The Trapped Knight - Numberphile
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- Опубліковано 15 чер 2024
- Featuring Neil Sloane... Check out Brilliant (and get 20% off their premium service): brilliant.org/numberphile (sponsor)
More links & stuff in full description below ↓↓↓
Trapped Knight T-Shirt: teespring.com/en-GB/numberphi...
Neil Sloane is creator of the On-Line Encyclopedia of Integer Sequences: oeis.org
For more on the sequences and diagrams in this video, you can start at: oeis.org/A316667
The sequence was submitted by and diagrams by Daniël Karssen.
More chess-related videos on Numberphile: bit.ly/chess_numberphile
More from Neil Sloane: bit.ly/Sloane_Numberphile
Discuss this video on Brady's subreddit: redd.it/ajeujk
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumberphile
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A T-Shirt based on the trapped Knight Tour --- t.co/bpqqnmUjM6
If you look on a video that was "why does youtube views freeze at 301 views. And you said " The views start counting back up again after a day or two but it's been 6 years.
It's really a shame that you've let your channel get political by supporting Patreon. A big thumbs down.
@@paulthompson9668
And how is that?
I would buy it, but I don't like the feel of American made shirts
@@paulthompson9668 they kinda have to with UA-cam giving the finger to every content creator
I'm currently testing the king, currently at square 14,456,283 and I'm taking a break. I suspect it's infinite but I'll keep trying.
Never give up
It can simply follow the spiral, right?
@@TheModestRat r/woosh
@@TheModestRat yeah it can (it was just a joke)
oh my comment made me think that maybe different tiling could result in more interesting results
That many knight moves in the opening loses tempo.
I just claim stalemate after the first thousand (I wanted to see if they'd get bored first...)
*cough* Carlsen-Caruana 2018 Game 6 *cough*
@@DraoxxMusic LOL
*tempi
Kramnik approves
I think this is a clear indication Magnus Carlsen will remain world champion until 2084.
Lol
He'll use the trapped knight to draw his games.
🤣
Caruana is gonna win next time
I'll admit, I'm a bit of a Carlsen fan. That said, I don't see why Caruana would win next championship. He definitely could win, but he can't be favored, and is probably quite the underdog - especially now before the candidates.
"The Trapped Knight" sounds like a scrapped Dark Souls boss.
Scrapped? Hah! You mean you didn't find him? Noob! ;)
@D.O.A. Thou shall perish in the twilight of Anor Londo.
Isn't he right after Abyssal Jeremy?
I thought it was a Batman graphic novel.
though "trapped" isn't quite as fantastically an adjective for Dark Souls :P They'd probably go for "Imprisoned" or "Confined"
I tried this with a pawn.
It's less interesting.
Nice!
I wonder if the king gets trapped too
@@dennismuller1141 hmmmmmmmmm
Dennis Müller The king and rook would just follow the number spiral. They will never get trapped.
Until it reaches the eighth rank and becomes a queen :)
not so fast i am still filling up the infinite board with numbers....
Let us know when you’ve got that finished.
Bro you can stop the video, dont panic
my infinite board is still shipping. :(
same, hold up guys
@ktbDash No, it's 1,000,000^900 (= 10^5400) in the long system or 1,000^(900 + 1) (=10^2703) in the (clearly broken!) short system that English has chosen to use.
What happens if you mark 2084 as already visited before you start the game? Will it still get trapped somewhere else?
That’s an interesting question???
@@numberphile where to find a simulation of this?
cool! run it and find the next one where it gets stuck... then mark that one as visited too and run it again.. and so on. That would be an interesting series!
@@Some.username.idk.0 Wait a few weeks and it will be on the coding train channel.
It will just get stuck one tile before it. Did I get your question wrong?
That pattern is absolutely spectacular. Somehow both natural and artificial in appearance. It's like a cyborg pattern.
I think you'd get an interesting infinite sequence if you posited a "knight with foresight". Basically, follow the same rules, except if you are ever trapped, scrap the last move from the sequence, mark the square that would have trapped you as "trap" and go to the next smallest square you can move to.
This would have to create an infinite sequence of knight moves and it would also create a complementary sequence (presumably infinite) of "trap squares".
Interesting idea!
Late reply I know but I wonder if that sequence actually would be infinite? Is one level of foresight enough, or will the knight eventually be "double trapped"? That is, a position where the only available position is a trap position. Given the large number of holes that were left, it seems like this could potentially happen. Thinking on it further, what level of foresight would be needed to never be trapped, or is there no finite amount of foresight that will never result in being trapped?
in that case just move back 2 steps. but that also brings along another question, what if the knight has extinguished ALL the possibilities EVER. from the starting of the game, to all the different locations it got trapped at. what to do then? @@thundersheild926
This guy's my new favorite numberphile guest.
More to come with Neil.
4:46 deal with it
Snoop dog!
OOH OOOOH 🔥🔥😂
"you love it don't you..."
"i do! i do haha! ha ha.. heh.. ye.."
That's just the (1,2) knight. I wonder which (x,y) knights get trapped and which don't. You can create a 'yes/no' graph with all of them. It would be interesting to see. Thank you.
Or the steps it takes before getting trapped, or the maximum number reached...
Well the (0,1) knight works.
2:24 OH NO THAT'S A TROLLFACE!
*Thanks, now I can't unsee it*
😂amazing bro just amazing
You mad?
24 January, the 'trapped Knight' incident
But if the knight is on a parker square?
A Parker Spiral?
Well from the thumbnail pic, it seems more likely the Knight is caught in Parker's concatenation.
the proof is left as an exercise to the viewer
That would be a parker square of an infinite chessboard.
Finite will do
I have no moves, so I must scream
You again
5th like on a Justin Y comment. Do I get a medal?
u subscribe this too?
Huh, you must've actually watched the video to make that one
Tell me, do you have a life?
I love this video, and I love this man; and I especially love that he's wearing a Jimi Hendrix t-shirt.
You forgot about loving numbers!
@@mateussilva635 Numbers are concrete and consistent, so much better than letters.
There is too much confusion...
@@imumsi I can't get no relief...
I would really appreciate a video specifically on the question: "Is pi essentially related to circles or is that just one of pi's aspects?" Is there always a circle hiding behind any occurrence of pi or not?
If you watch 3b1b's videos on infinite series that have pi in their solution - he says that there is always a circle hiding in there somewhere.
@@polyacov_yuryI would have to strongly disagree with 3b1b on that. There are ways of deriving the sine/cosine functions independent of circles; actually, it is entirely independent of geometry.
The association of pi with circles is an example of "if all you have is a hammer everything looks like a nail." Pi is more actually described as the cycle constant, but because humans are hardwired to think in terms of 2D/3D shapes and patterns, we try to tie every cycle back to the first simplest cycle we understand which is the circle.
@@amirabudubai2279 How is it any more correct to use the term "cycle" instead of "circle"? Both are analogous interpretations of the mathematics. We use representations, analogies, to understand and explain maths... how is this a problem?
@@amirabudubai2279 i dont think anyone would come up with pi if they didnt already know of it from circles. Pi is inextricably tied to a circle.
@@scottwhitman9868 No need to go into hypotheticals here, people have independently found Pi plenty of times without trying. It is closely tied to primes though the zeta function and it shows up in most complex exponential. Pi is also just as important to triangles as it is to circles. Sine and Cosine are also unavoidable parts of solving partial differential equations with pop up everywhere as a result of the principle of locality.
If there somehow existed a 1D world with intelligent life, there physicist would discover Pi because it would be a constant in every physical law. Just studying the concept that things only affect what is next to them is enough to require finding pi.
Calling Pi the circle constant is selling it short. It is considered the most important mathematical constant for a reason.
More videos with this guy. He is nice.
I've been completely obsessed with chess for the past year. So if numberphile is making a chess related video, nothings going to top this today!
This guy is the David Attenborough of numbers, love these interesting graphs he describes.
Love you numberphile
No idea if this is of any interest to anyone, but I programmed this up today and thought I'd try something that wasn't mentioned in the video... allow he knight to go to each square more than once. Not unexpectedly the knight gets trapped, but takes longer. Anyway below is the results I got up to allowing the knight to enter the square up to 8 times (I ran out of ram trying to go bigger lol)....
1: x=-23 y=10 value=2084 steps=2016 (15 15 -> 961)
2: x=176 y=128 value=124561 steps=244273 (164 -18 -> 108059)
3: x=-635 y=663 value=1756923 steps=4737265 (-182 584 -> 1362655)
4: x=-1341 y=2312 value=21375782 steps=98374180 (1113 -2251 -> 20271369)
5: x=3470 y=2524 value=48176535 steps=258063291 (2055 -3272 -> 42829264)
6: x=-5664 y=-4569 value=128322490 steps=836943142 (3853 -5520 -> 121890974)
7: x=-7013 y=-5657 value=196727321 steps=1531051657 (-6945 -5433 -> 192930589)
8: x=-7588 y=-6900 value=230310289 steps=1897092533 (-5902 7124 -> 202990035)
My spiral has 2 directly above 1 at the start. The first x,y are the coordinates relative to the center at the stopping point. The final numbers in brackets are the relative x,y coord and value of the smallest value square which have 0 visits (obviously 2 and above have other values but I haven't included them here). Somewhat interestingly the pattern of visited squares for 2 max visits and above makes a shape like a square with an off-centered indent along all sides. Anyway just FYI.
interest in trying another variation? I have an idea in mind on a different way of numbering the board that I'm interested if it will untrap the knight. Here's hoping!
In scenarios where multiple visits to the same square are allowed, did the knight avoid visiting a square multiple times unless it absolutely had to, or did it move to smaller squares that were already visited before larger squares that weren’t?
I looks like it has been about 5 years since you posted that comment, so I hope you see this response. How different does this pattern look and end if the initial square is zero instead of one? One interesting trait I noticed in both configurations is the locations of perfect squares.
4:49
Me when someone tells a dirty joke.
Dr Sloane is awesome. Such enthusiasm, such passion. The cartoon animations in these videos are also wonderful. Really makes me want to hang out with Dr Sloane in his office all day. A tribute to Notts University ! Anyway, as always, am now going to have to write a program to explore this Trapped Knight for myself ... a task that makes for a pleasant afternoon ! Then I'll be on to check out the Rook, Bishop, ....
My 6 year old son loves this channel. He discovered it on his own. He is a fluent reader and excels in mathematics. He writes equations on scrap papers for hours. He even teaches me the formulas after watching a new video. I am not sure he has 100% of the basics of math or calculus but he appears to extrapolate new equations from anything he can think of and use the formulas he sees on the videos as the basis for his “lectures” (where he goes into more detail about theoretical number combinations.
that's honestly awesome!
This was one of the most beautiful videos on this channel. You could see the passion and the joy in the eyes of Neil Sloane.
I’d love to see a simulation of this. Test it out with the same initial conditions that are given at the start of the video but instead of starting at 1, start at 2 and see what happens, then go to 3 and see what happens, and so forth. If these sequences are finite, would be interesting to see this sequence of results
I had the same thought, and scanned through the comments to find someone else who mentioned it. What about starting at 0? I think that might be my first test, but then I would definitely like to see it increase as you described.
Why would a rook get stuck, wouldn't it just follow the spiral?
What would a bishop do?
Sounds right to me? I'm not sure what a Castle piece is either. I chalk it up to him not playing since 14.
@@Tahgtahv Castle is another name for Rook. Depends where you live.
Numberphile only stay on odd numbers! if it starts at 1.
@@numberphile wouldn't it jump back and forth along the diagonal: 1, 3, 7, 13, …?
They didn't really say anything interesting about it though, did they? Just sort of, "Hey, you could do this thing! Until you can't anymore". Don't get me wrong the topic itself is interesting, but I expected a lot more depth on the topic or at least some kind of attempt at an explanation. It was also mentioned that this could be done with a rook? That doesn't make much sense, could you show us what you mean? Oh, no, now it's the Brilliant ad at the end of the video
The rook would just follow the spiral.
I agree, it's a bit short on answers ...
exactly what I felt.
Its fascinating how mathematicians come up with these new ways to use numbers, but this just looked like coincidence with not much maths behind it.
"Hey, you can do this thing! Until you can't anymore" is a sentence you can sum a lot of math with
I feel like this is the story of recent numberphile videos. I got hooked on the channel years ago when they talked about numbers more, like 3435, the only number where if you raise the digits to themselves you get the number.
Surely a rook, queen and even king would just follow the spiral. That is from 1 it goes to 2. Then to 3 and so on. The one in the corner of the board would perhaps be more interesting for these pieces
Check them. It’s also fun to check if different rules make a difference!
From the corner, the rook would just go along a straight line to the right (1, 2, 4, 7, 11, 16...), since futher to the right and bottom, the numbers would increase, and the next to the right is always less than the next below (since numbers increase toward the bottom-left).
If the numbers weren't always going from top-right to bottom-left along the diagonal, but instead alternating their direction, it would instead repeat right-down-left-down (or down-right-up-right) forever, leading to (1, 2, 5, 3, 4, 9, 12, 10...).
If, instead of diagonals, the numbers were arranged in reverse L shapes (always filling up squares), the board gets filled out:
Starting the numbers at alternating sides gives a trivial since adjavent numbers are always right next to each other. Always starting at the same side will gives a slightly more interesting shape, going (1, 2, 3, 4, 6, 5, 7, 8, 9...), always finishing one L, then jumping right to the next one and then up to the very top of the board before filling out that L. This is, of course, assuming we can skip over previously visited spaces. If we can't, things might get a bit more interesting.
Actually, with the L shapes and starting at the same side, if the rook is not allowed to skip previously visited spaces, it gets trapped after only a handful of moves, giving the sequence (1, 2, 3, 4, 9, 7, 5, 6, 11, 10, 17, 18, 19, 12, 13, 14, 15, 8).
We needed this without even knowing.
Props to the guy who wrote all the numbers on the board, that must've taken a really long time, especially with the larger numbers.
I love the passion of this professor. Do more interviews with him!
You should add an exception like if it get's stuck it goes to the last step and continues the game. that would be nice :D
It's not arbirtary - 2084 is the year of the Earth/Mars war of course.
also the robot revolt, Robotron 2084
Currently bingeing on Numberphile. I'm infinitely more interested in mathematics today than I ever was at school twenty years ago!
Thanks for sharing this video. I apprecieate seeing the wide viriety of things on your channel.
What a great guy, love the graphs and his enthusiasm.
I wonder: Does there exist a starting value that results in an infinite series? Also, given a different starting number, what is the lowest possible trapping number?
What do you mean starting number? If the numbers are still increasing in the same pattern that doesn’t affect anything
@@benjaminfischer6022 both examples they started on square 1. I was wondering what the trapping number becomes if you start on someplace like 2 or 37 or 2084.
After that, I wonder what the lowest possible trapping number is? Next, is there a starting value for which there is no trapping number and the knight moves on the infinite chessboard forever?
What if the target is the highest of the all the available options? I suspect in that case it will definitely be an infinite sequence, ever expanding, but nevertheless, it will be interesting to see what sort of pattern will appear out of it.
One of my favorite numberphile videos and it's definitely too short!
I need more videos with Neil Sloane, I just love everything he talks about ❤
Wonderful video and absolutely love that it's only 6 mins long. Actually have time to watch it :)
The bishop was always my favourite piece because it lives in this strange parallel dimension where it can only see half the board. The allied bishops will never get to meet each other :(
I love his enthusiasm!
I've been moving my knight without pre-numbered squares, but still trying to keep it within as compact an area as possible, whereby each new square is the least isolated from the old ones. Often easy to judge, but when not, then rule that a new square surrounded by n old squares in the middle of a 3x3 grid is less isolated than one surrounded by
What about a Knight on a 3D board?
you can have a flat 3d board... just saying.
What about a knight on a _cheese_ board?
How would you make the spiral?
@@Henrix1998 Maybe you could number them in a pattern similar to winding string around a ball?
That's what I was thinking, increase dimensions
I'm having a hard time thinking of a way to number the tiles as well. Maybe a spiral that increases in +Z and negative numbers in -Z
If you don't care about neighboring tiles being consecutive #'s, you could have one function getting the next tile closest to origin, then just assign it the next number
Neil Sloan is a delight.
I love this man and his topics! You certainly have the perfect mix of advance, fun, interesting and just pointless number-related things to watch!
I think it’s really interesting that even with a world of infinity the knight and the rules it is bound by it gets itself trapped unable to explore the rest of the world of numbers it lives in and is stuck at 2084
Mathematicians create their own problems and then try to solve them..😂😂
They said the same thing a few hundred years ago about Euler :)
meanwhile, soviet-bugs-bunny - Their problems will be our problems eventually
Sometimes they fudge numbers to do it, too…
For some reason that first spiral looks like a map of Syria.
that is a nice observation
This is how they plotted out Syria, maybe
@@thomasw4422 given the present geopolitical chess game, i guess you're right
chess is ISIS propaganda confirmed
thanks guys for all you do
We need to see more patterns of these as they're really cool.
hmmm.... seems pretty... arbitrary to me! But then, a lot of other maths did. Until suddenly some meaning jumped out of nowhere :)
4:49 when your teacher tells a cringey joke
Those coloured-line graphs are pure Angelic! 😍😍😍
I think an interesting variation of the Trapped Knight problem would be to select some numbering scheme other than the square spiral. That might produce some interesting results.
I deeply respect the love that Neil has towards math
This is unbelievably amazing! More chess-related videos, please!!!
What if you change the rules so that the knight alternates each move to pick the smallest and then the highest number possible on the next move and then back to the smallest the move after that... Start with picking the smallest first. Will it still get stuck? My guess is no.
i'd listen to this lad neil speaking for hours, he's so interesting, and seems a nice person too
*Bobby Fisher has reentered the chat*
Vishwanathan Anand has joined the chat
Capablanca has left the chat
Every chess grand master has entered the chat
Ok so the world will en in the year 2084
Could be. Though I've become suspicious of such predictions since the Mayan calendar failed to deliver the end in 2012. Also, my wall calendar assured me that the world would end on 31 December of that year. Maybe it's a problem with the technology of calendars that doesn't apply to Numberphile.
He is so happy
More videos of Neil Sloane please, this guy is thousand time more soothing than any ASMR videos.
"I have the picture of the spiral here" A really bored mathematicians
"Do you play chess?"
"No I retired..."
"It was taking too much of my time"
At the age of 14
The inherent complexity isn't in the movement of the knight but in the structure of the square spiral. That is what is most interesting to me. The sieve of Eratosthenes also lies on a square spiral. My mathematical intuition tells me that there is another Mandelbrot set in here hiding behind the patterns. I can't wait to get home and play with this. ^_^
I love the Dr Neil Sloane videos!
Since by definition a Knight move consists of either x+2y or 2x+y, it would be interesting to examine moves of 3x/3y and integers higher than just two.
With x and y being the unit vectors in the directions of the x- and y-axes, respectively? Well, that's incorrect, since you could negate the component in either direction and end up with another valid knight move.
04:49 haha, heheh, heh, yeah... Best sequence.
no doubt
What a charming man! I could listen to him for ages.
I love this dude! All excited and enthusiastic and rocking a Hendrix shirt!
These vids are great
Stopped playing chess because it takes too long but does math for the entirety of his life . Legend.
For rooks and similar the question is whether or not the piece is allowed to cross squares already visited, or whether those are blocked (as if by another piece). Otherwise, given that a rook has unlimited movement, it could always find a free square and never get stuck, by definition.
Yes, that would be a reasonable restriction for pieces with infinite range. Of course, for the rook in particular, this doesn't make a difference, as it just hits every square in sequence regardless.
This is absolutely magnificient
Actually the horse was exhausted...
And had to stop
Oh horses can move infinite distances...if they Cantor. :)
Someone send this to Jerry :-D
This made me think...what about if the Knight moved in 4*2 instead of 3*2? Or 4*3? Is there a point where making the L larger would make it be able to "escape"?
I always apprecaite a numberphile video that doesn't make me feel like an idiot.
I understand the poor knight, but rook and bishop could move to an infinite amount of squares, so how they can be trapped?
I think the rook can stick to the spiral so it never gets trapped, and the bishop ends up jumping along the NW/SE diagonal and likewise keeps on going forever.
Did he actually say that they would get trapped? I notice he said the queen wouldn't…
The rook moves 1,2,3,4,5,6,7... so it'll never get trapped until you run out of chess board.The bishop moves 1,3,11,9,23,7,19,5... so it's not immediately obvious, to me anyway, that it will never get trapped.
@@d5uncr bishop moves 1, 3, 7, 13, 21, 31, …
@@d5uncr the bishop has no restriction on how far it can move, and so in order to block it off you would need to have visited an infinite number of tiles, otherwise it would just continue along a diagonal until it reached an un-visited tile.
Nope, it can't jump over 1 to get from 3 to 7.
Think of it as someone putting a chess piece on every square you've visited. And you're not allowed to take any pieces.
Higher dimensions?
Would definitely like a 3D version of this.
i could listen to this guy for hours
I've been playing with knight's tours for a long time, but I haven't before come across this. To quote Spock, it's fascinating. Thanks!
But why?
It's there.
stalemate 2084
It's interesting that what amazes us the most and what we love the most are the things that we don't entirely comprehend
Top quality videos. The regular presenters are fabulous. I would love to see an episode or two on Fourier transformation. I find it fascinating and I think the rest of your viewers will too.
this is how *chess* works
Hitesh Sodhi oh yeah yeah
Oh yeah yeah
The sum of all of those numbers on that board is equal to -1/12
The limit of that sum is equal to -1/12... The sum is equal to aleph null.
The Major the sum doesnt converge so it has no limit
Only when discussing zeta functions. The actual sum is divergent.
@@palmomki Ikr, saying that the 'limit of the sum is equal to -1/12' is like he's trying to be as incorrect as possible so that there is not even a single interpretation of the statement that is correct. And the sum is not equal to me. No idea what you mean by that, since the sum is not the cardinality of a set.
I've actually discovered something about the bishop sequence (assuming the bishop can only move one diagonally at a time). it is self-encoded. for B(n) tells you how many steps the bishop will take to reach 2n+1 (let square 1 be step 1) and B^-1 (n) tells you the tile number (since they're all odd, it gives (n-1)/2 instead) for a given number of steps (where step 1 returns 0, representing square 1). Listing all the odd outputs of B(n) will give you the sequence of B^-1 (n). amazing!
A bit of a similar intuition: list off all pairs (step number, tile number) where (1,1) is the initial point. Then sort the points in order of (tile number). The odd terms in the sequence of (step number)s match the initial sequence! the odd terms also come in large chunks, and so can be used to predict farther out numbers. Ordering as i described:
1,8,6,4,3,11,9,23,7,19,5,15,14,12,28,10,24,46,22,20,40,18,16,34,33,13,29,53,27,25,47,77,45
notice the odd terms 1,3,11,9,23,7,19 are also the first odd numbers in the sequence A316884.
Fascinatingly beautiful!!
-1/12th
Nah fam, 42
Lately it seems your videos are just "hey this is a thing. Ok bye."
Seems like you don't make any connections or explain stuff as much any more.
But you sure don't forget the adverts.
Broken Wave Dummy, that's because this doesn't have connections to anything. The video is 6 damn minutes long. It's in the thumbnail. You could've realized this and chosen to not to click on it. People nowadays... complain about everything.
so what if he had a choice not to watch it? that has no effect on its quality
@@angelmendez-rivera351 Youre the one complaining...about my observation. I had no way to know it would be such a bland video until I watched it. 6 minutes is plenty of time for very interesting videos, connecting multiple subjects.
And of course this has connections to something. No mathematical or geometric phenomenon has zero connection to anything except itself.
P.S. Grow up.
The bookshelf behind him looks delicious! I'd love to see a video similar to the mathematicians' breakfasts one, except about their bookshelves.
I love this man’s enthusiasm 🥰