Magic Squares of Squares (are PROBABLY impossible) - Numberphile
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- เผยแพร่เมื่อ 12 มิ.ย. 2023
- Tony Várilly-Alvarado uses surprising mathematics to show that a 3x3 Magic Square of Squares is highly unlikely. See Matt Parker react to this video at: • Matt Parker Reacts to ... --- More links & stuff in full description below ↓↓↓
Tony's webpage at Rice University: math.rice.edu/~av15/
Paper by Nils Bruin, Jordan Thomas, Anthony Várilly-Alvarado: arxiv.org/abs/1912.08908
Christian Boyer, "Some notes on the magic squares of squares problem":
link.springer.com/article/10....
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Watch Matt "Parker Square" Parker react to this video: th-cam.com/video/U9dtpycbFSY/w-d-xo.html
It's Parkin' Time!
It is now part of his name 😂
5:25 you definitely don't have correct number for the failed diagonal, it's 38307, not 9409. Where did you even come up with 9409 there?
All of this talk of higher dimensions has convinced me we need a Parker brane.
This comment is me reacting to Brady's comment.
I feel like it’s worth mentioning that because of its faulty diagonal, the Parker Square isn’t even on the Parker Surface
But it gave it the best shot.
"the Parker Square isn’t even on the Parker Surface". That's it, i'm gonna call it the Parker paradox
Parkerdox
Just one more thing the Parker Square doesn’t quite succeed at.
Tony is trying so hard to give Matt all the credit for his attempt and Brady is not having it, this is amazing
Skilled pros want to encourage other people to share their passion. TH-camr friends just want to dunk on each other.
@@DanielHarveyDyeror he is just being playful
dunking on is playful@@raynermendes210
I can't tell if this man just became Matt Parker's best friend or his archnemesis.
LOL
Arch-nemesis definitely
Kismessis obviously :p
@@Ms.Pronounced_Name
so it’s more like a parkership…?
Arch-Frenemy
Let’s take a minute to consider that the Parker square is eventually, but surely, going to end up in very serious, very academic papers. Matt’s made it.
I mean, it doesn't really add anything new, unless mathematicians get very interested in semimagic squares with a single line of symmetry. At best it'll probably appear in papers like these as a sort of example, and may end up becoming the mathematical version of loss.
Parker finite fields
@@matthewstuckenbruck5834Mathematical version of loss 😱
@@matthewstuckenbruck5834 mathematical version of loss 😂
It already has
I absolutely love how Brady remembered that one of the diagonals of the Parker square is defective
Lol😅
Of course he would since it is the whole point of this video.
I remember it too, honestly
I dont think he ever forgot.
I haven’t seen Tony in a video before. Charming, cogent, patient, honest, and passionate about his subject. I look forward to more!
He looked so proud every time Brady asked very insightful questions. And simultaneously so excited that he was going to have to answer them. Great lecturer, so great.
I agree!
Yes
Lovely fellow!
100%
i secretly love that the production quality of these has not really improved over the years. It adds some continuity. It also adds a veneer of cinema verite/documentary. and it feels very authentic. Like, you just love this stuff and you wanna share it.
Well the one definite evolution is the complexity and depth of topics. I remember hearing brady complain about the epic circles video on an episode of hello internet years ago, and now he’s showing off some surprisingly deep stuff on the regular
If it ain’t broke don’t fix it! One of the thing I love about Brady’s channels is it’s so clear that he’s not chasing views or trying to make change for the sake of change. He just wants to get the point across as best as possible.
Almost all the improvements that have been made to the effects and animations have been in service of ease of understanding.
"has not improved" is not the kind of compliment you want it to sound like though
Has never needed to change. Numberphile videos are amazing!!!
Brady has improved quite a bit, but the technical standards are about the same.
I love how genuinely excited Tony gets every time Brady chimed in. So fun to watch these two
I really liked this dude, he was much fun and very insightful.
Parker magic square square needed
Tony is such an amazing communicator, hope he's on more
Geosquare, a perfect name for this video.
Surprisingly the best explanation for elliptic curves inside
I love how "(generously)" appears across the screen, roasting Matt further.
One of the best explainer you've had on this channel
I agree, Tony explained it well and you can feel his enthusiasm.
I had him as a professor in undergrad and he really is a great explainer! And his enthusiasm really comes across in his teaching, he's a really great professor :)
15:09 love the transparency and honesty in Tony's voice tone...
Tony Varilly-Alvarado was a legend in this video! I hope we see him again.
"Paper IV - A New Hope" Lol that was a nice pun!
3:30 CHRIST that "(generously)" is so so brutal
I really enjoyed how excited Tony got when Brady asked exactly the right leading question.
16:39 16:41 Is everyone forgetting that the Parker Square doesn't lie on the Parker Surface? Since it doesn't fulfill all conditions (the sum on one diagonal doesn't equal the sum on the other and the rows and columns), and all points on the Parker surface do fulfill this criterion!
nice video. But there is a mistake in Sallows' Square, the diagonal that does not work does not add up to 9407 but instead it adds up to 38307
Bit of a Parker Square edit
I caught that too. I instantly smelled that something was not right when I saw that supposedly the squares of the 3 biggest numbers add up to less than half of the magic number...
@@andrasszabo1570 yea exactly thats why i noticed it😂
Yes. 9409 is the number in the bottom right square, not the sum of the whole diagonal.
parker parker square
Phenomenal video. Tony's storytelling was great (more of him please!), the animations helped visualize the story and the quality of Brady's questions is impressive as always!
This reminds me of the search for the perfect Euler brick: a cuboid which has integer sides, diagonals, and space diagonals. The problem can be solved if you relax ONE of the constraints...
And rightly so! In fact, the article mentioned in the video has a very similar statement to make about the surface corresponding to the Euler brick.
For the Bremner Square, the first number in the second row should be 360721 instead of 366721. (The brown paper was correct, but the animation was not.)
The “missing” diagonal in Sallow’s Square was also incorrect. Should be 38,307.
Very entertaining, and such depth. Would love to see this guy back again.
Can we take a moment to appreciate his handwriting?
Permission granted.
On the Bremner square - Andrew Bremner was my professor for both group theory and number theory, and he is a fantastic man and professor. I cannot believe he got a shoutout in a numberphile video, how wild!
I still have my Parker Square t-shirt! After so many ears its exciting to see how far the Parker-Square has come! Always love to see updates on the magic square conundrum.
Videos like these make me wanna try and write a program/script that would try and workout the numbers, and "solve" the Parkersquare.
We went from tic tac toe to 8 dimensional planery
Awesome video. The explanations go so deep with no oversimplification and yet we are able to follow the discussion easily. I've been following this channel for many many years with great pleasure but this is actually one of my very favorite videos. It gave us such a good insight on what topics are actually interesting for mathematicians with such a good pedagogy. Thank you very much for bringing this to us.
Really liked Tony, cheerful and fun to follow. Also, the animations are very well done, my compliments to the animator.
15:09 Brady's love for naming things never ceases to bring me joy
I've seen this video twice now, and I must say that I loved Tony's energy and passion. I really hope to see more videos with him in the future!
Love the enthusiasm! Excellent video!
What a great teacher. I almost, kind of understood this one thanks to Tony. Good video!
Fantastic new guest on the channel! He has such amazing enthusiasm
What a pearl! I guess we have to start the Parker program to find all rational/elliptic curves in the Parker blob :-)
michael penn and numberphile both posting about magic squares?! this must be a miracle!
maybe its magic
I love this guy! Not only does he embrace Parker Lore, but he has nice blackpenredpen skills too! :)
I didn't think that i will watch another long video on this topic from beginning to end, but Tony was so engaging and it was presented in such a clear and interesting way that i'm in for several more of such videos. Please?
Very clear and interesting. Perfect balance between in-depth and vulgarisation.
No no no. A 2 dimensional surface that describes magic squares solutions?
That's a magic carpet!
I hope you realize that "Parker Surface" is going to become standard nomenclature. Or at least common lol. Because people will seek a way to refer to this surface, and they'll be like ... "well, like in that numberphile video, the Parker Surface" ... this is how terminology is born lol. It's like the semi-used thagomizer
@IanZainea As Parker squares are not elements of the surface, it would be more appropriate to call it a Non-Parker surface. Otherwise people could be confused.
@@rennleitung_7 fair! Lol
I love that Brady never stops trolling Matt Parker.
I love the light switches inside the bookshelf. I guess they had so many books but no space left, that they just built a bookshelf with cutouts for the switches. I can't look away after seeing them
I do that, for outlets, Thermostats, ceiling fan switches - books always have right-of-way!
Wow! This is instantly one of the best Numberphile videos ever, period
Much love for Tony, very clear explanations and clear excitement and passion for the subject. Matter of fact, he follows the rules of improv very well. The moment Brady offers a suggestion, he instantly affirms and rolls with it. Yes, we are setting up a monster equation, a set of them in fact. Yep, it's a Parker surface, and yes exactly it bumps up in dimension and becomes a Parker blob. Just nailing it.
This is the best Numberphile video for a while. I'm so excited at 06:34 to know what happens next!
The ending, which I will now call "A New Hope for Parker", strongly reminds me of the n-dimensional sphere packing problem, where some numbers of dimensions are "easy" and others are totally unknown "with current mathematical technology". Is '3' the only "hard" dimension, or are there others?
20:43 I believe there was a minor typo, where the x-coordinate should be 2t/(t^2 + 1) (rather than have the extra ^2)
Came to say the same : )
Came to say the same :) Worked through the derivation to generate those rational points on the circle from values for t and found this.
Just noticed that 2t^2 / (t^2 + 1) cannot be correct, without having to do a derivation. To create lines that intersect the circle at a third point, t > 1 or t < -1. Then, 2t^2 > t^2 + 1, meaning the x-coordinate is > 1, and the point would not be on the unit circle.
Do you guys still remember the 301 views video of this channel?? That video still has 301 views and 3m or 4m+ likes stunning!
If I remember rightly I believe the reason was because youtube agreed to manually freeze it at 301 views as a special case in the spirit of the video (I have no real way of knowing if that is true or not, though)
18:47 thank you for this question! Exactly what I'd been thinking.
PS, fun video format: I like how Tony is writing on paper, and we're (generally) seeing a tidier digital version of that paper, but can picture it being real
Love this guy's energy. A total joy to watch!
I've been pondering this problem for years ever since I learned about the Parker Square, and it's led me down some interesting rabbit holes like Pythagorean triples and modular arithmetic, but hearing about "blobs" is light years beyond anything I've considered about this problem
Cool
@@Macrotrophy-mq3wh why did you make this comment?
@@idontwantahandlethoughCool
LOVE a Parker Square callback. Long live the Parker Square!
I'm so happy I watched this whole thing. Really great, thought provoking stuff.
That was a really cool video, this man is interesting, funny and very clear
Great speaker! Very clear and amiable
A great way to see the link between algebra and geometry. He's a great speaker.
We need more Tony on numberphile. He ca explain complex phenomenon with ease.
The 368 solutions where two of the numbers are the same, but where all the diagonals match, seems like the closest to a magic square of squares. I'd be interested to see one of those.
Yea it would be interesteing if they could get something out of those as it seems it would still be better than any of the example attempts there have been previously
Why are there 368 solutions? That seems like it would be actually infinitely many solutions? Is it just so far we've found 368?
@@highviewbarbell In the video he says there are finitely many solutions. But there are more than 368, and they haven't determined the exact number.
@@vicarion just got to that part now, very interesting indeed, thanks
what an emotional roller-coaster of mathematics! First you think, well proving there _isn't_ a 3x3 magic square of squares might be cool, but then you learn why having one would be way cooler, and it only gets better from that.
I conclude from this that Parker-ness is a concept of great practical use in mathematics.
I nearly spat out my drink at 3:31. Brady you are hilarious! 😂
Is this the first video with Tony? Lovely video!
This is one of the clearest videos I've seen about a very abstract concept on this channel.
Excellent work, Brady!
I really enjoyed this, and this guy was really easy to listen to, and made sense. More from him please.
RIP the Parker Square
I was just thinking what sort of irrational set of points might be on the square then I just realised just the square roots of a normal magic square squared would love on this surface as a trivial example
This is exactly what my first thought was too: If the allowed numbers can be square roots, a normal magic square is a magic square of square of squares indeed.
Just had a carriage full of commuters give me a funny look as a burst out laughing to "Parker surface". Great video as always!
Paper IV, .A New Hope! I love it. Nice touch.
To a mathematician, having no points on the Parker surface is the same thing as having finite points until you can find a single point
'parker square shirts are now available' was the best punchline I've ever seen on this channel
This is why number theory is great, you can ask questions that feel like just about anybody can think of, yet they take math analogous to some of the math that pops up in string theory to actually get anywhere.
Looking forward to the N-Dimensional Parker Blob shirt, honestly sounds like a pretty great rock band name.
I love the way things happen on this channel.
In relation to the last comment of the professor, I think it would be useful to point out that in general there cannot be an algorithm that say wether or not a polynomial (in several variable) has an integer solution. That is Matiiassevitch's theorem.
Of course, for a specific polynomial we might find the answer.
Living up north, I pick computer projects to do over the winter. A few years ago. I picked this one. I could not find any solutions where all the numbers are under 2^30. I encountered an issue with sqr() and sqrt() large integers.
The interesting thing about the computational problem is you can start making assumptions that limit what you can test.
(Hint, the largest number has to be in a corner, the smallest number is on a side, and the average is in the middle. Knowing this, you can quickly discard a large set of numbers!)
ok, my hint was not accurate, because it has been a few years. My point is there are assumptions that can be made. Just finding three squares where one is the average quickly limits your selections.
I didn’t realize how appropriate my choice of shirt was this morning.
26:03 "But often finite can mean empty"
Maybe it's the beer talking. but man that's funny
@28:24 The 6-by-6 feels a bit unsatisfying because it includes all numbers 0 up to 36, except that it skips 30.
I had always thought that a video explaining basic concepts of algebraic geometry to a lay audience was essentially impossible, but here we are.
All thanks to the Parker Square.
Have any deeper searches for 4x4, 5x5 and so on square of squares been made? Would be interesting to see a few! Also magic cubes are known, is there a Parker cube?
I like how by now you can casually make statements like "this 6-dimensional surface is _obviously_ infinite".
I love Brady throwing shade to Matt instantly in this video
Tony is great at this!
Great video, but I noticed a mistake. On 4:25, the Bremner Square shows a 366721 which should be a 360721. No one will probably read this, but I couldn't stop seeing it once I noticed.
Very nice example of overdetermined problems
Incredible stuff
I wrote a program to sweep through square numbers to see if it can generate a magic square of squares. Got through about 50 million square numbers but could only ever generate 7. I’ve been trying to disprove it ever since
That's the spirit.
Did you see the email I sent you about it a few months back?
@@mackey_iiiIt generated SEVEN? How??
@@danielyuan9862 very carefully. However, the laptop that I had the program on is no longer with us and I haven't rewritten it yet.
Tony would be great for a video on the Euler Brick😁
small mistake: at 20:36 it's 2t/(t^2 + 1). Intuitively, you can see that if t
i like this guys enthusiasm
It is probably not in the spirit of the channel, but out of curiosity I ran a script to find out how often we have at least 1 diagonal. Apparently not so often - there are only 3 families of semi-magic squares with numbers below 2000: they happen at sums 21609, 257049, 1172889. Smallest square is
97 82 74
94 113 2
58 46 127
Also the Christian Boyer paper linked seems to be only available behind a paywall, unless there's an arxiv or other link.
Love this channel
If you allow integers instead of natural numbers, you can sort of fake using the same number twice (-2)^2 = 2^2 so that makes the problem a little easier than every number being unique.