Magic Chess Tours (with Knights and Kings) - Numberphile
ฝัง
- เผยแพร่เมื่อ 31 มี.ค. 2024
- 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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Video by Brady Haran and Pete McPartlan
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T-Shirts and merch based on the Kings Tours - numberphile.creator-spring.com/listing/symmetric-kings-tours-number
This is really cool when you see the pattern on the board like this! Thank you for sharing!
The Parker square still being referenced today is very funny
There was development in the story not so long ago
@@volodyadykun6490 Oh?
Silly goose, why would a mathematical law not be referenced?
Poor Matt tho 😢
@volodyadykun6490 you can't just leave us hanging.
"Who would call that a magic square?"
That's savage 😂
I've been working on a Bishop's Tour that hits all 64 squares for 25 years, but haven't succeeded yet.
Anything's possible! Don't give up!
When you’ve cracked that I think you should work on the pawn’s tour.
😂 keep at it bro
Don't try Rook's tour. I think it's too straightforward.
Did you try doing it on a Möbius board?
A Parker knight's tour on a Klein bottle that sums to -1/12. The ultimate Numberphile video.
But the path is first passed through an Enigma
1:31 Parker Square spotted!
Thanks just upgraded my phones unlock pattern !
📱🔓👍
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.
Exactly. It's the extra restriction on the Knight that makes it so much more impressive.
I think the fact the a magic square can be formed by each number adjacent to the previous is pretty amazing.
Massive shout out to Pete for the outstanding graphics!
Guy called Pete: "You rock".
Your mom rocks
I absolutely adore Ayliean MacDonald!
I sometimes sit for hours making art by methods she's shown on Numberphile and her own channel.
Thanks Pete ❤ 11:18
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!!
0:38 looks like a Nepo v Dubov game 😂
Waiting to see how many will get this reference
Hahaha wow very niche reference
I know who are Jan and Danila, but I don't know which game itiis about.
Imagine 3 fold repetition of knights tour.
Knights go brrr @@Filipnalepa
1:45 It's called Parker Square
"It's even cooler! If you look at the diagonals... April Fools!"
Thanks Pete
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."
How do you pronounce it?!
I pronounce it "toor".
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.
I do accept and love them both. Harmony. ❤
Relationship goals: me and my partner hopping wildly on an 8x8 grid in L shapes.
I know this wouldn't be a magic square, but the most obvious king's tour in the first place is the "snake path."
chess, magic squares and beautiful art... lovely combination!
That is super cool! Thanks for sharing! 👏
This is a visually beautiful video. Well done to the subject and the photographer.
for someone who loves both maths and chess, this is a win video
Some nice potential tattoo designs for Ayliean here! Love the 3D ones at the end!
Thanks for the animations Pete :)
A knights tour on a Mobius Strip.
That's it. That's the most perplexing thing I've ever seen.
Nice bit of -sunshade- fun shade thrown at Matt 1:32 LOL
This episode was extra magical, thank you!
Excited about the upcoming Parker Magic Tour
I love these math videos that are creating beautiful shapes, like this one and the one tile discovery
That rebelious squint smirk is my favorite
Loved this.
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.
I bet these tours would look especially nice as Bezier curves.
Matt Parker tries every year different method to calculate Pi, still he will be remembered for Parker Square 🤷♂️
Surely the room with those patterns on the walls was deliberately chosen. ❤ Ayliean
I saw Ayliean, I clicked ASAP
Aww thanks 🥰
This channel's maths crush! 😅@@Ayliean
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.
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! 🐴
Love that flash of the Parker Square
3:05 I immediately thought of tiling in the pattern of a Hilbert curve
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.
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
1:30 catching strays 😂
That's just fascinating.
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.
The last one looks like a DNA double helix. Blew my mind bro.
Ayliean is a gem!
Is there someplace online where we can view pictures of all the Knight's Tours and King's Tours?
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?
I chatted with Ayliean for 42 seconds in London last year. Highlight of my vacation.
Big fan of the intersection of numberphile videos and puzzles from professor layton games that traumatised me as a kid. Eight queens next?
I think we’ve done that.
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.
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 :)
I've been watching since the original Parker Square. It was very funny to see it referenced again.
Are there any underlying properties with the knot being made with this method?
Cool thing 😎 these Celtic patterns had some mathematical connection
Ayliean and chess? Oh this will be an amazing episode!
B2 looks great.
Now I want to make a belt and some border wallpaper with King's tour patterns.
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?
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.
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?
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.
The math speaks for itself.
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 😅
What are the RL applications to these tours besides it's pleasing to look at?
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
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.
I have collected these patterns as knots
I want those knight tour bracelets!
It took a while but I eventually managed to successfully achieve a tour for every type of chess piece on a 1x1 board!
How about a double bishops' tour?
Kinda boring I think
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.
I personally like "dizzy king tour": where king not allowed make move in the same direction twice in the row.
My tours with other pieces ran into problems when I got to bishops.
Your makeup looks so nice! Also thanks for the cool math knowledge
Thank you ☺️✨
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.
It's a Magical Chivalry Tour! (Roll up!)
Could we invent other moves? Could it work? Moves you don't find in chess, like 3-1. Fascinating as usual!
Nice house Ayliean has got! 😉
Could you invent a new 10x10 chess game with a special figurine (x4 + 4 extra pawns) with a special movement as well?
Fairy chess has plenty...
She shared the secret quite early on in the video! Is she sure we are her favorite kind of people????
What about the bishop? Does he get a magic tour?
Shots fired! Lol
⏺ graphic design/animation appreciation button!
Parker Square spotted in the wild 😂
Yay, Pete!
I wonder if they noticed the kings tours-like patterns on the wooden wall behind them…
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).
nepo and dubov likes this video...
6:15 it is not clear why this wouldn't change the row sums
This is mathematical wizardry 🧙
Wait, there's another Perth??
Parker Knight Tour
I’ve been working on the Pawn’s Tour for the last 30 years. What the heck? 😂😂☠️☠️
It speeds up a lot after the seventh move .....
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...
what if you made a new piece with its own moveset?
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..
Resonance frequency..
But how many of them form the S that everyone seemed to collectively draw in school??
3:33 could you make this in 3D‽
10:44 Cliff Stoll entered the chat
Remember kids, it's 'white on the right'. 😊