More Numberphile featuring Ayliean - th-cam.com/video/PGuRmqpr6Oo/w-d-xo.html T-Shirts and merch based on the Kings Tours - numberphile.creator-spring.com/listing/symmetric-kings-tours-number
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.
it's. not the fact that a closed path exists with the King It's the fact that it' s a magic square But it' s not that surprising, given that for a path with a certain kind of symmetry, each square and its matching square with respect to 64 ( each square at step n, and its twin at step 64 - n) are chosen to be on the same line That' s for the vertical symmetry. Any path that has a vertical symmetry will always have the sum of lines equal to 260 But I. can't intuitively see why it's also working for the columns when you have a symmetry with respect to the horizontal axis
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!!
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.
UK English Tour is "two-uh" BUT tournament is "tore-nah-muhnt" US English Tour is "tore" BUT tournament is "terr-nah-muhnt" Both inconsistent; some overlap; English is messed up whomsoever(?) may be using it.
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.
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.
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.
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?
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 :)
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?
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.
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.
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..
The Bitcoin Halving is approaching and I have a suggestion for a video: Some cripto wallets require a set of 12 or 24 words as a recovery passphrase, that you must keep secret. Without those words, your asset is lost forever. What if you keep those words safe, but get incapacitated and no one knows that you have cripto assets? How can you get a set of 6 of your closest friends and family to share a backup of those words in a way that not a single individual, nor a pair of two people would have access to all the words, but any combination of 3 people could unlock your assets (in case some of them loses their copy)? What that arrangement would be? Which words you should tell each person?
More Numberphile featuring Ayliean - th-cam.com/video/PGuRmqpr6Oo/w-d-xo.html
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!
"Who would call that a magic square?"
That's savage 😂
A parker would!
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.
1:31 Parker Square spotted!
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.
it's. not the fact that a closed path exists with the King
It's the fact that it' s a magic square
But it' s not that surprising, given that for a path with a certain kind of symmetry, each square and its matching square with respect to 64 ( each square at step n, and its twin at step 64 - n) are chosen to be on the same line
That' s for the vertical symmetry. Any path that has a vertical symmetry will always have the sum of lines equal to 260
But I. can't intuitively see why it's also working for the columns when you have a symmetry with respect to the horizontal axis
Okay, someone can revoke a point off my math nerd card, I did not know that a King's Tour always makes a Magic Square. That's pretty sweet
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
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!!
Massive shout out to Pete for the outstanding graphics!
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
Guy called Pete: "You rock".
Your mom rocks
Thanks Pete ❤ 11:18
1:45 It's called Parker Square
Thanks just upgraded my phones unlock pattern !
📱🔓👍
I absolutely adore Ayliean MacDonald!
I sometimes sit for hours making art by methods she's shown on Numberphile and her own channel.
😊
"It's even cooler! If you look at the diagonals... April Fools!"
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.
chess, magic squares and beautiful art... lovely combination!
Thanks Pete
That rebelious squint smirk is my favorite
3:41 The Awani Kumar Magic Knight's Tour reminds me of the Hilbert's Curve
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.
That is super cool! Thanks for sharing! 👏
A knights tour on a Mobius Strip.
That's it. That's the most perplexing thing I've ever seen.
This is a visually beautiful video. Well done to the subject and the photographer.
Some nice potential tattoo designs for Ayliean here! Love the 3D ones at the end!
Thanks for the animations Pete :)
The last one looks like a DNA double helix. Blew my mind bro.
I know this wouldn't be a magic square, but the most obvious king's tour in the first place is the "snake path."
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".
UK English
Tour is "two-uh" BUT tournament is "tore-nah-muhnt"
US English
Tour is "tore" BUT tournament is "terr-nah-muhnt"
Both inconsistent; some overlap; English is messed up whomsoever(?) may be using it.
Excited about the upcoming Parker Magic Tour
for someone who loves both maths and chess, this is a win video
This episode was extra magical, thank you!
I love these math videos that are creating beautiful shapes, like this one and the one tile discovery
I chatted with Ayliean for 42 seconds in London last year. Highlight of my vacation.
3:05 I immediately thought of tiling in the pattern of a Hilbert curve
I bet these tours would look especially nice as Bezier curves.
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.
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
Loved this.
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
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.
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.
1:30 catching strays 😂
Ayliean is a gem!
Matt Parker tries every year different method to calculate Pi, still he will be remembered for Parker Square 🤷♂️
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.
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.
Ayliean and chess? Oh this will be an amazing episode!
It took a while but I eventually managed to successfully achieve a tour for every type of chess piece on a 1x1 board!
Are there any underlying properties with the knot being made with this method?
I've been watching since the original Parker Square. It was very funny to see it referenced again.
6:15 it is not clear why this wouldn't change the row sums
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?
Love that flash of the Parker Square
Is there someplace online where we can view pictures of all the Knight's Tours and King's Tours?
Cool thing 😎 these Celtic patterns had some mathematical connection
Now I want to make a belt and some border wallpaper with King's tour patterns.
I personally like "dizzy king tour": where king not allowed make move in the same direction twice in the row.
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 😅
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.
Your makeup looks so nice! Also thanks for the cool math knowledge
Thank you ☺️✨
Yeah..like an artwork
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.
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?
How about a double bishops' tour?
Kinda boring I think
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.
3:33 could you make this in 3D‽
10:44 Cliff Stoll entered the chat
My tours with other pieces ran into problems when I got to bishops.
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 :)
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?
B2 looks great.
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.
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.
The math speaks for itself.
I have collected these patterns as knots
That's just fascinating.
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!
She shared the secret quite early on in the video! Is she sure we are her favorite kind of people????
I want those knight tour bracelets!
Parker Square spotted in the wild 😂
I wonder if they noticed the kings tours-like patterns on the wooden wall behind them…
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 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).
What about the bishop? Does he get a magic tour?
⏺ graphic design/animation appreciation button!
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...
what if you made a new piece with its own moveset?
Wait, there's another Perth??
Remember kids, it's 'white on the right'. 😊
This is mathematical wizardry 🧙
Will we ever get to meet Pete? (It's not me by the way)
nepo and dubov likes this video...
The Bitcoin Halving is approaching and I have a suggestion for a video:
Some cripto wallets require a set of 12 or 24 words as a recovery passphrase, that you must keep secret. Without those words, your asset is lost forever. What if you keep those words safe, but get incapacitated and no one knows that you have cripto assets? How can you get a set of 6 of your closest friends and family to share a backup of those words in a way that not a single individual, nor a pair of two people would have access to all the words, but any combination of 3 people could unlock your assets (in case some of them loses their copy)? What that arrangement would be? Which words you should tell each person?
Nice house Ayliean has got! 😉
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...
Nice bit of -sunshade- fun shade thrown at Matt 1:32 LOL
But how many of them form the S that everyone seemed to collectively draw in school??
Gonna assume the maths behind pawn's tours is pretty dull ;)
Only till it becomes a queen, and then it just zips around the rest of the board.
Parker Knight Tour