Euler Squares - Numberphile

Also known as Graeco-Latin Squares. Featuring Dr James Grime.
Extra footage at: • Euler Squares (extra) ...
Patreon: / numberphile
More links & stuff in full description below ↓↓↓
Dr James Grime's website: www.singingbanana.com
More James Grime videos on Numberphile: bit.ly/grimevideos
This video was coincidentally released the day of this news: www.news18.com/news/opinion/i...
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumberphile
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science. www.simonsfoundation.org/outr...
And support from Math For America - www.mathforamerica.org/
NUMBERPHILE
Website: www.numberphile.com/
Numberphile on Facebook: / numberphile
Numberphile tweets: / numberphile
Subscribe: bit.ly/Numberphile_Sub
Video by Brady Haran and Pete McPartlan
Numberphile T-Shirts and Merch: teespring.com/stores/numberphile
Brady's videos subreddit: / bradyharan
Brady's latest videos across all channels: www.bradyharanblog.com/
Sign up for (occasional) emails: eepurl.com/YdjL9
Thanks to these Patrons (and others)
Arjun Chakroborty
Ben Delo
Jeff Straathof
Jeremy Buchanan
Andy
Yana Chernobilsky
Mael Thépaut
Ken Baron
Christian Martel
Nat Tyce
Ben
Garrett Smith
Dr Jubal John
Andrei M Burke
Adam Savage
Matthew Schuster
Mitch Harding
James Bissonette
Zachary Knechtel
Ronald Hochsprung
Robert Donato
Steve Crutchfield
Valentin-Eugen Dobrota
Maxwell A Zelenka
Mirik Gogri
Charles Southerland
Arnas Valeika
Ian George Walker
Jerome Froelich
Bernd Sing
Tracy Parry
Igor Sokolov
Kermit Norlund
Alfred Wallace
Bodhisattva Debnath
Alex Khein
Wall of fame: www.numberphile.com/patrons

[@numberphile]

Extra footage at: kzbin.info/www/bejne/fqaso4ibhKmnjLM Patreon: www.patreon.com/numberphile

[@deepakkpradhan]

Sadly, sharadchandra shankar shrikhande one of the co-authors of the Euler's spoiler paper passed away recently.

[@NStripleseven]

deepak pradhan Wow, that's sad.

[@leif1075]

Wait i thought BOTH diagonals had to be different but for the 3x3 you have one diagonal as A A A ..at 8:20 isnt that wrong?

[@leif1075]

Also at 8:35 you have the same number in the diagonal...that's inconsistent too..cant have all 3s or all 1s..should be one of each..

[@RedwoodRhiadra]

@@leif1075 He explicitly says the diagonal restriction is only for the 4x4 cards puzzle and not for the general Latin Squares puzzle.

[@massimookissed1023]

Moral of story: _Even when Euler's wrong, he _*_still_*_ gets things named after him._ (That's gotta make Matt Parker feel better.)

[@electromika]

What a Parker naming system.

[@yourguard4]

Even things, which doesnt exists, need a name, so that everyone knows, what you are talking about :P

[@rcb3921]

The Euler Square O.O I can't believe that went over my head.

[@sumdumbmick]

nah, the moral of the story is that people who name discoveries after people are idiots.

[@fredg8328]

@@sumdumbmick Because people don't deserve being recognized for their work ?

[@K-o-R]

10:52 No-one brute-forces a problem like Gaston!

[@otakuribo]

When he was a lad he did 4 dozen trials every morning to help with the proof, And now that he's grown he does 5 dozen trials on his quest for mathematical truuuth!! 💪

[@BlokenArrow]

🤓

[@MichaelJamesActually]

K.o.R take my upvote

[@talonbraxton8394]

K.o.R are you a Ben and hollys little kingdom fan?

[@oz_jones]

Literally egghead Gaston.

[@jesusthroughmary]

4:00 "We don't actually need to match up the diagonals." 4:30 *diagonals match up anyway* The Anti-Parker Square

[@PanduPoluan]

So we have Parker (dud), Non-Parker (works), and Anti-Parker (works even in ways not required by original requirements).

[@teiermyler4926]

@@PanduPoluan Gold

[@mati.benapezo]

@@PanduPoluan Is there the Anti Non-Parker?

[@aidentoman-sager5527]

@@mati.benapezo so it fails even in ways not required? that's just a regular Parker

[@skorp5677]

In fact so special that Euler got involved xD We should award puzzles with the 'Euler tried' award

[@XHappyKillerX]

The Parkeuler

[@XHappyKillerX]

Now that I think about it, I just noticed how the title "Euler Squares" must have been a deliberate reference to the Parker Square, nice!

[@NStripleseven]

Yeah, this should be a thing. xD

[@salerio61]

There's still loads of awards going for a similar idea of "Erdos tried" puzzles. You even get the choice of accepting the monetary award or a cheque signed by Erdos to frame in your study

[@metleon]

Instead of 'college try' we should change it to 'Euler try'.

[@rahulujjal8245]

The youngest of the ‘Euler’s Spoilers’ is no more. He was 103. Indian mathematical genius, Sharadchandra Shankar Shrikhande, who along with his mentor late RC Bose and their colleague late ET Parker disproved way back in 1959 an 18th century mathematical conjecture, passed away at Vijaywada on April 21, bringing curtains to a glorious chapter from the world of statistics and mathematics.

[@alveolate]

indian mathematicians are amazing... if anyone has a nice long documentary with a bunch of indian mathematicians, please link!

[@amritkumar7488]

@@alveolate there is a documentary on SC Shrikhande tho. I saw it at a college. I don't recall the name. Try googling.

[@GerSHAK]

+

[@aquibalamLUMOS]

Respect

[@GrandMoffTarkinsTeaDispenser]

Title: Euler Thumbnail: James Grime Me: visible excitement

[@screamsinrussian5773]

Euler didn't respond to their calls :(

[@sam08g16]

Hotel: Trivago

[@darkphoenix0808]

Is that the speaker's name? I honestly don't know

[@squidly1369]

@@darkphoenix0808 yes it is he is the best out here I guess I mean his accent presence is awesome 😎🔥

[@Triantalex]

??

[@drewnardolillo9670]

James Grime: Oiler Spoilers Me, an intellectual: Euler Speulers

[@ianmoore5502]

I cannot stop giggling. Thank you.

[@sean3533]

Squares in order of importance 1. Parker Square 2. Euler Square 3. The Square

[@RogueLich]

4. 2^2

[@squidly1369]

@@RogueLich lol😂

[@alihesham8167]

@@RogueLich 5. 1

[@mattiacapuano2022]

The ping pong letters and numbers are adorable

[@auroralong5437]

I thought that too until I saw their tiny white pupils and now I think I'll have nightmares....

[@mattiacapuano2022]

@@auroralong5437 damn, you're right

[@xan1455]

Yay but 4

[@MiserableMidnight]

Nooooooooo

[@marcushendriksen8415]

Watch #4, it looks like it's teabagging

[@itwasinthispositionerinoag7414]

5:45 I'm mostly here for the twerking 4

[@Just_A_Baryonyx]

i noticed that too

[@Son-Of-Gillean]

@@Just_A_Baryonyx "Noticed" = fapped

[@hamiltonianpathondodecahed5236]

ikr

[@Just_A_Baryonyx]

@@Son-Of-Gillean no....

[@leadnitrate2194]

it was in this positionerino agadmatorino Hey you're not commenting on Agadmator's recent videos. What's wrong? I really enjoyed your work during the MC Invitational.

[@ajharris3486]

This was one of the best Numberphiles in a while for me! James really knows how to give information succinctly and interestingly. Bravo, chaps!

[@psyboyo]

14:02 "Never arrange a ping-pong tournament with six team members" -- I first understood "with sixteen members", I went crazy! WTF?!? And then I turn on the subtitles.

[@methethpropbut8519]

3:33 I realise not only the rows columns and diagonals, but the four 2×2 sub-squares also have one of each rank and suit!

[@inigo8740]

That makes it even more like a sudoku! Hooray!

[@andersyu4464]

also the middle 2x2 square

[@prich0382]

Which surly means that the "difficult" 6 by 6 example surly could be solved by doing the sub 3x3 sub squares no?

[@gargravarr2]

It follows by definition, since the other 3 squares of the sub-square are on the row, column and diagonal of the corner square.

[@gabor6259]

Also the 4 corners.

[@cerwe8861]

James talks about something from Euler, can there be something better?

[@cerwe8861]

@Carey Hunt what?

[@Pro_Triforcer]

You mean there's a square named after some other mathematician? Sounds almost exciting, but not quite

[@kasajizo8963]

Yes, there is something better. Matt Parker talking about squares.

[@cerwe8861]

@@kasajizo8963 that's also cool, but like a Parker Square not perfect xD

[@cerwe8861]

@Carey Hunt Thanks random guy from the Internet

[@rensvanampting234]

I love how team member "4" is animated at 5:46

[@microwave856]

hes a little silly

[@WaterCarrier07]

I always love when different pronunciations clash like one is correcting the other straight away... “ohh it’s a Sudoku” ... “yes a sudoku”

[@howard5992]

and each time the word " sudoku " is repeated more emphasis can be placed upon that work in the sentence until it can become a very happy shouting match !

[@marshmelows]

That brown paper on Graham's number signed by the very own Ron Graham is just amazing! 0:25

[@benjaminlum5894]

Agreed! I'd like that to hang on my wall as well

[@jesusthroughmary]

Wow, I didn't even notice, that is too cool

[@henriccarlsson9052]

The first thing I noticed also :)

[@balinttoth9287]

"it's so difficult Euler got involved"

[@tombomb_com555]

It would have been nice to talk about the link between this and magic squares: say instead of AKQJ and 1234 we used two sets of 0123, and made them into the same arrangement, we could then read off each number as a two-digit number in base 4, then those would be a valid magic square (excluding diagonals) or we could add 1 to every number and it would still work. For a 3x3 example (since I know that one well), [21,00,12;02,11,20;10,22,02] (excuse the formatting) becomes [7,0,5;2,4,6;3,8,1] or [8,1,6;3,5,7;4,9,2] which is a magic square. This logic works for all sizes too.

[@HansLemurson]

Wait, so does that mean there are no 6x6 magic squares?

[@khaitomretro]

@@HansLemurson No. The assertion is wrong. There are 6x6 magic squares but no 6x6 magic squares that take that form. You always end up with a square that repeats one of the base 6 digits in the rows.

[@khaitomretro]

If you can construct a double Latin square then you can use that to create a magic square. Euler's methods for creating double Latin squares can be used to create forms of magic square but won't find all of them, just a subset.

[@user-mc4zu9qt8u]

When he described the puzzle, I paused it, got some paper and a pen, and figured it out. And I solved it, hooray! It really is like doing double sudoku, lol. Cheers for the interesting video and fun little puzzle, Numberphile :)

[@TheZotmeister]

Check out the 2016 United States Puzzle Championship :)

[@helenanevrayeva]

Haha, I was so intrigued so I pulled out a stack of cards for this 😁 I did AKDB first, then it was easy to rearrange for ♠️♥️♦️♣️. Enjoyed it thoroughly!

[@DrKaii]

3:14 you think James is sped up here, but actually this is his normal speed, the rest of the video is slowed down

[@jessicawang6558]

After so many years I still get a smile when I see James Grime

[@haricashravi3900]

This video is like a tribute to SS Shrikhande who was part of the "Euler's Spoilers" - a bunch of three people at UNC-CH who disproved Euler's generalisation of this problem - who sadly passed away on the day of the release of this video.

[@numberphile]

Amazing coincidence.

[@alveolate]

@@numberphile your video --killed him-- satisfied his lifelong ambition of getting obliquely referenced in a numberphile video.

[@Alex-xk6sx]

He passed away on April 21, 2020 (at age 102 at that!), not on the day this video was released, though the dates are truly close to each other 😉 May he rest in peace.

[@haricashravi3900]

@@Alex-xk6sx that's weird because I clearly remember seeing the news where SSS's death was reported and a few hours later this video released... Could it be possible they had reuploaded/changed the video later?

[@Alex-xk6sx]

@@haricashravi3900 Maybe? Waiting for Numberphile to chime in.

[@AbiGail-ok7fc]

Back in high school (late '70s, early '80s), our math teacher had a large, handmade, quilt hanging from one of the walls, with a 10 x 10 Euler square as the pattern. Him telling the story behind is was the first time I heard about Euler.

[@Freytana]

Grime's passion is always very enjoyable to listen to and watch

[@bryanstanley3866]

I like the framed brown paper for Graham's number hanging in the background.

[@DuCaDo003]

One of your best videos in quite a while. Really enjoyable. James really knows how to explain things. Thanks!

[@ralfoide]

Dr. James Grime is such a joy to listen and watch at. Always with a big smile. We need more enthusiastic people like him :)

[@mjswart73]

This lockdown really hasn't cramped the style of the animator. Full marks. I love it.

[@mathguy7480]

Can we talk about that 4 for a second?

[@michaelmatter1222]

4 really went 4 it

[@free_h2o142]

4 goodness sake...

[@proloycodes]

4 4 a second*

[@Stephen_Lafferty]

It's great to see James again - I feel like I haven't seen him in a NP video for ages!

[@numberphile]

Do you have notifications on for our videos? Bash that bell 🔔

[@adrianfletcher8963]

I love how you never cease to stop making innovative animations

[@jonopriestley9461]

Classic James. What a mad lad. We gotta have Numberphile live-streams some time.

[@justbetty7334]

I love NumberPhile! I watch it all the time. It's one of the only this getting me through lockdown! 😀

[@zinniaward8549]

Mood

[@esotericVideos]

I do too! I'm a little concerned that they don't seem to be too socially distanced in their videos though. I don't want any of my Maths friends to get sick.

[@ragnkja]

esotericVideos I’m sure it was filmed well before the lockdown.

[@deadlyore3986]

I love that you've got the paper from the video with Ron Graham hung up on the wall. RIP

[@Codricmon]

That four day tournament was the greatest event of my life - the first game on the second day was just the bomb!

[@vj_henke]

11:12 "To be fair, he was a proper mathematician. But he also checked every case." Shaking my head

[@jodfrut771]

5:40 that four was flapping his privates LOL

[@edelcorrallira]

Oh man this remind me of playing around with multidimension Karnaugh maps. I love this channel so much, thank you guys for keeping science available for all

[@stv3qbhxjnmmqbw835]

I love this guy. The excitement, the energy, everything

[@akshatmishra7110]

What a coincidence....just when the Indian Mathematician who debunked Euler's Theory passed away! P.S. - He died today at the age of 103! His name was Shrikhande !

[@numberphile]

Amazing coincidence!!!

[@zstanojevic9574]

@@numberphile Also, a video tributing Conway's departure is visibly missing..

[@WatchingMyLifeFlashB]

The video showed an Indian Raj Bose as completing it successfully in the 1950's, '54 I believe it was. This Strickhande was he in the '20s that were later disproven until Bose, or was Strickhande later?

[@TheAntibozo]

James Grime is such a wonderful communicator.

[@zivush3d]

great that numberphile keeps uploading turring the lockedown. love this video.

[@jimdevilbiss9125]

This was an excellent presentation as all of yours are. Having taught statistics for years I never thought of using this with setting an Experiment thank you.

[@rubenleejohnsen2037]

I see Euler and James grime in title, i click.

[@hazimans]

watching this while currently having in sudoku mood. I suddenly thought of this sudoku variant, 2 sudokus (normal sudoku and wordoku) in one grid following regular rules with the extra rules mention in this video (each cell must have a unique combination of a letter and a number) would be interesting tho (and hard)

[@MM-cr7dq]

What a great story - so many twists - a strange pattern - thanks James

[@Elitematt74]

This video was a mental rollercoaster ride

[@avi12]

14:04 Fs in the chat

[@hubert6943]

F

[@helloiamenergyman]

F

[@tlep2979]

F

[@hauphan917]

F

[@mementomori7160]

F

[@zaphodbeeblebrox5511]

This reminds me of "The Schoolgirl Problem Puzzle" : In a boarding school there are fifteen schoolgirls who always take their daily walks in groups of three. How can it be arranged so that each schoolgirl walks in a group with two different companions every day for a week (7 days)?

[@woodchuk1]

ZaphoD Beeblebrox Isn’t that an instance of a Steiner Triple system of order 15, where there would be (15*14)/6, or 35 triples?

[@rosiefay7283]

Let's give Kirkman due credit for this problem. Another Numberphile video, perhaps?

[@omikronweapon]

@@rosiefay7283 sounds like one for Cliff. He loves Euler and stuff about taking walks.

[@ghost_cipher]

James got me interested in teaching myself better math skills that have laid dormant for years.....BIG THANKS!!!!

[@kungfutuber]

Just completed one with each row, column and corner diagonal. It's also nice to see the centre 4 are also one of each, as is each corner, including many 4 place patterns like B1, C1, B4 & C4 for example! :)

[@aidanwalter2823]

thanks JAMES

[@macaroon_nuggets8008]

your first.

[@RecursiveTriforce]

In my head I'm just singing to the "ABC Song" by Jackson 5: A B C pair them with 1 2 3. A B C 1 2 3, that's how easy maths can be!

[@michaelmatter1222]

LoL

[@saikat93ify]

Amazing video ! Thanks for making it ! It infused my enthusiasm of Combinatorics !

[@larissaketterer8653]

thank you so much James Grime, finally I´ve understood it!

[@prasanttwo281]

I bet after it was disproven, Euler's viewers started using the term to describe anything that was given a go but had something wrong in it. As in, "Oh look at that square number magic square Matt Parker came up with, it's such an Euler square of a solution!"

[@emoluv54865]

I am a simple man ,I see James. I suddenly love math... Until the video ends.

[@abracadabra8501]

James grime is so engaging I love it when he’s in the videos

[@jalepezo]

I heard of the latin square design by learning about experimental design, nice to see math at work!

[@XHappyKillerX]

What a ride.

[@tuftman6092]

I got really into these a couple of years ago. and I found another type of puzzle that is also cool. It's basically the same except instead of an n by n grid with 1 of n items in each row and column you have a 2n by 2n grid with exactly 2 of each item in each row and column. I was trying to figure out how many different possibilities there are, but it's harder to compute than the euler squares.

[@OW0974]

The starter puzzle was something I randomly thought of on a bored Saturday and I couldn't find a single solution, thanks for solving it for me

[@johninwaynenewjersey5253]

Throughout history there have been teachers that, through a combination of their passion and understanding for the subject and the way they present it, make learning easy to digest. James Grime is one of those and I envy the students that have studied under him.

[@dragonfruitdana]

I’ve had a puzzle like this ever since I was a child, with colours and numbers instead of card values and suits. Never knew it was called an Euler square :)

[@G33v3s]

I reckon they should have called them “Speulers”

[@eyadfromthesky]

Dr. Grime videos are probably the easiest to follow. I wish my school had professors like him

[@ronniesims547]

The four corners also constitute a four card set, as do the central four cards and each four card quadrant, plus others. If you were given all these conditions to meet at the start, it would seem more difficult to solve, but actually makes it easier.

[@menpee]

Ping-pong player B did literally nothing and still won the tournament. Impressive.

[@sudheerthunga2155]

Omg!! James Grime!! (The earliest I've been)

[@nightworg]

What an amazing story. Loved it.

[@YG-ub4dk]

A James Grime video I haven't seen? Oh right it was posted only minutes ago!

[@LinkAranGalacticHero]

7:12 Number 2 in third place *logic*: wait, that's illegal!

[@waldokind]

Waltlab Channel Zero based?

[@esotericVideos]

It's charming that James has a framed bit of numberphile paper in his house.

[@renerpho]

And not just any framed bit, but the paper from one of Numberphile's most iconic videos, signed by Ronald Graham himself. :-)

[@98081]

I think that's Brady's house.

[@CdFMasterVideo]

I feel like those dancing letters and numbers are gonne stay stuck in my mind for quite a long time. I'm not sure whether I should complain about it.

[@EebstertheGreat]

In 2012, this channel uploaded a video about a "special magic square" that remains magic after rotation or reflection. But this video provides the explanation. It is really two orthogonal 4x4 Latin squares with the digits 1, 2, 5, and 8: one for the tens place and one for the units place. These digits rotate or reflect to give 1, 5, 2, and 8, respectively, so the Latin square property still holds. So the total of every row, column, and diagonal must be 1 + 2 + 5 + 8 = 16 for both the unit and tens digits, giving a total 160 + 16 = 176, invariant under reflection or rotation by 180 degrees.

[@TrappedinaBrain]

I'd be interested in a way of judging "how wrong" a square is, and then seeing how many 6 squares exist that are the "least wrong"

[@Arthur0000100]

But WHY doesn't 6x6 work??

[@renerpho]

An example of the strong law of small numbers (2 and 6 are pretty small). They sometimes do weird things that aren't representative of the general behaviour.

[@jacobnugent6895]

@Nhật Nam Trần I agree. why only six? I feel like if six doesn't work, then it should manifest itself again at some point on the number line, causing some multiple of 6 to not work either.

[@cleliaalbano2184]

This video is delightful!! 💓

[@DrKaii]

00:02 James' puzzles always get me going

[@pasunurusaivineeth3739]

8:44 THE CUNNING SMILE

[@GuyNamedSean]

Like a giddy toddler that just pulled a fast one on his dad. I'll never stop loving James's cheeky enthusiasm for maths problems.

[@mattiviljanen8109]

Now that was a rollercoaster!

[@RobinWootton]

A satisfying anecdote, enthusiastically told!

[@mattiviljanen8109]

12:45 I would love to have that thing hanging on my wall! Upvote for new merch!

[@manioqqqq]

↑

[@LaGuerre19]

James 👏 Grime 👏 makes 👏 my 👏 day 👏

[@rosiefay7283]

3:43 Beautiful! I see that you've done more than Euler asked for, because you also have all four suits and all four denominations in 1) each diagonal broken two & two 2) each 2x2 quadrant 3) the central 2x2, and the four corners 4) the corners of each 3x3 block. (To get utterly magic square nerdy: your square is "complete", in magic square jargon, because any two cards that are two diagonal steps from one another are either two major suits (spades, hearts) or two minor, and either two high cards (ace, king) or two low.)

[@VyTran-tk8jq]

James is back!

[@char7114]

I know probably no one really cares, but I solved it all on my own and I'm really proud.

[@jedagelijksebraintraining]

14:04 you better don't want a tournement with 12 members (A, B, C, D, E, F, 1, 2, 3, 4, 5, 6), not 16.

[@abhijiths5237]

I think he said 6 team members not 16

[@jedagelijksebraintraining]

@@abhijiths5237 6 players do work because that is 3x3. Sorry if i missheard.

[@andersbendsen5931]

@@jedagelijksebraintraining 3 by 3 is nine. What they said was 6 by 6 won't work.

[@brianmarco5873]

@Je dagelijkse braintraining *** wiskunde-puzzels 6 team members with 2 teams make 12. As opposed to 6 player with 2 teams of 3 team members. The ping pong tournament described had 2 teams. I also misheard it as 16 though.

[@howardchan3244]

@@abhijiths5237 That's make sense. I am confused for a minute thinking I do not understand the problem.

[@ciangrant3042]

Working this out with the ABCD1234 notation and then watching James fill out the exact same square i had written down was very satisfying

[@grutshohawk1174]

You are the best host of numberphile.💖💖

[@robinsparrow1618]

14:52 wait, *that* adam savage? or just someone coincidentally named adam savage?

[@HanabiraKage]

Adam Savage is a massive fan of Numberphile, so I wouldn't be surprised if it was the actual guy himself.

[@screamsinrussian5773]

Yes.

[@timothymckane6362]

"I reject your identity, and substitute my own."

[@omikronweapon]

I mean, the dude díd just recently do a video with Matt Parker.

[@michaelmatter1222]

Some people say it's him, I say it's a Myth :D

[@Robi2009]

3:15 - dr James turned into Mickey Mouse on helium xD

[@fpereira77]

Nothing restores the balance of the universe like a Numberphile video featuring Dr James Grime.

[@amazingkook143]

I got goosebumps the second he showed his first solution because it was the solution I came up with right after pausing the video to see if I could do.

[@OlafDoschke]

Matt Parker rehabilitated.

[@chairwood]

I dont like that droopy bit on the 4 :(

[@ratsukutsi]

I particularly enjoyed the organ sound effects for cards example

[@Shlungoidwungus]

It looks like you can handle this puzzle pretty easily by solving just for suits and just for types, making sure your solutions for both are not isomorphic to one another, then combining them into one grid. Edit: Just watched a bit later where he pretty much explicitly mentions that. My brain is on airplane mode.