Magic Chess Tours (with Knights and Kings) - Numberphile

Ayliean MacDonald shows how KNIGHTS and KINGS can create MAGIC SQUARES on chess boards. More links & stuff in full description below ↓↓↓
This video features Ayliean MacDonald... More of her at / ayliean
And linktr.ee/Ayliean
More Numberphile featuring Ayliean - • Key to the Tower of Ha...
We created some T-Shirts and merch based on the Kings Tours - numberphile.creator-spring.co...
The Trapped Knight - • The Trapped Knight - N...
Knights Tour - • Knight's Tour - Number...
Parker Square - • The Parker Square - Nu...
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[@numberphile]

More Numberphile featuring Ayliean - kzbin.info/www/bejne/hnjYg6CnpddphdE T-Shirts and merch based on the Kings Tours - numberphile.creator-spring.com/listing/symmetric-kings-tours-number

[@AlSuChess]

This is really cool when you see the pattern on the board like this! Thank you for sharing!

[@the_blue_chicken]

The Parker square still being referenced today is very funny

[@volodyadykun6490]

There was development in the story not so long ago

[@aryst0krat]

@@volodyadykun6490 Oh?

[@racecarrik]

Silly goose, why would a mathematical law not be referenced?

[@eryqeryq]

Poor Matt tho 😢

[@cartatowegs5080]

​@volodyadykun6490 you can't just leave us hanging.

[@ericgoldman7533]

"Who would call that a magic square?" That's savage 😂

[@topherthe11th23]

I've been working on a Bishop's Tour that hits all 64 squares for 25 years, but haven't succeeded yet.

[@shinobu5359]

Anything's possible! Don't give up!

[@JohnSmith-nx7zj]

When you’ve cracked that I think you should work on the pawn’s tour.

[@N.I.R.A.T.I.A.S.]

😂 keep at it bro

[@JustAnotherCommenter]

Don't try Rook's tour. I think it's too straightforward.

[@mathijs58]

Did you try doing it on a Möbius board?

[@brianlane723]

A Parker knight's tour on a Klein bottle that sums to -1/12. The ultimate Numberphile video.

[@harmanpreetsingh7848]

But the path is first passed through an Enigma

[@deliciousrose]

1:31 Parker Square spotted!

[@user-hr7po5tn5i]

Thanks just upgraded my phones unlock pattern ! 📱🔓👍

[@noahblack914]

9:04 Knight's Tours almost _have_ to be more awesome. There's nothing surprising about a piece that moves 1 space at a time being able to visit every square. The weird movement of the Knight is what makes it interesting.

[@U014B]

Exactly. It's the extra restriction on the Knight that makes it so much more impressive.

[@TheArtOfBeingANerd]

I think the fact the a magic square can be formed by each number adjacent to the previous is pretty amazing.

[@DeclanMBrennan]

Guy called Pete: "You rock".

[@thenoobalmighty8790]

Your mom rocks

[@simonf8370]

Massive shout out to Pete for the outstanding graphics!

[@iseriver3982]

0:38 looks like a Nepo v Dubov game 😂

[@prathamesh413]

Waiting to see how many will get this reference

[@Matthew-bu7fg]

Hahaha wow very niche reference

[@Filipnalepa]

I know who are Jan and Danila, but I don't know which game itiis about.

[@jeronbaxter]

Imagine 3 fold repetition of knights tour.

[@I_am_Itay]

Knights go brrr ​@@Filipnalepa

[@Sons1717]

I love her comment on obsessions of drawing these mathematical objects! I'm a postdoc in theoretical physics, and I definitely questioned myself multiple times in the past, "Do I actually like physics, or do I just like drawing shapes?". It's really nice to see someone who emphasizes the same sentiment!!

[@OneTrueBadShoe]

I absolutely adore Ayliean MacDonald! I sometimes sit for hours making art by methods she's shown on Numberphile and her own channel.

[@SeanKennedy]

Thanks Pete ❤ 11:18

[@christopherpellerito5883]

I like how "tour" comes out as "tewer" in Ayliean's Scottish lilt. By the end of the video, Brady is also calling it a "tewer."

[@TomDarlington]

How do you pronounce it?!

[@Nightriser271828]

I pronounce it "toor".

[@Axacqk]

"It's even cooler! If you look at the diagonals... April Fools!"

[@IAmTheGreekMind]

Thanks Pete

[@EmilioBPedrollo]

1:45 It's called Parker Square

[@wyattstevens8574]

I know this wouldn't be a magic square, but the most obvious king's tour in the first place is the "snake path."

[@Matthew-bu7fg]

chess, magic squares and beautiful art... lovely combination!

[@ChessforCharity]

That is super cool! Thanks for sharing! 👏

[@IvanDobski]

Some nice potential tattoo designs for Ayliean here! Love the 3D ones at the end!

[@SquareWaveHeaven]

9:49 Look at them... they're having the time of their lives together... and you're just gonna have to learn to accept that.

[@BooleanDisorder]

I do accept and love them both. Harmony. ❤

[@LimeGreenTeknii]

Relationship goals: me and my partner hopping wildly on an 8x8 grid in L shapes.

[@Zentusichernun]

A knights tour on a Mobius Strip. That's it. That's the most perplexing thing I've ever seen.

[@paulvanegeren1227]

This is a visually beautiful video. Well done to the subject and the photographer.

[@AroundTheBlockAgain]

This episode was extra magical, thank you!

[@AgentM124]

Thanks for the animations Pete :)

[@Censeo]

I love these math videos that are creating beautiful shapes, like this one and the one tile discovery

[@sngosne]

for someone who loves both maths and chess, this is a win video

[@IrishEye]

Loved this.

[@likebot.]

Nice bit of -sunshade- fun shade thrown at Matt 1:32 LOL

[@KilgoreTroutAsf]

Excited about the upcoming Parker Magic Tour

[@yoram_snir]

Matt Parker tries every year different method to calculate Pi, still he will be remembered for Parker Square 🤷‍♂️

[@KaushikAdhikari]

I saw Ayliean, I clicked ASAP

[@Ayliean]

Aww thanks 🥰

[@lessgoofyone]

This channel's maths crush! 😅​@@Ayliean

[@MeriaDuck]

Surely the room with those patterns on the walls was deliberately chosen. ❤ Ayliean

[@user-et5ct1dk6f]

9:49 this I find very similar to that 'synchronously dancing bears' gif. Probably cuz they both have the same pace of movements and also the angle of view.

[@NickCombs]

I bet these tours would look especially nice as Bezier curves.

[@walterfristoe4643]

More than 25 years ago, I became somewhat entranced with knight's tours, and composed a few dozen of them that were very beautiful. I concentrated on the symmetrical ones, because I was looking for beauty. I even made a chessboard of knight's tours, which used 32 tours twice, mirroring each other. Each square of the chessboard was 2 inches, so the whole thing was 16 square inches. And it was a closed tour. I also made what I called modular tours, dividing the board into sections and then connecting the sections. It was loads of fun to play with something I had read about 50 years ago! 🐴

[@adipy8912]

On the sponsor screen before the video recommendations i heard Neil's beautiful voice. I miss his sequence videos so much. Hope he return some day

[@gillfortytwo]

Anyone else notice that the 12x12 magic and semimagic knight's tours follow space filling curves? Super cool the fully magic one is a Hilbert curve, and that's why it translates up.

[@PatrickPease]

That rebelious squint smirk is my favorite

[@marklonergan3898]

I think it's funny that you gave an example of a closed one before an open one, given that the closed one IS an open one 1 move before you close it.

[@frankharr9466]

That's just fascinating.

[@jesuizanmich]

7:10 was gonna say, that looks exactly like something you'd find in the Book of Kells, a very old church, or weaved into an aran jumper.

[@yeoman588]

Is there someplace online where we can view pictures of all the Knight's Tours and King's Tours?

[@deject]

Ayliean is a gem!

[@_rlb]

I chatted with Ayliean for 42 seconds in London last year. Highlight of my vacation.

[@estherstreet4582]

Big fan of the intersection of numberphile videos and puzzles from professor layton games that traumatised me as a kid. Eight queens next?

[@numberphile]

I think we’ve done that.

[@mikew6644]

1:30 catching strays 😂

[@emulationemperor8924]

I've been watching since the original Parker Square. It was very funny to see it referenced again.

[@serta5727]

Cool thing 😎 these Celtic patterns had some mathematical connection

[@JefferyMewtamer]

Are any of the magic, symmetric King's tours pan diagonally magic? Also, I find myself wondering about Queen's tours where you forbid King's moves and require alternation between Bishop and rook moves. Are any magic and symmetric... and how big can one make the smallest step and still complete a queen's tour? And what about tours using non-standard chess pieces or on a hex or triangular grid?

[@luketurner314]

3:05 I immediately thought of tiling in the pattern of a Hilbert curve

[@theassailer18]

What are the RL applications to these tours besides it's pleasing to look at?

[@theguyshetellsunottoworryabout]

Nothing more for the moment I think Centuries ago mathematicians were playing with numbers developing what we call number theory today, ignoring that few centuries later we would use them for the security and cryptography of your credit card, or write the code source of your mobile phone or computer Soooooooo nothing for the moment I think, maybe one day it will have some And if not that's still beautiful enough to be published in my opinion

[@nekrataali]

Chess is still unsolved. Specific board states of chess have been solved, but starting from White's turn 1, we're still mostly in the dark. Given there are more possible games of chess than there are atoms in the observable universe, chess is excellent for training computers and testing their limits. Research into topics like this could help us eventually solve chess, which would also result in solving problems using large or infinite numbers. If you can prove specific moves always leads to a win, you'd also be proving stuff about 10⁷⁸. It'd be like proving the last 10 digits of pi.

[@brololler]

Are there any underlying properties with the knot being made with this method?

[@Finn-OskarMikkelsen]

Love that flash of the Parker Square

[@Phymacss]

Ayliean and chess? Oh this will be an amazing episode!

[@rmsgrey]

Obviously you can start a closed tour from any square (you can start it at any point on the entire loop) but are there open tours that start at any given square? For a knight's tour, you obviously have to alternate colours, but if you pick any white square and any black square, is there always a tour that starts at one and ends at the other? I'm sure the answers are known, but they're still obvious questions to ask :)

[@sail2temporary]

The last one looks like a DNA double helix. Blew my mind bro.

[@Z0M8I3D]

IDK, seems like king's tours & Celtic knots naturally divide a space with a line of connections. Sounds like a way to encrypt with complexity.

[@gwalla]

The patterns made by the magic king's tours make me think of knot theory. Also, I wonder if the fact that magic tours are possible on 8x8 with a king but not a knight has anything directly to do with the fact that a knight is strictly color-switching and a king isn't? Would you get the same results as the king with a piece with the same number of possible moves that is similarly divided between colorbound and color-switching, like a wazir+alfil?

[@SwordQuake2]

B2 looks great.

[@somewinner8229]

I wish there was an option to see a pawn's tour... which promotes to a knight when it reaches the end of the board 😅

[@flamencoprof]

Now I want to make a belt and some border wallpaper with King's tour patterns.

[@cabbageman]

The magic knights tours seem to me to resemble a Hilbert curve shape. I wonder if this is a mathematical connection there. Both space filling curves?

[@Holdem17]

The math speaks for itself.

[@zecuse]

Just wanted to throw out there that these tours can be represented as a Hamiltonian path. Finding new tours could be done by changing which 2 vertexes connect to each other and then working to remake a new Hamiltonian path from that.

[@SaveSoilSaveSoil]

I want those knight tour bracelets!

[@stephanemami]

Could we invent other moves? Could it work? Moves you don't find in chess, like 3-1. Fascinating as usual!

[@pierreabbat6157]

How about a double bishops' tour?

[@jiaan100]

Kinda boring I think

[@thisnthat3530]

It took a while but I eventually managed to successfully achieve a tour for every type of chess piece on a 1x1 board!

[@Rubrickety]

Nice house Ayliean has got! 😉

[@EebstertheGreat]

2:54 In fact, there are no knight's tours _at all_ on a 4x4 board, let alone magic knight's tours. In general, there are clearly no knight's tours on 1xn or 2xn boards (except 1x1), and it turns out there are also no tours on 3x3, 3x5, 3x6, or 4x4 boards.

[@r75shell]

I personally like "dizzy king tour": where king not allowed make move in the same direction twice in the row.

[@Neptoid]

I have collected these patterns as knots

[@mathijs58]

She shared the secret quite early on in the video! Is she sure we are her favorite kind of people????

[@lafcursiax]

It's a Magical Chivalry Tour! (Roll up!)

[@ZoonCrypticon]

Could you invent a new 10x10 chess game with a special figurine (x4 + 4 extra pawns) with a special movement as well?

[@landsgevaer]

Fairy chess has plenty...

[@antonholt3236]

Your makeup looks so nice! Also thanks for the cool math knowledge

[@Ayliean]

Thank you ☺️✨

[@ferretyluv]

Do the diagonals really all have to look like that? Why not just have a big Snake-style squiggle? Just go horizontally over each row.

[@chuckgaydos5387]

My tours with other pieces ran into problems when I got to bishops.

[@Nick-Lab]

Shots fired! Lol

[@faxhandle9715]

I’ve been working on the Pawn’s Tour for the last 30 years. What the heck? 😂😂☠️☠️

[@bluerizlagirl]

It speeds up a lot after the seventh move .....

[@iabervon]

You should try a bishop's tour. I've been working on that, and it's going great! I'm almost half done, and no problems so far...

[@wzdew]

Doesn't make a lot of difference in this context (though it definitely does in chess), but the bottom right square should be a light square if the board is set up correctly.

[@coconuts2513]

Parker Square spotted in the wild 😂

[@JamesGuillochon]

I wonder if they noticed the kings tours-like patterns on the wooden wall behind them…

[@obiwanpez]

Yay, Pete!

[@Darilon12]

What about the bishop? Does he get a magic tour?

[@arneperschel]

⏺ graphic design/animation appreciation button!

[@robertolson7304]

So basicaly it a multiplication chart diagonal. Set zero (1,3,5,7,9) set one (1,3,5,7,9), set two ( 1,3,5,7,9) set three ( 1,3,5,7,9). It just repeats over and over. Now for 8. set zero (1,5,3,1) set one (1,3,5,1). That my friends is D/C and A/C. How that effects your outcomes is up to the user. In this type of system 8 x 8 will never be read diagonally as AC. If 8x8 was AC what would 12 x 12 be? Here you go.1,9,7,5,3,1 and 1,3,5,7,9,1. Its D/C.. it has the same frequency as diagonal multiplication table..

[@robertolson7304]

Resonance frequency..

[@serinadersiova1599]

nepo and dubov likes this video...

[@robinbrowne5419]

This is mathematical wizardry 🧙

[@coloneldookie7222]

But what about Return to Zork's "Survivor" board game? There's no real data to find online of PvP games, just player vs AI (the game).

[@RichardWinskill]

Gonna assume the maths behind pawn's tours is pretty dull ;)

[@bluerizlagirl]

Only till it becomes a queen, and then it just zips around the rest of the board.

[@iseriver3982]

Remember kids, it's 'white on the right'. 😊

[@lingannong3102]

Perth represent!

[@johnrichardson7629]

Parker Knight Tour

[@boerhae]

what if you made a new piece with its own moveset?

[@bscutajar]

6:15 it is not clear why this wouldn't change the row sums

[@RadioactiveLobster]

But how many of them form the S that everyone seemed to collectively draw in school??