Superpermutations - Numberphile
Vložit
- čas přidán 28. 01. 2018
- The Great Courses Plus (free trial): ow.ly/C3FE30hIvhc
This video features Dr James Grime.
More links & stuff in full description below ↓↓↓
More James Grime on Numberphile: bit.ly/grimevideos
James Grime: singingbanana.com
Tackling the Minimal Superpermutation Problem: arxiv.org/abs/1408.5108
(Apparently I mucked up the copy/paste for the 1-6 superpermutation and, amazingly, people actually check this stuff... The full number is in the paper linked above)
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumberphile
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science.
NUMBERPHILE
Website: www.numberphile.com/
Numberphile on Facebook: / numberphile
Numberphile tweets: / numberphile
Subscribe: bit.ly/Numberphile_Sub
Videos by Brady Haran
Patreon: / numberphile
Brady's videos subreddit: / bradyharan
Brady's latest videos across all channels: www.bradyharanblog.com/
Sign up for (occasional) emails: eepurl.com/YdjL9
Note: The Great Courses Plus is currently available to watch through a web browser to almost anyone in the world and optimized for the US market. The Great Courses Plus is currently working to both optimize the product globally and accept credit card payments globally. - Věda a technologie
Let's check...... YES, 1 contains all the permutations of 1.
My 2nd favorite moment in this video.
😂😂😂😂
He didn't check that it was the minimal one, though...
*Second* favorite? xD
@Josny13 - maybe Mathloger will make a video about it
I love that moment at 3:38
"We don't know."
"...no!"
Brandon Steele (no) factorial? Um. Well. That’s a problem I don’t know how to solve.
@@notottomedic The factorial of (no)
Well No or Aleph Null is a smaller kind of infinity which contains all the full numbers
so the factorial is all the numbers down which is really weird cause it'd be bigger than Aleph Null.
Let's just say its that
That was my reaction too
wE dOn't nOeEEe
@@BLRSharpLight what do you mean we dont know. just look at the pattern. 1, 3, 9, 33, 153, 873, 5913, 46233, 409113, 4037913. The formula is easy. just take the number you want to know the superpermutation and calculate its factorial then add number of digits from the previous numbers superpermutation. for the 6 you supposedly dont know, 6!=720 and 5's superpermutation has 153 digits. so 720+153=873, therefore the number of digits in the superpermutaion of 123456 is 873. is the formula would be n=1!+2!+3!...+n! where n is the number of digits.
Dr Grime: "For 6 we don't know"
Brady in total shock and awe: "No!?"
Hilarious hahah gotta love the emotions one can get on these topics
what do you mean we dont know. just look at the pattern. 1, 3, 9, 33, 153, 873, 5913, 46233, 409113, 4037913. The formula is easy. just take the number you want to know the superpermutation and calculate its factorial then add number of digits from the previous numbers superpermutation. for the 6 you supposedly dont know, 6!=720 and 5's superpermutation has 153 digits. so 720+153=873, therefore the number of digits in the superpermutaion of 123456 is 873. is the formula would be n=1!+2!+3!...+n! where n is the number of digits.
@@ephemera2
He said in the video that the factorial summation is just a guess.
The 1 - 6 permutation was proven to be one less than the factorial summation, at 872.
@@ephemera2 L
I don't know how this reply got into this comment set but I watched the rest of the video I saw that the pattern doesn't necessarily follow.
@@ephemera2 and here we see why you need to watch the entire video before posting a comment.
Whenever James checks the easy n=1 cases. I am always 1) slightly amused, and 2) very happy that he is so diligent about checking bases cases. There are a surprising number of wrong proofs that fail because the base case was not checked!
??
@@Triantalex ???
6: Parker's superpermutation
Stop trying to make Parker Stuff a thing! It's not a thing!
Too bad it's actually a meme.
It's ALMOST a thing, but not quite...
+Triumvirate888 +Heulerado
It's a Parker thing.
It's a Parker meme?
"I counted the digits 10 times now, and it is still 1 digit short!"
"This can't be true, count again, idiot!"
One less than expected after five steps? Sounds a lot like how the maximal number of divisions in a circle separated by n chords starts out as 2^n up to n = 5 but then falls away because it's actually the sum of five binomial coefficients. If, by some _incredible_ miracle, this is analogous (and hey, binomial coefficients _are_ factorial-related), we'd expect the solution for six to have length 2!+3!+4!+5!+6! = 872 and the solution for seven to have length 3!+4!+5!+6!+7! = 5910. That would be neat.
Lagraig O'Moof thanks that is what I was noticed glad you put it so elegantly. Do we know if 7 is 2!-7! Or 3!-7!
Lagraig O'Moof My intuition thinks you might be right
You probably already know now, but the answer for 7 is at most 5906, so that pattern doesn't hold
"Superpermutations" sounds like a 90's math rock band
I don't know of any math rock bands from the 90's. Except for the Quadratics, but they were fictional. Probably still are.
The quadratics are too irrational to be considered fictional.
Like my old favourite, The Permutations of the USA
I really like how much Dr. Grime seems to enjoy talking about numbers.
Here is the scramble of James' Rubik's cube
D B2 R2 F2 U R2 U' R2 U R2 B2 L' F D' F' B' R' D F2 L' U'
Sébastien Verdier nice
Sébastien Verdier wauw.
Nice
51090942171709440000
I stg of course you did this
Two years ago, I created the following sequence:
00110212203132330414243440515253545506162636465660717273747576770818283848586878809192939495969798990
I didn't know what to call it or whether it has been created yet, but now I do!
The sequence contains all two digit numbers from 00 - 99 only once.
There is a pattern and I've been trying to do it with three digit numbers.
suPERmutations
This is what I originally read it as.
Paul J. Morton How did they miss that?? Mathematicians upset me
That's a bad portmanteau, because it ends up just looking like super-mutations.
Nice try though.
You said "bad portmanteau" and now I need a safe space. Don't blame the victim!
Badmanteau
And I was thinking, "permutations aren't that interesting pretty much everything is known about it" My reaction when James said we don't know about 6 I was like no way. Numberphile never disappoints me
I bet they counted to 872 alot of times.
Why don't we call it supermutations? :'(
Daniel Chmiel it would be much more fitting as well as it leverages the overlaps
But it would sound like sub-permutation
Koha Raisevo super mutation
The problem is that "super mutations" would mean/imply a different thing to "super permutations", so the portmanteau version would be ambiguous.
Because we're creationists, you heretic!
Must do work, must not spend afternoon numbercrunching
I absolutely love those sequences that start to escalate very quickly and then there's some factor when James (or any host for that matter) turns to the camera and says "WE DON'T YET KNOW" *cue the thriller music*
EDIT: and this time it's even better than some humongous undescribable number such as Graham's Number or TREE(3)! Amazing video and subject!
So, in case anyone is curious why 6 is so hard, from what I can tell the best algorithm for exaustively checking all possible answers is of the order 2^(2*(n!)) steps, where n is the number of symbols, which is a very big number even for a supercomputer to check when n = 6.
Joshua Hillerup what do you mean with "all possible answers"? Do you mean all possible superpermutations? Could you give some information about that algorithm?
Specifically, they used something for solving what's called a symmetric travelling salesman problem, where the best known algorithm has the complexity of what I posted.
There are 8 solutions to n=5 and 96 expected solutions to n=6 so far documented that I have found. Expected because this video is pointing out the solution that broke the rule. Easily found too, it is linked out from the wiki on it.
Joshua, for every n there are an infinite number of possible answers (permutations) so they can never be exhaustively searched. Presumably an acceptable lower and upper bound for the string length, and possibly some other restrictions, would have to be figured out first to define a mere subset of the answers to actually test..
MrDannydoodah it's not infinite for the minimum strings.
James Grime is one of my favourite hosts on this channel! Lovely video :)
It's so weird when you have a problem where you can evaluate the simplest cases almost by hand and you just add one or two and boom it becomes an unsolved problem.
It would still be quite hard to find it for small numbers
like TREE (3)
Dr James is back! Yay!
I think the founder of Superpermutations missed the perfect idea to call them "SuPERmutations". Would have sounded much cooler.
Not only would it have sounded cooler but it would've fit in perfectly with the whole idea of superpermutations in the first place
@@kevina5337 yeah that's the joke
Wow, That's genius!!!
"I want to learn about Super Mutations!"
@@anawesomepet Xavier's ears start ringing
I was excited to learn about supermutations
When most people think of grime they think of stormzy, skepta etc but I think of my man James!
Met James the other day at an outreach event. Greatest day of my life.
Dr James is the best
this is SUCH a classic numberphile video. loved it
This is also been recently dubbed as "The Haruhi Problem", as a major breakthrough in solving the problem of "how long is the shortest string" was actually discovered recently by 4chaners trying to optimize watching Haruhi in every possible order.
I wish I was joking.
The Haruhi problem was solved in 2011. Before this video was made. Why would you wish to be joking every mathematical discovery helps, whats more impressive is that this discovery immediately had a use, to work out how many episodes of haruhi you would have to watch to see it in every possible order. I think 4chan and channers get a bad rep because unlike other forums there are no rules governing the discussion and this is the end result. When your opinions and ideas have to stand on their own merit without the pomp of an identity behind it, all you have is your argument and everything you say is peer reviewed.
Yeah! More James Grime.
The best of this video is your happiness about something that no one needs ... still I find it VERY interesting and always LOVE to watch your videos!!!! =)
Along with Matt, James is my favourite Numberphiler!
Dr. James is back!
Is it just me, or is it far too long since we last saw James?
It's always too long since the last one.
i think this one is also filmed long time ago,
Before the pictures got hung.
FrogSkull could have be using a text only browsers.
It is just you.
The way Dr. James Grime explains these very complicated concepts makes it very easy to understand what the concept at hand is.
now THIS is quality content! I love it!
I've been watching Numberphile for quite a few years now...I get most excited when I see a video from Dr. James Grime. I also loved the videos where you talk to the famous mathematicians like Dr. Eisenbud , Dr. Fefferman, or Dr.Isenberg. For some reason though, I don't get as excited about the Klein bottle videos or calculator unboxing and alike.
All of the shortest superpermutations for the digits 1 through 5 are numerical palindromes: they read the same backwards as forwards. The shorter one for 6 does not...
There are also non-palindromic shortest superpermutations for 5.
Here is one: 123451324153241352413254132451342513452134512341523412534123541231452314253142351423154213542153421543214532143521432514321542312453124351243152431254312
Thanks for that. Very clever.
the ones that follow the pattern are palindromes
Tristen Roddenberry: follow what pattern?
ThePraceKingdom .. noo !!!!
It's always fun to see James videos
James Grime is back!! 😃
Edit: Real OG's know about the scrambled rubik's cube in the background.
He didn't say noombah :(
Love how there’s a rubies cube in the background of a permutations video
No matter what he is talking about, be it factorials or other stuff, he is so cheerful and hyped!!
There’s just something about going and learning math on your own that’s more envigorating than sitting in a classroom and memorizing stuff.
Should I watch this??? Oh it's James. Mind made up
Do you like James, James?
Halosty yep lol
I did notice that the superpermutations for 1 through 5 both start and end with 1, but the sequence for 6 doesn't. So if you allow for the sequences to 'wrap around', you can get rid of the last '1', and the pattern holds as 2! + ... + n! : 1, 2, 8, 32, 152, 872,...
Actually, this pattern breaks down at 5. There is a superpermutation for 1-5 of length 153 that starts and ends with 12, so if you allow the sequences to wrap around then you can combine the 12 at the end with the 12 at the start, and the circular sequence is shorter than expected.
Here it is! 123451234152341253412354123145231425314235142315421352413521435213452135421534215432154231245321453241532451325413251432513425132453124351243152431254312
If I'm not mistaken couldn't "reverse" the numbers and the pattern would still hold? Like for 5, change all 5 -> 1, 4 -> 2, 2 -> 4, 1 -> 5? Then it would start and end with 5's instead and would be the same length.
+jv3hLgYg18Ys Ah, damn. It was worth a shot :) Cheers.
I love that guy. Please - PLEASE - make way more videos with him.
I missed James!
Numberphile is back with my favorite professor!!
Awesome to see Grime again!!!!!!!!
Another Nice Video 👍 whenever I see these kind of videos I feel more fascinated by Mathematics. (I am sure now, I will do major in Mathematics definitely)
The Haruhi problem has recently solved a decades old problem, being written by an anon on 4chan and being discovered by mathematicians in October 2018. They have discovered the lower bound of super permutations using the question what is the lowest amount of episodes in the original Haruhi Suzumiya you can watch in any order.
Dr James Grime is my favorite of everyone to watch here, anyone else agree or have a fav of your own ?
The Disabled Gamer, he is the best
In no special order, but I really enjoy James Grime, Matt Parker, and Cliff Stoll.
Hannah Fry! But James is a close 2nd.
Neil Sloane
Oh excelent, more James Grime pls
NUMBERPHILE IS BACK!
We uploaded on Friday. Maybe the algorithm is withholding us from you!?
I missed james as much as a singing banana!
Bananas in Pajamas.....
Nice
??
It's such a relief to know what we still don't know
thank you Numberphile
Thank you so much!! Ive been thinking of asking for a video about this for so long! My old car used to have a key pad entry where anytime the buttons are hit in order it opens and ive always wondered how to hit the fewest buttons to hit all combinations.
Oh my god that is a very clever application.
put this on mildly unsatisfying.
because omg why. it was almost perfect
The correct quote is "In the beginning the Universe was created. This has made a lot of people very angry and been widely regarded as a bad move." Your paraphrasing angers me almost as much as the creation of the universe.
It's a James video! What a pleasant surprise
n!+(n-1)!+(n-2)!+(n-3)!+(n-4)! then? for negative factorials just conventionally assumed to be zero.
4:52 - That quivering anticipation is the mark of a top-class educator and teacher 🙌
I love permutations! Dell Logic Problems magazines often have puzzles dealing with permutations.
he's back!
brown chocolate 😛
You guys should do videos on the histories of famous mathematicians. I’m always hearing about little sneak peaks of the works of Euclid, Gauss, Euler, and Ramanujan, but I never hear about their history as a whole. Would be very interesting!
Adam Bergen Also Fermat.
I click on the video, I see James, I click like before CZcams even has time to start playing. He has yet to dissapoint. More James Grime I say!
I saw the thumbnail and I somehow thought this was a Mathologer video.
me too! cheeky from numberphile xd
Bright primary colors on a white background? Yep, Mathologer.
love dr James 4 ever
I have a magnificent proof that the superpermutation of 7! is 1!+2!+3!+4!+5!+6!+7! - 2, but the comment box is just too small for it.
Haha! I'm curious how many will get the reference.
Probably most, since this is a math channel (albeit a popular one), and the quote is quite well-know in pop culture
write it in many comments.
I'm sad I don't get it
It is a reference to Fermat's Last Theorem, and has become a rather overused math comment thread joke.
Noooooooooo, you have to call them SUpermutations instead of SUPERpermutations
but then it sounds like he's talking about some super mutant organism.
Fun Stuff ! Reminds me of many, many moons ago. I used to make reel strips for slot machines. The mechanics of a slot machine is based on the reel strips and pay-off percentages. You had to know what the percent the casino wanted to keep. The reel strips created were based on that. Machines were honest, but not all created equal. They kept from 3% to 95% of the money fed into them. Three reel machines, with 23 to 26 stops, was easy. I honestly did most of the math while driving down the Nevada highways. I had a notepad to write results, the computation was in the head. Four reel machines, or more stops, got a bit more complicated.
2:40 - that is some Parker level of humor ;)
I like what the pattern does when you allow the number wrap around itself. Drawing it around a circle. How few numbers are required then. Take a moment and try it out.
Nail and Gear for the win, Tims.
This is excellent! This exactly addresses a question that I casually started thinking about a while ago: "what is the shortest length of a superpermutation [obviously, I did not use that term at the time] of length n?" Now that I know that we don't yet know a general formula ... and that it seems like it requires quite a bit of brute-force math .... uh, nevermind. I'll move on to another problem.
However, the de Bruijn sequences look interesting. I may look more into those!
Just watched it. Super cool as always
DUDE you cant possibly watch 9 mins vid in 50 seconds
Anyway, he's right. It's an amazing video.
I also like the nail and gear in the background
I like the subtle Nail and Gear poster in the back...
I love how the 121 in the middle is the centre point and everything after that centrepoint is reversed.
what's up with those frequent uploads? almost gets me excited about math.
I like these videos where it shows things that could go wrong like how the pattern did not hold.
I read the title as "Supermutations" and wondered what that had to do with maths, was not disappointed with the vid though.
Loving the nail and gear picture
Isn't that actually a framed flag?
K.o.R, it’s from cgp grey, i think
I would prefer THE ☭ HAMMER ☭ AND ☭ SICKLE.
The paper has stripes instead of being solid brown. This upsets me for some reason.
It's almost brown, but not quite.
You could say that it's...
Parker Brown
Mister Apple, you are rotten!
Ya the paper they use has always bothered me too but I've gotten used to it because I like these videos
This conversation +++
Brown paper always has stripes 😱
This is like maths shower thoughts...I like it!
im so glad the original numberphile guy came back
What if you read the number right to left as well as left to right? The number would be smaller.
So let's start with 123. The Super Permutation is 123121321. If you allow reading to be both ways you get 12312. The options are 123, 132, 213, 231, 312, 321. All 6 options are still there.
Just want to point out an update to the superpermutation problem:
An anonymous 4chan user found a lower bound in 2011 inspired by an anime problem, and this result was rediscovered and written up by mathematicians in October 2018
Its now called the "Haruki problem" after the anime which asked the same question but for watching all episodes in every order
I am a simple man. I see Dr Grime; I click like.
Me too. James Grime is the bee's knees.
M: "Hello, random stranger that I just met in the subway."
S: "Hello."
M: "I just liked a video about the minimum length of superpermutations for the numbers 1 to n. At first it seemed like the length is always equal to the sum of all the factorials from 1 to n, but then at n = 6 there came a shocking surprise. The currently shortest known superpermutation for n = 6 is 1 shorter than expected. Isn't that super strange and weird? Oh and by the way, would you say I am a simple man?"
S: "Well, I guess that depends..."
M: "On what?"
S: "Was there Dr. Grime in the video?"
Nail and gear sneaking into the background, nice
I love this ranga
Ay it's a Grimey one. Nice!
Love the Tommyball reference at the end of the video.
I see dr Grime i upvote. I am a simple man.
According to Wikipedia, De Bruijn numbers are cyclic - which is another difference to Superpermutations along with allow 'repeats' of the same number.
Literally low key an easy problem. i found out the numbers 6,7,8,9, and 10, with a simple code. took like 5 hours not even. Also did that like 2 years ago, i didn't think it was that big of deal so i stopped at ten, now that i see this vid, I'm running the program currently.
Thank you! I solve this when I have nothing else to do and there is nothing watchable in TV.
Oh, it's the vidoe's 1 year anniversary, I thought this came out today bc of Matt Parker's video about 4chan improving on the maths like yesterday
Shout out Houston, he's in Matt's vid and he's the one who found the 872 permutations
Thinking through a brute force solution (taking 2! as an example):
1) Build combinations (2! = 2 = [1,2], [2,1])
2) Build string by concatenating values (if match is not already present in string, while collapsing any adjacent numbers) (1221, 2112). With collapse of adjacent numbers = (121, 212)
4) sort strings by length ascending (121, 212)
5) pick first string to get the superpermutation (121) (in this case there are two equal length solutions)
It seems like you need to do a permutation on the products of the first permutation. So doing 2! means an additional permutation of 2!. But doing 3! = 6, and 6! = 720 different strings to construct using the steps above.
At 5! you have to do 120! = 6.689502913×10¹⁹⁸ . This is probably something that can not be done in a reasonable amount of time on a personal computer. 6! requires building 720! strings which is probably why it hasn't been solved yet. I get a precision error on my calculator just trying to solve 720!. You would need the memory space to store that construct many strings, and sort them to pick the shortest length.
James Grime makes be happy
superpermutation = sum of n! from n = 1 to n = x minus floor( x / 6)
for x being the number on which to permutate
this works so far :p
3:39 No. :c