The Trapped Knight - Numberphile

[@numberphile]

A T-Shirt based on the trapped Knight Tour --- t.co/bpqqnmUjM6

[@Kariis327]

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.

[@paulthompson9668]

It's really a shame that you've let your channel get political by supporting Patreon. A big thumbs down.

[@argh1989]

@@paulthompson9668 And how is that?

[@MegaMGstudios]

I would buy it, but I don't like the feel of American made shirts

[@MegaMGstudios]

@@paulthompson9668 they kinda have to with KZbin giving the finger to every content creator

[@ongbonga9025]

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.

[@numberphile]

Never give up

[@TheModestRat]

It can simply follow the spiral, right?

[@rogersong9406]

@@TheModestRat r/woosh

[@bassett_green]

@@TheModestRat yeah it can (it was just a joke)

[@skeletonrowdie1768]

oh my comment made me think that maybe different tiling could result in more interesting results

[@JJ-kl7eq]

I think this is a clear indication Magnus Carlsen will remain world champion until 2084.

[@pietro9801]

Lol

[@ais4185]

He'll use the trapped knight to draw his games.

[@arnaud78]

🤣

[@tinytim8173]

Caruana is gonna win next time

[@andrewfoust3227]

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.

[@matteussilvestre8583]

"The Trapped Knight" sounds like a scrapped Dark Souls boss.

[@RFC3514]

Scrapped? Hah! You mean you didn't find him? Noob! ;)

[@donkosaurus]

​@D.O.A. Thou shall perish in the twilight of Anor Londo.

[@SatariusFymir]

Isn't he right after Abyssal Jeremy?

[@matthewwriter9539]

I thought it was a Batman graphic novel.

[@omikronweapon]

though "trapped" isn't quite as fantastically an adjective for Dark Souls :P They'd probably go for "Imprisoned" or "Confined"

[@BLooDCoMPleX]

That many knight moves in the opening loses tempo.

[@rmsgrey]

I just claim stalemate after the first thousand (I wanted to see if they'd get bored first...)

[@DraoxxMusic]

*cough* Carlsen-Caruana 2018 Game 6 *cough*

[@joaoluizmoraesgomes7778]

@@DraoxxMusic LOL

[@yrrahyrrah]

*tempi

[@secretpromisanceplayer3316]

Kramnik approves

[@maksiiiskam2]

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".

[@hannesdewinter1458]

Interesting idea!

[@thundersheild926]

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?

[@Cen_t1369]

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

[@smtsmt8678]

What happens if you mark 2084 as already visited before you start the game? Will it still get trapped somewhere else?

[@numberphile]

That’s an interesting question???

[@Some.username.idk.0]

@@numberphile where to find a simulation of this?

[@AuroCords]

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!

[@Djaian2]

@@Some.username.idk.0 Wait a few weeks and it will be on the coding train channel.

[@sinisternightcore3489]

It will just get stuck one tile before it. Did I get your question wrong?

[@massimookissed1023]

I tried this with a pawn. It's less interesting.

[@derAlphabet]

Nice!

[@dennismuller1141]

I wonder if the king gets trapped too

[@skya6863]

@@dennismuller1141 hmmmmmmmmm

[@jianxinhuang7068]

Dennis Müller The king and rook would just follow the number spiral. They will never get trapped.

[@rosiefay7283]

Until it reaches the eighth rank and becomes a queen :)

[@Nick-ym4gh]

not so fast i am still filling up the infinite board with numbers....

[@isaiahdobesh5109]

Let us know when you’ve got that finished.

[@pandrzewko2780]

Bro you can stop the video, dont panic

[@sofia.eris.bauhaus]

my infinite board is still shipping. :(

[@PersonManManManMan]

same, hold up guys

[@therealax6]

@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.

[@2Cerealbox]

This guy's my new favorite numberphile guest.

[@numberphile]

More to come with Neil.

[@warrenpribula7265]

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.

[@vsm1456]

that's honestly awesome!

[@eelkedeboer1724]

4:46 deal with it

[@matthewpalermo4957]

Snoop dog!

[@Majestic469]

OOH OOOOH 🔥🔥😂

[@MrFrak0207]

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!

[@severed6s]

"you love it don't you..." "i do! i do haha! ha ha.. heh.. ye.."

[@k3dr1]

More videos with this guy. He is nice.

[@patrickhanlon932]

I love this video, and I love this man; and I especially love that he's wearing a Jimi Hendrix t-shirt.

[@mateussilva635]

You forgot about loving numbers!

[@patrickhanlon932]

@@mateussilva635 Numbers are concrete and consistent, so much better than letters.

[@imumsi]

There is too much confusion...

[@pseudo4914]

@@imumsi I can't get no relief...

[@JustinY.]

I have no moves, so I must scream

[@torque390]

You again

[@karlovcg1823]

5th like on a Justin Y comment. Do I get a medal?

[@morphmoprg4810]

u subscribe this too?

[@WorthlessWinner]

Huh, you must've actually watched the video to make that one

[@MrSanches97]

Tell me, do you have a life?

[@jacobbaartz7710]

This guy is the David Attenborough of numbers, love these interesting graphs he describes.

[@MoosesValley]

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, ....

[@ailblentyn]

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?

[@polyacov_yury]

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.

[@amirabudubai2279]

@@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.

[@trequor]

@@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?

[@scottwhitman9868]

@@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.

[@amirabudubai2279]

@@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.

[@babulalyogi1952]

Love you numberphile

[@eduardocavalcanti]

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.

[@SBGif]

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.

[@eezz9522]

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!

[@daleftuprightatsoldierfield]

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?

[@freemanthompson706]

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.

[@chrisg3030]

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

[@nestorgames]

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.

[@nestorgames]

Or the steps it takes before getting trapped, or the maximum number reached...

[@not2tired]

Well the (0,1) knight works.

[@PC_Simo]

It’s quite interesting that the knight can keep moving for longer, on the ”quadrant-board”, than on the ”full infinite board”; even though, there’s obviously ”less space / freedom to move around”, in a sense 😮🤔.

[@lovebuzz4116]

We needed this without even knowing.

[@MrRyanroberson1]

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.

[@AlisonBryen]

Currently bingeing on Numberphile. I'm infinitely more interested in mathematics today than I ever was at school twenty years ago!

[@Anchor9Studios]

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

[@freemanthompson706]

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.

[@Dimitri_gdr]

I simulated it and I get (2084, 711, 3915, 556, 3915, 556, 3915, 3380, 2086, 1339) starting from 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 3, 5 and 7 are the same because their path meet

[@bartholomewdan]

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.

[@valscripted]

One of my favorite numberphile videos and it's definitely too short!

[@Euquila]

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 :(

[@7zaxo]

Thanks for sharing this video. I apprecieate seeing the wide viriety of things on your channel.

[@alfiechenery4146]

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

[@numberphile]

Check them. It’s also fun to check if different rules make a difference!

[@AlienValkyrie]

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.

[@AlienValkyrie]

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).

[@tae.eun.translates]

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

[@viscinium]

The rook would just follow the spiral.

[@Morbacounet]

I agree, it's a bit short on answers ...

[@majinpe]

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.

[@zatarraagain7496]

"Hey, you can do this thing! Until you can't anymore" is a sentence you can sum a lot of math with

[@lexnellis4869]

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.

[@p11111]

Wonderful video and absolutely love that it's only 6 mins long. Actually have time to watch it :)

[@MisterMajister]

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!

[@viscinium]

But if the knight is on a parker square?

[@minor_edit]

A Parker Spiral?

[@MX-S]

Well from the thumbnail pic, it seems more likely the Knight is caught in Parker's concatenation.

[@sheikchilli8670]

the proof is left as an exercise to the viewer

[@munawwarabdulmuneer5877]

That would be a parker square of an infinite chessboard.

[@the_multus]

Finite will do

[@Aciek25]

I love the passion of this professor. Do more interviews with him!

[@TheMelopeus]

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

[@jsmunroe]

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. ^_^

[@kubo8248]

2:24 OH NO THAT'S A TROLLFACE!

[@muhammadriedhoramadhansyaf5086]

*Thanks, now I can't unsee it*

[@anaysasane6186]

😂amazing bro just amazing

[@vol10O000]

You mad?

[@KaziNafisAhmad-cx2eq]

24 January, the 'trapped Knight' incident

[@prosimulate]

What a great guy, love the graphs and his enthusiasm.

[@ALifeOfWine]

Neil Sloan is a delight.

[@DrOnlyDeath]

I need more videos with Neil Sloane, I just love everything he talks about ❤

[@squaremarco]

I deeply respect the love that Neil has towards math

[@ethanrossignol4718]

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

[@cemerson]

It's not arbirtary - 2084 is the year of the Earth/Mars war of course.

[@kev4241]

also the robot revolt, Robotron 2084

[@catherinedesrochers]

More videos of Neil Sloane please, this guy is thousand time more soothing than any ASMR videos.

[@armanhaikia]

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.

[@panulli4]

4:49 Me when someone tells a dirty joke.

[@Michaelminecraft1]

Why would a rook get stuck, wouldn't it just follow the spiral?

[@numberphile]

What would a bishop do?

[@Tahgtahv]

Sounds right to me? I'm not sure what a Castle piece is either. I chalk it up to him not playing since 14.

[@TheBioRules]

@@Tahgtahv Castle is another name for Rook. Depends where you live.

[@joeydunn930]

Numberphile only stay on odd numbers! if it starts at 1.

[@PhilBoswell]

@@numberphile wouldn't it jump back and forth along the diagonal: 1, 3, 7, 13, …?

[@ballaurina8367]

I love his enthusiasm!

[@TomPVideo]

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?

[@benjaminfischer6022]

What do you mean starting number? If the numbers are still increasing in the same pattern that doesn’t affect anything

[@TomPVideo]

@@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?

[@cletus993]

i'd listen to this lad neil speaking for hours, he's so interesting, and seems a nice person too

[@AdityaX2703]

*Bobby Fisher has reentered the chat*

[@Majestic469]

Vishwanathan Anand has joined the chat

[@Kobs.A]

Capablanca has left the chat

[@iDunnoMC]

Every chess grand master has entered the chat

[@gedstrom]

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.

[@quill444]

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.

[@isavenewspapers8890]

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.

[@afgto79]

Has anybody already asked what would happen if the knight uses a doped-up horse moving one square further ? Let's say a (1,4) move instead of a (1,3) ? Is there a point where the knight evades to infinity ? Are they remarkable (a,b) moves ?

[@schnipsikabel]

hmmm.... seems pretty... arbitrary to me! But then, a lot of other maths did. Until suddenly some meaning jumped out of nowhere :)

[@maxbramwell.1598]

I always apprecaite a numberphile video that doesn't make me feel like an idiot.

[@LinkAranGalacticHero]

This is unbelievably amazing! More chess-related videos, please!!!

[@KplusU]

thanks guys for all you do

[@evaristegalois6282]

The sum of all of those numbers on that board is equal to -1/12

[@livedandletdie]

The limit of that sum is equal to -1/12... The sum is equal to aleph null.

[@lucascisneros8147]

The Major the sum doesnt converge so it has no limit

[@litigioussociety4249]

Only when discussing zeta functions. The actual sum is divergent.

[@alephnull4044]

@@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.

[@super_7710]

We need to see more patterns of these as they're really cool.

[@TheAstronomyDude]

What about a Knight on a 3D board?

[@BlackWhiteCloud]

you can have a flat 3d board... just saying.

[@RFC3514]

What about a knight on a _cheese_ board?

[@Henrix1998]

How would you make the spiral?

[@RebelKeithy]

@@Henrix1998 Maybe you could number them in a pattern similar to winding string around a ball?

[@PerfectlyNormalBeast]

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

[@Liliou]

It's interesting that what amazes us the most and what we love the most are the things that we don't entirely comprehend

[@vaishalibanerjee7343]

Mathematicians create their own problems and then try to solve them..😂😂

[@TheJuggtron]

They said the same thing a few hundred years ago about Euler :)

[@hamiltonianpathondodecahed5236]

meanwhile, soviet-bugs-bunny - Their problems will be our problems eventually

[@Mr.HotDogShirtGuy]

Sometimes they fudge numbers to do it, too…

[@sirvalimont9614]

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.

[@isavenewspapers8890]

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.

[@maximmatusevich3971]

For some reason that first spiral looks like a map of Syria.

[@TheMelopeus]

that is a nice observation

[@1TW1-m5i]

This is how they plotted out Syria, maybe

[@maximmatusevich3971]

@@1TW1-m5i given the present geopolitical chess game, i guess you're right

[@skyscall]

chess is ISIS propaganda confirmed

[@lordtrollalot8707]

i think the rooks problem is, that he need a kind of curve to move free ... if you take the numbers as area 3217 for excample this gives 64 diametre yes it is wonderful indeed ...

[@dondovahkiin7899]

"Do you play chess?" "No I retired..."

[@oz_jones]

"It was taking too much of my time"

[@dertyp6833]

At the age of 14

[@elfinthekitchen]

Those coloured-line graphs are pure Angelic! 😍😍😍

[@sportsgamingcubing1860]

4:49 when your teacher tells a cringey joke

[@TheSagasser]

"Takes up too much time" ... proceeds jumping around with knight hundreds of times until it gets stuck

[@gokul7542]

Stopped playing chess because it takes too long but does math for the entirety of his life . Legend.

[@MrDannyDetail]

I tried it for a variation where the knight is not permitted to land on squares that have a prime number, but otherwise still chooses the smallest of the other available numbers. The sequence is: 1, 10, 49, 24 27, 12, 9, 4, 33, 16, 39, 6, 15, 18, 35, 14. The sequence dies once you reach 14, because there are 6 primes, out of the 8 available moves, and the other two squares have already been visited. I'm in two minds about whether I should have started at 1 though, as I was banning primes, but if anything 1 is less divisible than the primes, so should probably be considered as 'more than prime'.

[@_MrMoney]

Ok so the world will en in the year 2084

[@blindleader42]

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.

[@SimpleAmadeus]

Bishops, Rooks, and Queens don't get stuck because they have infinite range and there will always be more numbers in any direction. Pawns won't get stuck because they can't retrace their steps in any way. The King is the only other piece with limited range that can revisit previous squares, but on the spiral it would just travel all the numbers in order. On the corner setup it also forms an orderly pattern that hits every single square, but there might be boards where the King could get trapped.

[@theyhaventfedmesince]

"I have the picture of the spiral here" A really bored mathematicians

[@laskieg]

I was hoping it was going to stop at 2084 on the other board too, but that would have been too good I suppose. Neil's enthusiasm is contagious.

[@caiheang]

04:49 haha, heheh, heh, yeah... Best sequence.

[@sohamdatey22]

no doubt

[@sortagoodish8491]

In tamerlane chess there's a piece called the cammel: it moves like a night, except instead of moving 1 in one direction and 2 in the other, it moves 1 in one direction and 3 in the other. I wonder if a camel can get trapped? On the one hand, it moves a little faster than a knight, but on the other, it's "color-locked" like a bishop, meaning it can only even reach half of all the squares on the board.

[@jackmiller2614]

Higher dimensions?

[@DemoniteBL]

Would definitely like a 3D version of this.

[@presumedlivingston9384]

There's something about an old mathematician wearing a Hendrix T-shirt that just warms the soul.

[@rakhimondal5949]

Actually the horse was exhausted... And had to stop

[@glowingfish]

Oh horses can move infinite distances...if they Cantor. :)

[@Terminus316]

I believe this sequence is a mathematical testament to the harsh limitations the Knight sets upon itself, a code of honor

[@theoldbarbarian]

I understand the poor knight, but rook and bishop could move to an infinite amount of squares, so how they can be trapped?

[@PhilBoswell]

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…

[@d5uncr]

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.

[@PhilBoswell]

@@d5uncr bishop moves 1, 3, 7, 13, 21, 31, …

[@krisjaniswhittaker-lee6149]

@@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.

[@d5uncr]

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.

[@ginismoja2459]

What a charming man! I could listen to him for ages.

[@AgentSmith911]

Someone send this to Jerry :-D

[@gamingbutnotreally6077]

Ah yes the knight, truly one of the most interesting chess pieces

[@pedronunes3063]

It's sad that 2 knights can't force Checkmate against a lone king

[@matteogauthier7750]

stalemate 2084

[@abangfarhan1]

I tried this on a cartesian coordinate. Starting from the origin (0, 0). I generate the next positions starting from the northeast side of the point, clockwise. Sort the points starting from the closest one to the origin (0, 0). Then use the first point that has not already been visited. Spoiler alert, the knight is also trapped. This time it's at the 20156 step. The final position is (-78, -3). The pattern is more or less circular/concentric.

[@skauge6848]

But why?

[@allyourcode]

It's there.

[@elyas3876]

Magnus shouldve shown this when Andrea asked “how does the knight move?”

[@brown_note4710]

These vids are great

[@sumanchakravarty9567]

Oooo! I tested ... 1. Go to the highest available square and it explodes in a straight line - 1, 24, 79, 166, 285, 436, ... 2. Go to the lowest available square that is higher than the current - The knight meanders twice and then takes off in 2 parallel lines 1, 10, 23, 44, 71, 74, 109, 112, 155, 158, .... 3. Testing (Jump to the square that has the smallest difference)...

[@hitesh1297]

this is how *chess* works

[@woshdndndj2103]

Hitesh Sodhi oh yeah yeah

[@lukesumberg9182]

Oh yeah yeah

[@riteshbhartiya6155]

0:14, 4th row from top looks a bit different. And it seems that there is a joint b/w 10th and 11th column !!