Look-and-Say Numbers (feat John Conway) - Numberphile
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- เผยแพร่เมื่อ 5 มิ.ย. 2024
- The legendary John H. Conway on properties he discovered within the so-called Look-and-Say Sequence. Conway Playlist: bit.ly/ConwayNumberphile
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'None of the boys wore long party dresses, or if they did they weren't distinguishable from girls'. Best quote ever.
"Anyway, one of them came up to me..."
Wait...is he talking about one of the girls in the long party dresses, one of the boys not in a party dress, or a boy in a party dress?
The world may never know!
Colonel Dookie it had to have been a boy, Conway says "he said.." before beginning the sequence explanation!
+nrh756 But the boy wasn't wearing a long party dress.
how did you know :)
SABER - FICTIONAL CHARACTER you forgot the intertogation mark
"I hope I'm getting it right..."
I hope I get a constant named after me, only so that I can personally get it wrong.
That's rude :/
@@sriruparoy4946 it was a witty remark. Not rude
@@sriruparoy4946 It's not rude, lol. I mean this man accomplished so much in his life that he actually forgot (or didn't even really forget, just mixed up some digits) in a constant named after him. I mean if most people had a constant named after them they'd remember it for the rest of their life because it'd likely be the highlight of their career, but it was essentially just one achievement in a lifetime full of them for Conway.
"I couldn't get it", then proceeds to find an irrational root of a 71 degree polynomial and the 92 seeds of the sequence.
Just when you thought you got him.
RIP John Conway
John Conway passed away on 11 april 2020... A great mind has left us. :'(
that's sad
Noooo seriously? I think I had put this video in the Watch Later playlist when he was still alive then...
He died on a Doomsday.
A victim of corona virus.
@@ganondorfchampin Wow, it's true! Sadly, most people won't understand.
R.I.P., by the way
That's so cute that instead of just skipping class they threw a party!
0:45 - "Some of the girls wore nice long party dresses. None of the boys wore long party dresses - or if they did they were indistinguishable from girls."
The deadpan delivery nails it. XD
Rest in peace. One of the most brilliant minds of our time...
Rip
Immense respect.
rip
false.
I actually "discovered" this series myself in middle school, and noticed that 22 was the only number that didn't generate a string...
22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22...
Oh and 2|3, 12|13, 1112|1113, 3112|3113, 132112|132113, 1113122112|1113122113, 311311222112|311311222113, 13211321322112|13211321322113, is interesting because the first half is just 1 less than the second half always.
He missed an opportunity; 22 should have been Helium; since it's a 'noble sequence'
@@livedandletdie Can you prove it?
@@itchykami But helium has 2 protons, so it's like it can decay into an hydrogen, but hydrogen can't decay.
But did you really find this yourself?
Because I knew about this kind of sequence myself, but did we knew it from Conway, even if we didn't know who he was when we were kids, this kind of knowledge was spreading around even before the internet. Maybe we saw it from someone else.
So I wonder, did we knew about the sequence independently of, or because of Conway?
1 1 was a race horse,
2 2 was 1 2,
1 1 1 1 race,
2 2 1 1 2.
I don't get it
@Deus Vult Fun thing is I was binge watching Vsauce.
So Vsauce actually makes you smart 😂
I love this.
I vaguely remember it from a while ago. I was very young then.
@@randomdude9135 its a quote of sorts
Knew about this sequence before it got viral. like a boss.
+Gio Eufshi this video came with a very strong dejavu feeling. i think a lot of people may have played with numbers like this before. :)
Rest in peace Mr Conway. This was my favorite math “party” stumper as far back as the 8th grade and has always been a fun thing to look at and study (or even just to write down). Your work will live on in (in)famy for history as some of the most interesting problems-turned-games for even the those who least enjoy math. You’ve left an impression that is everlasting and we won’t soon forget.
Wow, nice nicee! The real question is how the heck did they find out the Conway constant is the root of that 71-degree polynomial?! The polynomial doesn't seem to have any structure in it really. The plus and minus signs alternate chaotically as well as the quantities of X's. I would really like to know more about this.
Each term in the sequence depends on the last one. So it's a kind of recurrence relation, like the Fibbonacci sequence. If you write the recurrence as a 92-by-92 matrix, the polynomial should show up in the matrix's characteristic polynomial.
Wolframalpha
@@ignorantFid With more detail, as each element has a certain number of digits, to get the number of digits in the sequence you only need to know the number of times each element appears in the sequence, not the position. This allows the relevant information for the proof to be captured with a 92 dimensional vector, even though the order information is missing. This also defines a metric on the vector space (form of L1 norm) to get the length of the sequence. Each element generates a specific number of specific other elements regardless of where it is positioned, so the transformation from the vector summarizing one sequence to the next is linear and thus can be represented by a matrix. I'm guessing the determinant must give the average impact on the L1 norm after the transformation is applied, which is where the characteristic polynomial and Conway's constant show up.
R.I.P. John Horton Conway, born 26 December 1937, died of COVID-19 on 11 April 2020 (aged 82).
I remember being given this problem as an extra credit problem in my geometry class. Of the ~25 students, two of us got it right (this was back in HTML1.1 days). That evening I got home and began playing with it. It consumed me for a whole month. Even writing my first REAL bit of code on a machine my uncle gave me to print out on screen the sequence and navigate it forwards and backwards and to store sequences and highlight them in other numbers. This stupid little puzzle my teacher gave me 5points of credit for solidified in earnest my fascination with computers and programming that has endured to this day. Ahhh the nostalgia, THANKS BRADY!
Reminds me of the "only number that describes itself": 10233221, with the hundreds thousands' place being independent. Read audibly, it says: " One Zero, Two threes, Three twos, Two ones.", describing itself. Sourced from some book somewhere.
What about 22?
Could it be Finding Moonshine? I'm not really sure but that's where I read about John Conway for the first time
@@smiley_1000 Actually, 22, 4444, 666666, 88888888 should all work.
@@davinchristino No. Because 4444 gives "44" and then "24" and then "1214" and so on. 666666 gives "66" and 88888888 gives "88". It's only 22 that continues forever : 22, 22, 22, 22...
Did the student get credited in the paper?
For what?
For giving Conway the sequece@@alejrandom6592
Brady you should spend a week with this man!
...and Dr. Holly K.
doccleveland you doctors!
"none of the boys wore long party dresses, or if they did they were indistinguishable form girls" - So funny.
When I saw that some of the numbers contained 3113, for a moment I thought the 1's were Knuth up-arrows.
Lol
the PNG file format uses this technique to compress images.
Let's say you have an image with a bunch of red pixels, and then a bunch of blue pixels. You can store the images as *amount**color*, so you would go for example 20Red, 40blue, etc. It uses some more tricks for stuff like gradients, but in essence it's the same technique.
Run Length Encoding.
And that is why PNG is smaller (and crisper) for screenshots of an application but JPG is smaller (and not so noticeably compressed) for photos and such.
Actually it doesn't. PNG uses deflate compression in combination with prediction. Basically, it replaces patterns it has seen before with pointers to those patterns. There's a computerphile video on the compression technique used.
I don't care if Professor Conway reads the phone book, more please. Many more please :-)
I would tend to agree but if I recall correctly, Conway at one point in a past episode didn't try to be polite about the fact that the interview bored him to death so I wouldn't count on it.
Penny Lane Yeah, that was heartbreaking.
Penny Lane After watching it once more I think it was meant in an ironic way. He says he’s doing interesting things and clarifies that he means doing mathematics and not what he’s literally doing at the moment (giving an interview). I guess that guy simply has a very odd and dry sense of humour.
that's gonna be a bit difficult now
Yeah, guys like Matt Parker are great educators and attract new "followers"... but there's this strange magnetism in those "old professors" like Conway
The roman numeral "IIII" at the end made me think of counting in Unary (base one). If you count in Unary, all the numbers in this sequence are in exact sequential order.
I've tried this but a bit differently, instead of counting all the same consecutive digits I counted all the same digits in the whole number, and discovered that it eventually starts alternating between 2 numbers for a seed of 1:
1
11
21
1211
3112 (here is the difference, I count all the 1s in the whole number at once, not only the first one)
132112
311322
232122
421311
14123113
41141223
24312213
32142321
23322114
32232114
23322114
I was intrigued by this and implemented a generator for this sequence in python. It seems every single-digit input produces one of two cycles:
1,2,3,4: They all produce this exact same pattern after a few iterations: 32232114 23322114, also, all four reach 23322114 first.
5-9 and 0: They all produce constant sequences after a few iterations, and all limits look the same except for the last digit (the input digit): 332231141X, with X being the input digit (5-9 or 0).
I tried a few other random integers
Franz Luggin that's awesome
This is known as a "pea pattern"
There's two mathematicians named John Conway born in the 30s. Stop it universe, just stop it.
There would have to be a prime number of John Conways, wouldn't there!
Sorry, but isn't 2 a prime number?
Though this Conway is older, the two were birthed independently.
The sequence of John Conway just split in the 30s, and both halves continued correctly.
K.o.R Underrated reply :)
This was really interesting! These kind of video's make me increasingly interested in the mystery of numbers.
Rest in peace, legend!
I heard this puzzle long ago on a radio show called "Car Talk" - would have been 1987 or later, though, when the show went national.
Thank you for this fun and fascinating video. If I may make a suggestion, I think a Numberphile series on Surreal Numbers with Professor Conway would be an excellent topic.
COnway videos are the best! i really RELLY hope we will see more in the future!
I managed to work out the sequence before Conway revealed the secret, but to be honest, I think the title "Look-and-Say Numbers" somewhat made it easier to deduce, than it would have been for Conway in the original setting, not having a snazzy video title to clue him in :)
The legendary John Conway on TH-cam! What great times!
the fact that this recursion has a characteristic in another chaotic seeming recursive matrix(life or so , in conway's terms) is beautiful. and , such interlinks with no relations whatsoever, "catch my fancy". colouring such numbers, also brings them to life, bringing out patterns, with "no relation whatsoever".
Numberphile, you do a lot of videos on imaginary and complex numbers, but have you thought about doing on e about quaternions (a + bi + cj + dk)and their properties and uses?
I fully admit I don't understand a lot of the videos on this page.... I can see the concept and believe the speaker. But I enjoy them all none the less. I was excited when this one started because I was on board and was keeping up right up until he started putting chemistry in it. Then I was only partially on board. lol.
1 MILLION SUBCRIBERS! Congratz, guys!
When I first saw the sequence in the form of a problem I was around 11, and I though I was the only one who though deeply about it. Of course the only real conclusion my 11-year-old self reached was that there could never be a 4.
skele56 Same
It's my favorite sequence of numbers and I spend many times in school after tests when I was bored and I had much fun with it.
Rest in peace John Conway.
This is a good version of the "petals around the rose" problem. Since I was already acquainted with that one, solving this one took me one second.
This is fantastic! Thanks for sharing!
I'm seriously loving this..
I'm happy he got to talk about something other than the Game of Life.
Congrats On 1 Million Subs!
it's not to a million yet! give it an hour or so haha
Wow, i missed this one. Great episode!
Brilliant video as always!
If you've got John Conway handy for any upcoming filming sessions, would you mind asking him to explain that Blue Eyed Islander riddle? Would you film a Numberphile on the topic with the man himself?
As far as I know Mr. Conway provided the first solution (certainly one of the most accepted solutions), he used a concept of common knowledge. I've read solutions, they go: “if you know a person knows, you know he's got blue eyes”, and then I can't understand anything. It's rather logic-based, as opposed to arithmetic.
It's an excellent riddle and the solution is genius. If Mr. Conway and yourself have time to film such a session, it would be a historic moment in the riddle's history, and something I think the world would cherish forever.
Either way, thanks for all your work Brady and Mr. Conway!
Absolutely amazing!
advent of code brought me here
Yay, I love this sequence!
I like videos like this, they give me a rare glimpse into just how smart some people are.
I love this Conway stuff
Came on 1/11/21 (to us Americans) to pay respects. RIP Professor Conway
Very cool. I'm gonna try this on my friends.
this channel consists of the nerdiest of nerds... I love it
I'm so glad I'm alive at the same time this man is.
Fascinating!
SURPRISE!!
Grats on 1 million subs!!!
I had fun typing this out over several rows. And if you align the text, you also see a nice curve (or curves it's centre align) appear.
I wish the name of this sequence wasn't in the video title; I've never run across it before, and it ruined the puzzle :(
A teacher put this on the board when I was in high school. Took me about a minute to get up and add the next line, and no one else in the class seemed to get it.
So, how does it feel to rediscover this sequence 2 years after ? ^^
I like John.
This may seem rude, but I'd like to see more old mathematicians on here.
They're like wise, yet befuddled, wizards. There is something about minds like that that I would like to find a connection too. Beyond the adorableness of it. Maybe it's their personality, still intact, their lack of common characteristics of communication, shining through coupled with the deterioration of their bodies, in this case, the vocal cords and the face. However, I imagine there is something more going on upstairs, inside. Maybe they've spent their lives investing in the neocortex, and I recall that being the first thing to go as the brain deteriorates. While it's the newest part of the brain for mammals like us, and the one that makes us so richly complex, it doesn't seem to stay intact as long as the older, more basic parts.
My apologies if that sounds morbid or disrespectful, but I like brains. All brains are interesting but it's the outliers that are particularly fascinating.
Liking brains doesn't make you disrespectful, it makes you a zombie.
.
cringe..
I’m guessing it’d be impossible to have a 4 in the sequence because that would require say “1111” which would require you to count the same number twice back to back when you could just say “21”.
It's very funny to me that someone could do a whole lecture (and have it attended by highly educated people) on the interesting mathematical properties of a riddle.
Really awesome results for such a simple math game-like problem.
@delwoodbarker, the precise words to explain the logic, chapeau...
Rest In Peace
wow, I didn't know this one, but when he showed the 4th one, I knew it already :D this is a cool sequence
RIP 😭
My dad showed this to me when i was in junior high. I filled a note book. There is also a growing pattern in the front of the number as well as the back. In the front it's every other. In the back it's every third. This pattern grows from the inside out. Try coloring the pattern green as you go down you will see what i mean.
RIP professor Conway. A truly great man.
RIP. A legend
Rest in peace, John Conway.
the question is, at witch number will you see the first 4
4 is impossible.
Imagine if you saw a 4 somewhere in the sequence, for example XXX41XXX
In this case, the previous sequence must have contained 4 1s in a row.
However, 4 1s in a row (1111) wouldn't exist, because 1111 describes the same as 21.
Samuel Hunt Unless you start with a "seed number" that either contains a number greater than 3 or contains a sequence of more than 3 of the same digit. If your seed is 4 or 1111, for instance, then you'll get 4s.
Starting with "1" as the seed? Never.
The sequence "12" gives "1112" as "y22xx" gives "y222x" & "y333xxx" gives "y333x"
There is no sequence of "y's" that will add a matching initial digit to make these "1111x" or "2222x" or "3333x"
Any such "y" would merge with the prior iteration to remove the grouping of three in the resultant set.
• 1
• 11
• 21
• 1211
• 111221
• 312211
• 13112221
• 1113213211
• 31131212131221
• 132113111211121213112211
• 11131221133112311211121113212221
• 31131122212321121321123112311312113211
• 13211321321112131221121113122112132112132113111221131221
tried everything, but yes, 4 can never achived
Matthew Prorok And even that will rapidly decay to a single 4 per "element."
I really wasn’t expecting this 😂
Is there any meaning associated with the other (complex) roots of the Conway polynomial?
This is identical to an old and simple as dirt data compression algorithm called Run-Length Encoding.
You can see that certain sequences get shorter, like 4442222 -> 3442. But random data tends to get longer, which is why these sequences always grow in length.
Nice, I've never seen this sequence before. Also: Will you maybe make a video with John Conway about FRACTRAN? Would be awesome :)
Seeing a relationship between different number bases would be cool too! like with hex, e.g. 4dd8 has 142d18, that could make the polynomial constant representation enourmous!!!
You'll be missed, sir...
1
11
21
1112
3112
211213
312213
212223
114213
31121314
41122314
31221324
21322314
21322314
21322314
i did a little adjustment and sorted the numbers instead just saying what number comes next i added all 1s and 2s ... and wrote them in that order and the last one will always repeat itself
That's really interesting. I did some tests and it looks like seeds 1, 2, 3, and 4 all lead to the same stable sequence, while any other single-digit number x leads to 312233141x.
This one is known as the "pea pattern."
Philip why
Cocodufable Haha, I haven't figured that one out either.
+TheNefari I added them over the previous two in the sequence rather than just the previous one, and I found something which goes on a lot longer and I'm not sure if it ever stops
1
11
31
3113
3133
3153
215315
31123335
41225335
3132631435
The furthest one can go with decimal is the 44th iteration:
219253247526275829
I kept going with hexadecimal, but that can only go as far as iteration 55:
31F2A324252627282292A3B5C2D
...spark of curiosity ignied the flame of ingenuity. It's interesting to see how mathematicians mind thinks over a number "problem" by relating it to a known physics phenomena...:)
If you put a gun to my head and demanded that I explain how to recreate this sequence. I think I'd end up with a new hole in my head. I don't understand how you get the third sequence.
Wow... this was very interesting... Thx
so close to 1,000,000 cant wait for their number vid on 1 million :D
We played this in 6th-7th grade a lot, starting with different numbers...
Never thought there was hard math behind! XD
John Conway-Rest In Peace
If I remember correctly, in early roman numerals, they wrote IIII instead of IV. This would probably lead to something different :)
Brady can you make a video with John Conway talking about his 'chained arrow notation'? It's like a much bigger version of Knuth's Arrow Notation.
It turns out this is a really boring sequence in unary.
Its unary what did you expect?
littlebigphil it one the prize for the most boring sequence
Lol
1
11
111
1111
11111
111111
1111111
11111111
111111111
1111111111
Haha, there's only one atom and it's 1. The polynomial is: x.
NonTwinBrothers It's*
A genius !
Thanks
RIP John Conway, what a ledgend
I think of John Conway as a mathematical mystic. What a mind.
Rest in Peace Mr. Conway...
This sequence appeared in a video game called "Please Don't Touch Anything".
He has a wonderful sense of humor.
WOW they just hit 100,000,000 subscribers!!!
Beautiful. It's like kids who never grew up and wanted to carry on playing with their lego, etch-a-sketch, etc. Just like some boys loved Tonka toys and now work on building sites...... Girls and their dollies and modelling, that's a joke......
This was already my favorite pattern but now my fave mathematician over analyzed it and nade it better
I think the name ot the video gave it away for me. Or maybe i have just seen it before.
I really need to start playing with the ideas in my head like prof.conway lol :)
We need more of the wizards!
(Conway, the heptadecagon man whose name I don’t know, and Stoll.)