Magic Squares of Squares (are PROBABLY impossible) - Numberphile

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....
The Parker Square: • The Parker Square - Nu...
Finite Fields & Return of The Parker Square: • Finite Fields & Return...
A Special Magic Square: • Special Magic Square -...
Magic Square Party Trick: • Magic Square Party Tri...
Parker Square Tee: numberphile.creator-spring.co...
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[@numberphile]

Watch Matt "Parker Square" Parker react to this video: kzbin.info/www/bejne/i2rHpaOvmMd5ibs

[@Jarx246]

It's Parkin' Time!

[@crazilycrazy29]

It is now part of his name 😂

[@Dakerthandark]

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?

[@Rank-Amateur]

All of this talk of higher dimensions has convinced me we need a Parker brane.

[@standupmaths]

This comment is me reacting to Brady's comment.

[@UltraCboy]

I feel like it’s worth mentioning that because of its faulty diagonal, the Parker Square isn’t even on the Parker Surface

[@TheKilogram1000]

But it gave it the best shot.

[@anhhoanginh4763]

"the Parker Square isn’t even on the Parker Surface". That's it, i'm gonna call it the Parker paradox

[@DavidBeddard]

Parkerdox

[@chucknovak]

Just one more thing the Parker Square doesn’t quite succeed at.

[@ericvilas]

Tony is trying so hard to give Matt all the credit for his attempt and Brady is not having it, this is amazing

[@DanielHarveyDyer]

Skilled pros want to encourage other people to share their passion. KZbinr friends just want to dunk on each other.

[@raynermendes210]

@@DanielHarveyDyeror he is just being playful

[@WillToWinvlog]

dunking on is playful@@raynermendes210

[@TheLastWanderingBard]

I can't tell if this man just became Matt Parker's best friend or his archnemesis.

[@Macrotrophy-mq3wh]

LOL

[@SwordQuake2]

Arch-nemesis definitely

[@UnknownCleric2420]

Kismessis obviously :p

[@redsalmon9966]

@@Ms.Pronounced_Name so it’s more like a parkership…?

[@maxw565]

Arch-Frenemy

[@davidconnell1959]

I haven’t seen Tony in a video before. Charming, cogent, patient, honest, and passionate about his subject. I look forward to more!

[@JoQeZzZ]

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.

[@DemianNuur]

I agree!

[@peterflom6878]

Yes

[@oscarn-]

Lovely fellow!

[@gazfpl7438]

100%

[@andrearaimondi882]

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.

[@matthewstuckenbruck5834]

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.

[@k0pstl939]

Parker finite fields

[@hnr9lt-pz7bn]

​@@matthewstuckenbruck5834Mathematical version of loss 😱

[@brianjones9780]

​@@matthewstuckenbruck5834 mathematical version of loss 😂

[@TheFreeBro]

It already has

[@johnchessant3012]

I absolutely love how Brady remembered that one of the diagonals of the Parker square is defective

[@hnr9lt-pz7bn]

Lol😅

[@wesleydeng71]

Of course he would since it is the whole point of this video.

[@danielyuan9862]

I remember it too, honestly

[@cihanbuyukbas7333]

I dont think he ever forgot.

[@Seymour_Sunshine]

I love how genuinely excited Tony gets every time Brady chimed in. So fun to watch these two

[@HasekuraIsuna]

I really liked this dude, he was much fun and very insightful.

[@IanZainea1990]

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.

[@crimsonvale7337]

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

[@stuiesmb]

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.

[@Irondragon1945]

"has not improved" is not the kind of compliment you want it to sound like though

[@awestwood3955]

Has never needed to change. Numberphile videos are amazing!!!

[@GynxShinx]

Brady has improved quite a bit, but the technical standards are about the same.

[@MrAmalasan]

Parker magic square square needed

[@Swampy293]

Surprisingly the best explanation for elliptic curves inside

[@Geosquare8128]

Tony is such an amazing communicator, hope he's on more

[@asheep7797]

Geosquare, a perfect name for this video.

[@MonsieurBiga]

One of the best explainer you've had on this channel

[@MrCheeze]

I agree, Tony explained it well and you can feel his enthusiasm.

[@bootypopper420]

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 :)

[@josda1000]

I love how "(generously)" appears across the screen, roasting Matt further.

[@borisnot]

15:09 love the transparency and honesty in Tony's voice tone...

[@soundscape5650]

Tony Varilly-Alvarado was a legend in this video! I hope we see him again.

[@jacovisscher]

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!

[@SplittingField]

I really enjoyed how excited Tony got when Brady asked exactly the right leading question.

[@blak4831]

3:30 CHRIST that "(generously)" is so so brutal

[@jonathansperry7974]

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.)

[@M31-ZERO]

The “missing” diagonal in Sallow’s Square was also incorrect. Should be 38,307.

[@yanhei9285]

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

[@quinn7894]

Bit of a Parker Square edit

[@andrasszabo1570]

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...

[@yanhei9285]

@@andrasszabo1570 yea exactly thats why i noticed it😂

[@tulliusexmisc2191]

Yes. 9409 is the number in the bottom right square, not the sum of the whole diagonal.

[@Pablo360able]

parker parker square

[@zh84]

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...

[@NilsBruin-ws8pv]

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.

[@arhythmic1]

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!

[@GoldfishWaterCooler]

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!

[@sammartano22]

I love that Brady never stops trolling Matt Parker.

[@batmanuk1810]

We went from tic tac toe to 8 dimensional planery

[@AmmoBoks]

"Paper IV - A New Hope" Lol that was a nice pun!

[@jakobwilns3006]

Can we take a moment to appreciate his handwriting?

[@backwashjoe7864]

Permission granted.

[@backwashjoe7864]

I love this guy! Not only does he embrace Parker Lore, but he has nice blackpenredpen skills too! :)

[@DouweHummeling]

Videos like these make me wanna try and write a program/script that would try and workout the numbers, and "solve" the Parkersquare.

[@noahblack914]

15:09 Brady's love for naming things never ceases to bring me joy

What a pearl! I guess we have to start the Parker program to find all rational/elliptic curves in the Parker blob :-)

[@wasko92]

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.

Really liked Tony, cheerful and fun to follow. Also, the animations are very well done, my compliments to the animator.

[@gracenc]

michael penn and numberphile both posting about magic squares?! this must be a miracle!

[@wyboo2019]

maybe its magic

[@torlachrush]

Very entertaining, and such depth. Would love to see this guy back again.

[@fuxpremier]

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.

[@Eye1hoe]

Love the enthusiasm! Excellent video!

[@colinfew6570]

What a great teacher. I almost, kind of understood this one thanks to Tony. Good video!

[@subjectline]

This is the best Numberphile video for a while. I'm so excited at 06:34 to know what happens next!

[@lrwerewolf]

No no no. A 2 dimensional surface that describes magic squares solutions? That's a magic carpet!

[@IanZainea1990]

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

[@rennleitung_7]

@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.

[@IanZainea1990]

@@rennleitung_7 fair! Lol

[@kikoerops]

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!

[@TarenNauxen]

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

[@Macrotrophy-mq3wh]

Cool

[@idontwantahandlethough]

@@Macrotrophy-mq3wh why did you make this comment?

[@want-diversecontent3887]

@@idontwantahandlethoughCool

[@max5183]

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

[@mcv2178]

I do that, for outlets, Thermostats, ceiling fan switches - books always have right-of-way!

[@DizzyPlayez]

Do you guys still remember the 301 views video of this channel?? That video still has 301 views and 3m or 4m+ likes stunning!

[@Casowsky]

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)

[@flymypg]

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?

[@Arc125]

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.

[@GregHillPoet]

LOVE a Parker Square callback. Long live the Parker Square!

[@kaushikmohan3304]

Fantastic new guest on the channel! He has such amazing enthusiasm

[@SebBrosig]

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.

[@Brawler_1337]

RIP the Parker Square

[@subjectline]

I conclude from this that Parker-ness is a concept of great practical use in mathematics.

[@kindiakmath]

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)

[@olivierbegassat851]

Came to say the same : )

[@backwashjoe7864]

Came to say the same :) Worked through the derivation to generate those rational points on the circle from values for t and found this.

[@backwashjoe7864]

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.

[@pinkraven4402]

Wow! This is instantly one of the best Numberphile videos ever, period

[@Alexand3ry]

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

[@suan22]

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?

[@igNights77]

Very clear and interesting. Perfect balance between in-depth and vulgarisation.

[@vicarion]

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.

[@jh-ec7si]

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

[@highviewbarbell]

Why are there 368 solutions? That seems like it would be actually infinitely many solutions? Is it just so far we've found 368?

[@vicarion]

@@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.

[@highviewbarbell]

@@vicarion just got to that part now, very interesting indeed, thanks

[@anirbanbiswas]

We need more Tony on numberphile. He ca explain complex phenomenon with ease.

[@mikedoe1737]

Love this guy's energy. A total joy to watch!

[@microwave221]

'parker square shirts are now available' was the best punchline I've ever seen on this channel

[@fk319fk]

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!)

[@fk319fk]

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.

[@matthewdodd1262]

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

[@WelshPortato]

Great speaker! Very clear and amiable

[@KevinHorecka]

I'm so happy I watched this whole thing. Really great, thought provoking stuff.

[@pifdemestre7066]

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.

[@andrewchapman2039]

Looking forward to the N-Dimensional Parker Blob shirt, honestly sounds like a pretty great rock band name.

[@kaushikmohan3304]

I nearly spat out my drink at 3:31. Brady you are hilarious! 😂

[@_ajweir]

A great way to see the link between algebra and geometry. He's a great speaker.

[@Smaug_le_dore]

That was a really cool video, this man is interesting, funny and very clear

[@MattGodbolt]

Just had a carriage full of commuters give me a funny look as a burst out laughing to "Parker surface". Great video as always!

[@dougdimmedome5552]

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.

[@igorstarfouk]

Excellent work, Brady!

[@GODDAMNLETMEJOIN]

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

[@erwinmulder1338]

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.

[@brine1986]

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

[@mmburgess11]

Paper IV, .A New Hope! I love it. Nice touch.

[@CorrectHorseBatteryStaple472]

26:03 "But often finite can mean empty" Maybe it's the beer talking. but man that's funny

I like how by now you can casually make statements like "this 6-dimensional surface is _obviously_ infinite".

[@glowingfish]

This is one of the clearest videos I've seen about a very abstract concept on this channel.

[@mrcpu9999]

I really enjoyed this, and this guy was really easy to listen to, and made sense. More from him please.

[@dehb1ue]

I didn’t realize how appropriate my choice of shirt was this morning.

[@jd9119]

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.

[@Veptis]

Is this the first video with Tony? Lovely video!

[@Marco-ti8sx]

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.

[@Nethershaw]

I love the way things happen on this channel.

[@arnerob123]

small mistake: at 20:36 it's 2t/(t^2 + 1). Intuitively, you can see that if t

[@Aiden-xn6wo]

At 5:26, the number in red is 97^2, not the sum of the whole diagonal. The correct sum is 38307.

[@mackey_iii]

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

[@standupmaths]

That's the spirit.

[@mackey_iii]

Did you see the email I sent you about it a few months back?

[@danielyuan9862]

​@@mackey_iiiIt generated SEVEN? How??

[@mackey_iii]

@@danielyuan9862 very carefully. However, the laptop that I had the program on is no longer with us and I haven't rewritten it yet.

[@CynicKnowsBest]

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.

[@kylesedate237]

I took a stab at this problem, and I also failed to find a square passing all conditions. The closest (proportionally) that I managed to find is: 441,058 436,894 675,473 736,303 529,538 138,274 324,034 608,593 605,218 When squared, all sums come out to 841,672,300,329 with the exception of the main diagonal which sums 841,231,480,332. Another fun attempt is : 26,128,319 43,422,722 19,112,842 25,739,602 31,303,601 35,931,602 39,853,562 8,250,718 35,737,121 The first eight sums come out to 2,933,522,568,972,009, and the main diagonal 2,939,746,306,701,603. This is a fun problem to work through. Thanks for introducing it on this channel.

[@micahpalin6136]

would you mind sharing your method of creating these? I get that your first attempt is just a scale up by 79^2 of every square compared to the one shown in the video, but how did you generate the other one?

[@kylesedate237]

@@micahpalin6136 I spent some time optimizing an algorithm that found a decent set of smaller ones. From there, it was recognizing patterns within the factors and orientation (which factors are often present, which are never present?). This allowed me to make further optimizations and scale up the numbers considerably.

[@micahpalin6136]

@@kylesedate237 ah thats cool, thanks!

[@andrewwalker7276]

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?

[@Toobula]

Tony is great at this!

[@pierQRzt180]

I am a simple man, I see "parker" attempts and I upvote.

[@patcheskipp]

I love Brady throwing shade to Matt instantly in this video

[@dj-maxus]

Very nice example of overdetermined problems

[@hellopio]

I think the Lang-Vojta Conjecture implies that there can't be a solution with all rational coordinates outside of the rational and elliptic curves, as once you have one such solution you can use it to define infinitely many such solutions through scaling.

[@umbralreaver]

I came into the comments thinking exactly the same thing and hoping anyone else had noticed. I wonder if there will be a follow up to this!

[@MatthewWeathers]

@28:24 The 6-by-6 feels a bit unsatisfying because it includes all numbers 0 up to 36, except that it skips 30.

[@CraigChrist8239]

Shameless self-promotion: I took an attempt at this problem. The best way I could find was to generate co-equal pairs of square sums, and then arrange them on the outside of the square. This sounds hard to do, but its actually easier than you might think. Basically you just take a number and factor it, and then using the factors and imaginary numbers, its possible to generate all pairs of square sums equal to the number. Essentially this reduces the problem to just a factoring issue. It'd make a great video actually I searched into the trillions (meaning pairs of square sums in the trillions), and no luck. When we're able to factor numbers in linear time, in only 10 years(TM), should be no problem Its available on my github