I think we might have a new “neatest brown paper” champion.

@Affews100

2 жыл бұрын

That handwriting, so amazing

@callumroy8899

2 жыл бұрын

For sure a podium place

@user-lk3zc8xj4m

2 жыл бұрын

Who is the old champion? Fedrico?

@GraniteGeek

2 жыл бұрын

My thought, as well: A great video ("Serpinski Arrow" - cool!) with truly excellent handwriting

@dtitco69

2 жыл бұрын

That is indeed very tidy brown paper

It's one of those puzzles which the longer you analyze it, the more amazing properties you find. That Sierpiński triangle really surprised me!

@telectronix1368

2 жыл бұрын

I was trying to work out what the pattern would be for a 4 disc tower rather than the 3 used there.

@Fanny-Fanny

2 жыл бұрын

klaatu barada nikto

@SpencerTwiddy

2 жыл бұрын

@@telectronix1368 it’s a bigger Sierpinski triangle (4th iteration)

@YOM2_UB

2 жыл бұрын

@@telectronix1368 Start with an n ring graph. To construct an n+1 ring graph: - Place an n graph at the top, with A appended to each vertex string. - Take another copy of the n graph, append B to each vertex, rotate it 120 degrees clockwise, and place it to the bottom-left. - Take a third copy of the n graph, append C, rotate it 120 degrees counter-clockwise, and place it to the bottom-right. - Add three edges connecting: - the A graph's bottom-left vertex with the B graph's top vertex - the A graph's bottom-right vertex with the C graph's top vertex - the B graph's bottom-left vertex with the C graph's bottom-right vertex

@ygalel

2 жыл бұрын

IKR Absolutely mindblowing

Haven't watched Numberphile in a while, now there's a scottish person wearing a fractal hoody talking about my favourite maths puzzle? Feels good to be back.

@steamer1

2 жыл бұрын

Not a hoodie. Sorry for being Hanoing.

@robertveith6383

2 жыл бұрын

* Scottish

@vigilantcosmicpenguin8721

2 жыл бұрын

I like how the fractal t-shirt was foreshadowing.

@rjrastapopoulos1595

2 жыл бұрын

Same here.

@MissionHomeowner

2 жыл бұрын

The Scotch are a strange people, so loving towards England they voted against freedom from it. They are not yet ready for self-government, but they are great at explaining things.

That was one of the best videos I’ve seen in a while. Love this hidden art in math stuff.

@numberphile

2 жыл бұрын

Cheers

@1994AustinSmith

Жыл бұрын

Immediately clicked when mentioned how much the smallest piece moves.

@Triantalex

6 ай бұрын

??

My favourite solution is to assign each ring to a digit of binary and start counting from zero, every time you switch a digit from 0 to 1 you move that corresponding ring to the next available spot. This also gets you the optimal solution. This also shows very obviously why the solution is based on powers of two and that each piece moves twice as often as the next largest ring.

@ancientswordrage

2 жыл бұрын

That would make a great video

@denny141196

2 жыл бұрын

@@ancientswordrage Boy, do I have news for you. Go look up 3b1b's video on the same topic

@kantpredict

2 жыл бұрын

I immediately thought of binary when I heard the musical pattern. I taught myself to count on my fingers in binary a while ago but have since forgotten...

@g.ricepad9470

2 жыл бұрын

Fun fact: this has an application on Super Mario 64 A button challenge

@YOM2_UB

2 жыл бұрын

@@g.ricepad9470 Do tell.

I literally cannot get over how perfect the optimal tower of hanoi solve fits into 4/4 time with a single beat always missing at the end perfection

@jnbplaysgames

2 жыл бұрын

Plot twist: last note is a half note 😉

Waaaaaay back in 1979 I wrote a program to solve arbitrary size towers of hanoi puzzles in BASIC for my Computer Studies O level course work :) I've always had a soft spot for this puzzle.

@numberphile

2 жыл бұрын

Nice one.

First ever Tower of Hanoi video ever to not mention recursion! Also the most musical one...

@Toobula

2 жыл бұрын

In my opinion, not showing the recursive solution makes everything else pointless. The recursive solition is the very soul of the puzzle.

@Koyasi78

2 жыл бұрын

Why mention when you can show the beauty of recursive thinking. A simple pattern solving complex problems with attractiveness and soul. Can you dig it?

@neiljf1089

2 жыл бұрын

It doesn't mention recursion directly but it is implied by the fact you get a fractal structure

@pectenmaximus231

2 жыл бұрын

@@Toobula there are so many videos covering the recursive solution, so I think this acts as a complement, rather than treading same ground and turning 14 minute video into say a 20 minute video

@PhilBagels

2 жыл бұрын

In order to understand recursion, you must first understand recursion.

Please get Ayliean back on the channel… never thought about a tower of Hanoi musically before and have to say I could listen to it all day! Thanks for the great content :)

@ZandarKoad

2 жыл бұрын

Do you have a link to an hour long (or long) Tower of Hanoi KZread music version?

I cannot believe it. Our professor showed the tower of Hanoi problem to us this morning. What a coincidence!

@MrCommentGod

2 жыл бұрын

Cool

@subliminalvibes

2 жыл бұрын

You'll notice a global "curricular trend" amongst your favourite KZreadrs from time to time.

@lukasmiller8531

2 жыл бұрын

Did you look at recursion or time complexity?

@karthikwasudevan

2 жыл бұрын

@@subliminalvibes why is that?

@deathhog

2 жыл бұрын

Coincidentally, I was thinking about the tower of Hanoi last night as well. "How do I calculate the minimum number of moves... I never did bother to figure that out as a kid."

Me 5 seconds into the video: Ooh cool jumper she's wearing Me at 10 minutes into the video: HEY WAIT A MINUTE

That was great - more from Ayliean please! Loved the secret jumper spoiler.

@murphygreen8484

2 жыл бұрын

+

My favourite incidence of this puzzle is when it cropped up in Professor Layton while I was relaxing after a long day on a business trip. It was the one night of my life I have spent *in* Hanoi.

@szkoclaw

2 жыл бұрын

Uhm, Google knows where you are/

@davidlyford-tilley1598

2 жыл бұрын

@@szkoclaw Sure, but Professor Layton on the DS doesn't :p

@CrazyDW00

2 жыл бұрын

Never will I see this puzzle and not think of stacks of pancakes…

@pratyushkumarsahoo8591

2 жыл бұрын

Which Layton Game it was?

@TheRabbitPoet

2 жыл бұрын

@@pratyushkumarsahoo8591 Pandora's box if I'm not mistaken

I absolutely loved it. Ayliean is so incredibly enthusiastic about the puzzle that it's contagious - you can literally feel her joy as she talks about it

I love it when patterns are easier to detect musically/rhythmically rather than visually.

I love how Numberphile can be so relevant to my life at times. I planned an activity around the Tower of Hanoi concept the other day so it was great to see it analysed in video!

@Joey-rs7uq

2 жыл бұрын

Its probably the youtube algorithm buying info from your facebook or something. xp

Me seeing this video in my feed: Cmon it's a matter of odds and evens. Me finishing this video: Mind blown.

You know you've found a real mathematician when they stop to think about 64 minus 1. "Now, what number system are we using...?"

@SpencerTwiddy

2 жыл бұрын

Actually, in all number systems 64 - 1 is still written 63

@SpencerTwiddy

2 жыл бұрын

The value of the number depends on base, and we could be in any base above seximal

@digitig

2 жыл бұрын

@@SpencerTwiddy Oh, if you only have to worry about *base*, sure... :)

@SpencerTwiddy

2 жыл бұрын

@@digitig I guess. I might have missed the joke in your comment, but to me I feel like not doing 64-1 quickly is exactly what a NON-mathematician would do

I've never drawn the state-space graph of the Hanoi towers. Had no idea it has something to do with Sierpinsky triangles. Thank you :-)

@vigilantcosmicpenguin8721

2 жыл бұрын

Wow, I thought everyone in the world had drawn the state-space graph of the Hanoi towers!

I love how this made like 5 new connections about concepts I already knew.

@WestExplainsBest

2 жыл бұрын

The great thing about math is the number of connections that can be made. Its beauty comes from the complexity derived from perceived simplicity!

I can't believe they didn't explain *why* all these elegant mathematical relationships are there. Let's take the 6 tower as an example. When you want to solve that, how do you do it? You need to move the base from the spot it's on (let's call that A) to the new spot (C). To do that you need to move all the ones above it off and restack them on spot B. Then, once you've moved the base to C, you can restack everything else again, but this time on C as well. So to solve a 6 height tower, you're solving the 5 tower, moving the base, and solving the 5 tower again. And when you solve the 5 tower, you're really solving the 4 tower twice, and so on. That's why each new disk doubles the time it takes to solve it. That's why the Sierpinski triangle map works: a Sierpinski triangle is composed of a smaller Sierpinski triangle on top (solving the n-1 tower the first time) and two more on each side (solving the n-1 tower the second time, but either in the B or C slot). Both the Hanoi tower and Sierpinski triangle are self-similar, i.e. made up of smaller versions of themselves. It's why you get those patterns in the movement of each disk. I think the math is a lot more beautiful when you explain why it all works

@SpencerTwiddy

2 жыл бұрын

3Blue1Brown already has a video on exactly this!

@MrAlRats

2 жыл бұрын

@@SpencerTwiddy Which one?

@SpencerTwiddy

2 жыл бұрын

@@MrAlRats It's a 2-part video titled "Binary, Hanoi, and Sierpinski" imo the best ever made on the subject

When you did it again without the music, the notes played in my head automatically and I love this so much....this gives me such an appreciation for the makers of this video

I love arpeggios, and I gotta say that optimal solve slapped.

@SpencerTwiddy

2 жыл бұрын

There weren’t any arpeggios in it but it did sound pretty🤪

This is so beautiful! Definitely one of my favorite videos from Numberphile!!

@numberphile

2 жыл бұрын

Thank you

I did not expect this video to be so interesting. I thought, Tower of Hanoi, done to death, but no, this kicked the door down.

Ah dang, she pulled a sneaky one on us! I thought that was just a cool shirt!

I never knew how to play this when I was young. I just thought it's supposed to be stacked randomly for fun

@Koyasi78

2 жыл бұрын

The philosophy in this quote is amazing!

@vigilantcosmicpenguin8721

2 жыл бұрын

Ah, to be young and carefree.

1:18 My brain kinda expected the music to go for 2 more notes

Have you noticed that this starts with the Dies Irae? The first four notes, C B C A, are a very famous and very old theme that is called “Day of Wrath” in Latin and is often found associated with death in movies!

@SpencerTwiddy

2 жыл бұрын

Hear for yourself at 1:42 - 1:45

*flashbacks of the tower of Hanoi puzzle from the Noveria mission in Mass Effect 1*

I love things that unexpectedly end up being the Sierpinski triangle.

Great episode! I love the "visualisation" using music. What is that called? Audiation?

@musik350

2 жыл бұрын

Sonification :)

@peterhansen5804

2 жыл бұрын

Notification ;-)

@felipevasconcelos6736

2 жыл бұрын

It’s called sonification or auralization.

Loved the musical element to this demonstration. We had an eight-piece tower growing up in the 1970s. All the wooden disks were brown. PS - She has beautiful penmanship.

Now I can compose music without any music lessons. Thank you.

This was the project I did a week ago! I solved it by using an algorithm where odd-numbered discs must sit on even-numbered discs. In case there exists an empty peg, the disc must be moved there. Like that.

She's great, hope to see more of Ayliean in the future.

I learned the proof in Further Maths - we now play the game with Year 6 students joining our school to see if they can notice any patterns. UKC does it as part of their secondary school enrichment days too. Kids seem to love it!

This was the most intuitive explanation of Towers of Hanoi I've seen so far. Back when I was taking Computer Science at school, I learned about the Towers of Hanoi, but my teacher never quite managed to communicate how exactly to solve them, other than stating "you need recursion to solve this problem". He never really elaborated on that much further and I've never quite grasped that. I think I understand a lot better now.

@shadowshedinja6124

2 жыл бұрын

It's recursive because in order to move the bottom piece of a Tower of Hanoi of size N, you must first move all of the other pieces into one stack, which is the same process as solving an N-1 tower. Once you move the bottom piece, you solve the N-1 tower again.

@1994AustinSmith

Жыл бұрын

@@shadowshedinja6124 Knowing the human trick to it, I'd just loop (psudo-code) Starting from left peg $smallest = left(odd)/right(even) 1 If peg left of $smallest is greater/less than peg right of $smallest {next valid move} Repeat

WoW! That a puzzle I did as a kid ends up generating a Sierpiński triangle... AND in a way that I can actually UNDERSTAND... mind blown!

Omg best Numberphile yet, love your content Ayliean

Love the Tower of Hanoi! It's so simple and profound. I keep learning things from it and it seems that the learning can never stop!

It’s actually a pretty nice puzzle to try and figure the method to solve it by yourself. I remember doing that in problem solving class in middle school. It is not as hard as it looks, you just have to spend some time

That's also really cool because you can see how the Serpinski triangle keeps growing as you add more disks! Imagine adding an A to the end of every node of the one MacDonald drew out, which would represent having 4 disks, and then starting from the BBB and CCC corners (now BBBA and CCCA), you can now move the bottom disk - and from there you get identical copies of the original triangle except now the last letter is something else - BBBA opens up BBBC (and a copy of the original triangle with "C" as the last letter) and CCCA opens up CCCB (and the "B" ending triangle!)

The Tower of Hanoi was the subject of a classic Doctor Who episode in 1966 and I remember discussing with my maths teacher at school the optimal solution (without music) which I had worked out.

This was so good! I hope we get to see Ayliean again. The jumper tie-in made me so happy haha

Mathologer's video on the topic is highly recommended. It's called "The ultimate algorithm".

It gives you goosebumps when she is unfolding Sierpiński triangle. Math is awesome

I knew it from the start that Sierpinski Triangle will come up. I mean, I’m old enough to notice how thoughtful mathematician with their tee shirt. Amazing video btw, great representation using notes on tower of Hanoi Love it!!!

Wow. Thee music of that alone gave away the inevitability of the solution.

Mass Effect 1 fans will know

its actually quite simple There are a total of f(N+1)=2f(N)+1 operations that needs to be done This hanoi works in a recursive manner. To move every disk from 1st location to 3rd location for N+1 Disks you would need to apply below 3 steps in a recursive manner: 1-You would first solve the problem for N disks and move all of them to 2nd location. (f(N) operations) 2-You would move the N+1th disk to third location (1 Operation) 3-You would move N disks on the 2nd location to 3rd location. (f(N) operations)

This is fantastic. Tempted to go plug a few instances of this pattern into my sequencer and see what comes out of the synths...

Every time i come upon these in a video game from now on I'm gonna watch this video again 👍👍👍

The moves also increase in a 2x+1 Pattern where x is the moves it take for the previous number of disks.

Beautiful way to show this puzzle. Now I know Ayliean and after a few videos of her on TikTok I'm listening "The Less I Know The Better" what a nice afternoon :) Thank you for all your excellent work.

That musical note thing was absolutely amazing.

As soon as I saw her shirt, I was thinking: "Oh, it is going to be one of THOSE!" - I was not disappointed. Great video. I love seeing familiar things pop up in unexpected ways :)

We programmed the problem in class a week ago, its really simple using recursion.

@chrisseddon5823

2 жыл бұрын

Obama, still making the world a little better every day.

@lukasmiller8531

2 жыл бұрын

the classic!

@XtecHubble

2 жыл бұрын

lots of free time now..

@chrisseddon5823

2 жыл бұрын

@@XtecHubble Michelle would disagree.

A great episode Brady. Thanks to you and Ayliean! This was a novel take (for me) on Towers of Hanoi. The graph representation of the permissable moves is so interesting!

This is the most amazing video on tower of hanoi!! OMG how crystal clear explanation!!!

Super great video ! The idea of using a tune to materialize the moves of each block was brilliant ! Another proof maths and fractals can turn musical !

@numberphile

2 жыл бұрын

Thanks

I am actually born in Hanoi, and so happy that have a puzzle named it. And now that puzzle is actually on Numberphile, one of the best Math KZread channels ever. I can't believe that this can happen!

7 with 127 steps could be pretty interesting musically. Certainly gonna work that one out on my guitar. Should shred something wicked at 127 bpm and a gnarly fuzz! Gonna take some practice though... 3Blue1Brown did one on Tower of Hanoi too. The optimal solve is the same as counting up to 15 in binary.

@ZandarKoad

2 жыл бұрын

Do you have a link to a lengthy musical rendition of this? Would love to listen while I'm working...

@awandererfromys1680

2 жыл бұрын

@@ZandarKoad No, and I can't garantee anything really. Got no decent recording hardware atm. But here are all 127 notes, for whoever is interested: C-D-C-E C-D-C-F C-D-C-E C-D-C-G C-D-C-E C-D-C-F C-D-C-E C-D-C-A C-D-C-E C-D-C-F C-D-C-E C-D-C-G C-D-C-E C-D-C-F C-D-C-E C-D-C-B C-D-C-E C-D-C-F C-D-C-E C-D-C-G C-D-C-E C-D-C-F C-D-C-E C-D-C-A C-D-C-E C-D-C-F C-D-C-E C-D-C-G C-D-C-E C-D-C-F C-D-C-E C-D-C Quite a fun little finger exercise once you get your head around the pattern.

Putting it to musical notes was interesting. I kept thinking about the preserved segment from one of the lost Dr. Who episodes where the Dr. is challenged by the Toy Master who challenges him to the Tower of Hanoi in 10 steps.

That music sounded really good, thanks!

I like to imagine Ayliean has spent years determining optimal Tower of Hanoi solutions in this old brick room (perhaps in a tower itself) with its 1960's radiator and Christmas lights which have remained on since 2006. As soon as she solves a 50-disk problem, Eric Liddell himself comes and frees her.

it would be neat if we could hear the musical sequence for a 12-disc solve so we could hear what it sounds like with all the notes of the 12-tone equal temperament system

@ZandarKoad

2 жыл бұрын

Has someone done this? Can you link to it? Would love this as background music.

I love these math nerds that are super nerdy to a very very specific sub category and are so passionate about it. Reminds me of the guy with a basement full of Prince Rupert drops. Meet one of these beautiful bastards early enough in life and you're a math nerd too for ever!

I learned most of this from experimentation, but learning its shape and mathematics was truly enlightening. Thanks for sharing!

WOW. From Hanoi tower to Serpinski triangle and Hamiltonian path. Mind-blowing.

Thank you! I did the musical notes, using a pentatonic scale and six disks, and programmed a four-track synthesizer with different note values for each track, 1/8, 1/4, 1/2, 1. It sounds so cool.

I had an exhibition and I'm in the maths club Your video made me easy to do it Thank u soo much❤❤❤

That puzzle must be way harder when you're trying to solve it in time with the music! I like Ayliean's accent; it takes me right back to my youth.

I usually just think of this puzzle inductively. If you need to move an n-stack from position 1 to position 3, simply move the top (n-1)-stack to position 2, then move the bottom piece to 3, then the (n-1)-stack to 3. This gives the same solution but its easier to remember and easier to reconstruct if you forget (though maybe worse for speed solving?).

@PanduPoluan

2 жыл бұрын

That's a recursive solution in CS. hanoi_solve(n) => hanoi_solve(n-1) + move_bottom + hanoi_solve(n-1) hanoi_solve(0) => do_nothing

It's been a while since we last had a new face on Numberphile. Great video in every way!

In a past job I did a lot of work with PostScript. One of my customer demos was the Towers of Hanoi: feed a PostScript file to the printer, it thinks for a second, then it prints a page with the moves on it.

Move the tower above the largest disc (may be a tower of 0 discs) to the spot that is not the destination, move the bottom disk to the destination, then move the remaining tower to the destination. This is actually enough of an algorithm to solve it. Just apply it recursively. Or, to make it a bit simpler to understand: If we are starting in A and need to go to C, the bottommost disc needs to go to C, the one above to B, the one above that to C, and so on.

Kudos to the editor for audio & video note work

Another game could be to start with the stack alternating colours, use 4 pegs/spaces and aim to build two towers, each containing all of one colour.

Brilliant! Thank you!

Any KOTOR fans here who remember this puzzle at the sith temple?

@TheChumm

2 жыл бұрын

In the Old Republic MMO, one of the first raids has a mechanic where multiple players solve a 3 disc tower multiple times over the course of a boss fight. The three switches are spaced out too far for one person to do it, so it's either one person calling out the puzzle to a couple others, or 2-3 people silently working it out together. It works really well as a collaborative puzzle, especially when you're rewarded for doing it faster

@AdamLeuer

2 жыл бұрын

Almost every BioWare game has a Tower of Hanoi-themed puzzle in it at some point.

I just sent this video to my Dad because he texted me complaining that he was playing Mass Effect and couldn't get past "the puzzle in Noveria".

I instantly recognized the optimal solve music as "The Ruler Song", a melody that my friends and I "discovered" in grade school while looking at (American) measuring sticks. Every inch was marked with a long line, every half-inch with a shorter line, every quarter-inch with an even shorter line, and so forth down to the sixteenths of an inch. To play The Ruler Song, simply "read" the ruler from left to right and play a note at each marking: the longer the line, the lower the note. So cool to hear the same tune again after all these years in a new mathematical context that's also related to the powers of two.

I thoroughly enjoyed that. I had no idea where it was going at the start.

Always love it when Math + Music are combined.

I would love to hear a version of that piece walking down the whole scale.

The audio representation of the optimal solution was amazing. Nice job on that and the animations Numberphile ! I would really love to hear the audio version of solutions for N > 6. If you produced an algorithm to generate these audio files, please share a few more solutions where N > 6. I think others would like that too.

Mind blown each time this is explained.

its so cool seeing how maths patterns translate into different forms of itself, from numbers to nature to music. its so cool. almost would be cool to have a small 100 hour stack sitting in a room as a symbolic sculpture, a reminder upon looking at it of the hard to imagine time something you can so easily see could take.

@BuShips

9 ай бұрын

Have you ever seen Close Encounters of the Third Kind?

How on earth can so many people dislike this video?! It's brilliant! Great upload, thank you.

If you want to solve by having the tower on the C position, or how programmed versions tend to check for completion: Redefine positions ABC as 123. Playing with an odd number of blocks: odd block moves to {position number + 2 mod 3}, even block moves to {position number +1 mod 3} Playing with an even number of blocks: odd block moves to {position number + 1 mod 3}, even block moves to {position number +2 mod 3}

12:52 Once you can hear it as 'hanoitonian' path, you cannot unhear.

Very cool. Love the music. thanks for posting! :D

This is very interesting and engaging. Well done!

Your hair is amazing, and it matches perfectly with your tower😍

the sonified result sounds a lot like the main theme of Michel Corrette's first concerto for organ!

Watching you solve it with musical notes was beautiful.

Rather than arranging the pieces in a row arrangle at the points of a triangle. You notice that odd numbered pieces travel in one direction and even numbered pieces travel in the opposite.

Some 70 years ago my grandparents had this puzzle which fascinated this 5 year-old. When was it called Tower of Hanoi, I have never heard this name before?