ah that's what I thought too.
Hi@@MarkVersteegh, Yes, I got that... but that was not put as part of proof so I was wondering. I have a different proof based on that fact. I am still not able to wrap my head around this proof though.
The proof I got:
Side of square of larger cross = √2 * side of square of smaller cross. Then if we cut all the four outer squares of the large cross by their diagonal, we will only need 4 cuts to cut all of them. The center square wont' be cut yet, so we just use one remaining cut horizontally on it and we now have all the pieces required to rebuild the 2 smaller crosses.
Link to solution image:
www.linkpicture.com/q/Cross-Puzzle-Solution.png
35:09
The fractions we get by truncation when put into the left side of the Pell equation alternatingly yield 1 and -1 because we alternate between cutting off rectangles where the longer side aligns with the longer side of the original rectangle and rectangles where the longer side is orthogonal to the longer side of the original rectangle.
If it aligns, the truncation means to make the denominator slightly smaller because we rescale the shorter side of the big rectangle to be reduced by the width of the truncated rectangle. This makes the truncated fraction bigger than the number we approximate and so the equation yields 1 .
If it is orthogonal, the truncation means to make the numerator slightly smaller since we rescale the longer side of the big rectangle to be reduced by the width of the truncated rectangle. This makes the truncated fraction slightly smaller than the number we approximate and the equation yields -1 .
Edit: In terms of 40:03 :
L^2 - 2*S^2 = ±1
=> (2*S + L)^2 - 2*(S + L)^2 = 4*S^2 + 4*S*L + L^2 - 2*S^2 - 4*S*L - 2*L^2
= 2*S^2 - L^2 = (-1)*(L^2 - 2*S^2) = ∓1 .
Also worth noting about this sequence (the first few terms are shown at 41:11) is that the odd-numbered terms produce all the Pythagorean triples in which the legs of the right triangle differ by one:
1/1: 1 = 0 + 1; 1² + 0² = 1² (trivial case to get started)
7/5: 7 = 3 + 4; 3² + 4² = 5²
41/29: 41 = 20 + 21; 20² + 21² = 29²
239/169: 239 = 119 + 120; 119² + 120² = 169²
...and so on. Every Pythagorean triple of the form x² + (x + 1)² = y² is hit.
That's not a coincidence. If you do the challenge at 13:37, you will find that the nth partial sum s_n in terms of the (n-1)th partial sum s_(n-1) is ((s_n)+2)/((s_n)+1). Keep in mind that this is the continued fraction for sqrt(2), we will use that fact later. Expressing s_(n-1) as a fraction in lowest terms p/q, we get s_n = (p+2q)/(p+q) (which is actually the same rule described at your timestamp now that I look at it 😅)
Edit: Or just skip to 39:30 for the relation.
In case you haven't noticed, all of the numerators are odd (which makes sense, otherwise constructing a Pythagorean triple from it whose legs differ by one is clearly impossible). To be rigorous however, we first have to prove that the numerator p+2q is always odd which will be done by induction.
Base case: Consider the second partial sum 1/(1+1/2). This simplifies to 3/2 and 3 is odd.
The case for all n: Assume s_n = p/q. Then s_(n+1) = (p+2q)/(p+q). If p is odd, then clearly p+2q is odd, no matter the parity of q. Since in the base case, p=3, which is odd, p should be odd for all partial sums.
This ensures that when we attempt to construct the Pythagorean triple from its corresponding partial sum, the legs are integers.
Next, we express the actual elements of the Pythagorean triple in terms of p and q. The 2 legs are (p-1)/2 and (p+1)/2 and the hypotenuse is q.
The sum of squares of the legs are ((p-1)^2+(p+1)^2)/4 = (p^2+1)/2 and by the Pythagorean theorem, is equal to q^2.
Multiplying both sides by 2 and moving the 1, we end up with the Pell equation 2q^2-p^2 = 1. However, since p/q is the nth partial sum for sqrt(2) (I told you we would use it :D), p and q are indeed solutions of that Pell equation (check 17:00). I forget the proof though, sorry 😞. Please have mercy on me, I am just a grade 9 student with no social life.
Edit: As an aside, you may have noticed that the Pell equation provided at 17:00 is actually 2q^2-p^2 = -1 and not positive 1. Actually, s_n only produces a Pythagorean triple if n is odd because p and q satisfy the other Pell equation for even n. This is the reason why there is no Pythagorean triple for 3/2, 17/12, 99/70 etc; the sum of the squares of the legs is actually 1 greater than the denominator squared. I could prove this to you, but I am sure you are tired of this rambling and I am getting tired of typing. Also 14:30 is exactly what I just said 😅.
This is a masterpiece!
This was one your best videos ever Mathologer; thank you. I'm curious if anyone has the answer to the puzzle about the Red Cross at 2:18? Cheers.
Someone posted this solutions imgur.com/JfpClXR
very nice! I tried to solve it for almost 40 mins with no success
36:29 there is always more to dig on every subject in math, this is a never ending quest.
School maths should teach more like this , love the geometric drawing conceptualiseations.
13:43 the numerators follow the pattern a1=1, a2=3, a(n)=2*a(n-1)+a(n-2)
the denonimators follow the same pattern sorry forgot to mention that
except that b2=2 instead of 3
I was supposed to study english for an exam tomorrow, but this is just way too fascinating
Great fan of your work sir .
I just love to see your videos
The way you explain concepts is just amazing.
Love from India sir
24:42 is not only a beautiful palindrome timestamp, but in a flash it gave me a deep understanding of both the Euclidian algorithm and of continued fractions. Thank you!
That's the reaction I was hoping for :)
Yay!
I think I need to invest in Mathologer T-Shirts!
Even if ur videos are long still it keeps us engaged. Good job 👍 appreciable ❤️love from india 🇮🇳 Ramanujan was great🙏
The way Mathologer pronounces ramanujan makes me so happy
Yesterday I saw Stand-up Math's video on how to approximately calculate the perimeter of an ellipse. And lo and behold Ramanujan was in there
And today, I meet Ramanujan once again 😀
really liked this one!
Wonderful video Every maths lover must watch.
Perhaps the (legendary) fellow that upset the Pythagoreans so much by proving the irrationality of root 2 may not have suffered such a tragic fate if he had been able to demonstrate root 2 as an infinite continued fraction.
The Pythagoreans also hated infinity
A Strand Is A Part Of A Rope Or Bond, While Stranded Means Being Washed Up On The Shore, And Being Stranded Is When You Can't Go Home.
The real question is whether Mathologer got the reference, or just gave this comment a heart because he's giving all the early comments on his video a heart...
Death Stranding?
@@OMGclueless Actually, I had to look it up :)
@@OMGclueless, what's the reference?
@@JNCressey Apparently to a game called "Death Stranding."
36:57 "Outramanujan" is now my new favourite verb!
I liked that
It’s gonna be interesting
8 houses could still be a "long street" just depends how close the houses are 😆
Ramanujan has no formal education,he taught himself and made himself a genius
Wow
@@ViratKohli-jj3wj are you really virat sir I am your huge fan🌷🌷🌷🌷
He was BORN genius. You can't make yourself genius.
@@homoxymoronomatura nope, it doesn't work like that
@@mrappu2884 It does work like that, unfortunately
I wish my mind could deliver answers in a flash like that.
Love from India 🇮🇳 Great research.
Huge fan of MATHOLOGER ❤
Big respect from #MathsPathshala
Sir loved your 22/7-Π T-Shirt which is equal to an integral(+ve value) Already asked in our 🇮🇳 IITJEE entrance.
This the the only _strand_-type math problem
:)
Wow
Upset I had to scroll so far to find this
@@Jop_pop Same
To me the fact that you can get from one convergent to the next is amazing. At first it looks like putting one more coefficient means you need to calculate the fraction again from scratch. But the picture makes the quick way so obvious!
Yes, it's really an amazingly insightful way of looking at these infinite fraction :) Hardly ever taught though :(
Actually, if you look at the continued fraction, it's reasonably obvious that adding one, flipping it, and adding one again will push everything down a layer. The hard part is noticing that pushing it down is going to work, due to the numbers being the same all the way down.
@19:04: to get the sequence of fractions, start with 1/1 = a/b. Then, given the current faction is a/b, the next fraction is (a+2b) / (a+b).
@39:19 So there is a geometric proof of what i saw in the numbers.
Infinitely mesmerised by the guitar music. Then I consider, is the tablature some kind of code?
If it's code, it's not my code :) The music in this video is by Chris Haugen, Fresh Fallen Snow (playing in the video) and Morning Mandolin (for the credits) and Nate Blaze 'Tis the season (chapters), all from the free TH-cam music library.
Mathologer it’s beautiful. Keep more of it coming!
I love the 1,1 solution 😂 technically correct is my favorite kind of correct
Yes, very cute, isn't it ?
@@MathologerYes it is! I'm so honored to receive a reply from you 😍😁 keep up the great math content!
These are the exact same answrs that ehen your friend says they are wrong you can say "well you see, technically...."
I do it all the time. Now i don't have friends 😂
@Lily Warner l'm in two minds about the 1,1 solution, for and against.
For: if we have a street of just one house long then the house numbers on each side must add up to 0 since there are no house numbers on each side. Moreover, even if we're talking about an infinitely long street, with an infinite number of solutions, the sum of the house numbers to the left of 1 is 0. We now have to find consecutive numbers to the right which also sum to 0. Since there are no such numbers, they too sum to 0. So 1 is the smallest solution. Is that what you're thinking? Fair enough.
Against: 1+2+3+4+5 = 15 and 7+8 =15 has just one whole number between the left and right side of of the summed sequences, namely 6. Call this "skip 1". What if there's no number at all, but the sequences on each side still add up to the same number? Example: 1+...+14 = 105 and so does 15...20. The smallest example of this "skip 0" as far as I can see is 1+2 on the left and 3 on the right. OK, we've already dealt with skip 1, now what about skip 2, the smallest example being 1...8 = 36 = 11...15 with 2 in-between numbers 9 and 10. The smallest skip 3 goes 1...11= 66=15...18.
Now line up the last numbers on the left hand sequence for each smallest skip sequence.They go: for skip 0 it's 2, for skip 2 it's 8, for skip 3 it's 11, skip 4 it's 14. They go up by 3 each time.
So for skip 1 it should be 5, not 1. We have 1...5=15=7...8 as the smallest skip 1 solution with 6 as the in-between number, not 1.
Make sense?
@@chrisg3030 yes I see what you're saying. I like how you are thinking about generalizing the problem
I do think that we should think about the framing of the original problem. The problem asked for a man living on a street, and that summing the house numbers on either side results in equal numbers. If you consider a street with one house a valid input, then yes the sum is equal on both sides of them.
If you define it instead as you have, by having consecutive number sequence sums with N skips, then 1 is not a solution.
It's all about defining the problem. Similar to the discussions on whether 1 is prime or if 0 is even
6:56 if the houses are spaced widely, 8 houses could still be a long street.
:)
It is a damn long street for a elementary school kid living in the countryside 😀
42:50 That time when you see Notch in the Paypals section
No. Notch doesn't deserve any heart.
@Giovanni Both Markus and Persson are common Swedish names, it would surprise me greatly if he was the only one.
29:45 "the proof is left as an exercise to the reader"
I like way when you get pumped we too get pumped.... And that's what I think is showing true love to THE MATHEMATICS
Ramanujan was God gifted not like me, A normal students wasting time in Indian Schools to memorize everything for exam.😢😢
Put in your effort. Nothing goes to waste. We Hindus believe ramanujan's gift was due to his hardwork may not be totally in his lifetime but lifetimes before.
Cosmic justice absolution.
Are u in 12th
"Checking by hand would be way too taxing for people, who grew up punching buttons on the calculator". Like this line. What about people, who grew up punching keys on the computer keyboard? :)
As always T-shirt on point
23:09 Turn on subtitles here
He said the numbers came to him from another place in his mind, not of this world.
I think this is the pure form of intelligence that had occurred
was this the first STRAND type puzzle?
Not sure what you mean by this, but one possible answer to your question is 'No' and another possible answer to your question is `Yes'. Anyway, have a look, there is a complete digital archive of The Strand magazine. You can find the page with the the puzzle that I show in here: tinyurl.com/y2lnb8xf (page 790)
@@Mathologer oh no, I'm making a reference to the video game director Hideo Kojima, who labeled his game Death Stranding as the first "STRAND type game", which, that statement has become somewhat of a meme in video game discourse.
@@SprDrumio64 Everyone knows Ramanujumbo genius. Therefore, the puzzle is brilliant
@@SprDrumio64 I see :) Will have to check out Death stranding.
Seems doubtful considering that this puzzle is interesting and the first strand type game is not LOLOLOLOLOLOL
"That's a good time to stop" 42:35
wouldn´t it rather be 42:30 ???
Catchy phrace. Mathologer should ''penn'' it.
all the time i am trying to wrap my head around on the thing that is printed on your shirt
Here is a version of the t-shirt I am wearing: tinyurl.com/y5vgo7zb This one is about that other famous Ramanujan anecdote: tinyurl.com/orezs2p
1729 actually features prominently in another one of my videos.
Nothing better than seeing a new video when wake up.
Where in the world are you to be waking up now ?
@@Mathologer maybe in another planet
@@Mathologer I live in Hong Kong
@@paknincheung8692 Cool, pretty close to my time zone then :)
Proud to be Indian❤️🇮🇳🙏🏻
Dear Sir🙏🏻, There is vedic Mathematic also
You should also teach vedic Mathematic here