343867 and Tetrahedral Numbers - Numberphile

[@numberphile]

Bug Byte puzzle from Jane Street at bit.ly/janestreet-bugbyte and programs at bit.ly/janestreet-programs (episode sponsor)

[@CheckmateSurvivor]

120 is also a triangular number that I am using in Pyramid Chess, a pyramid of 120 hexagons.

[@OwlRTA]

seems more like a bean dish puzzle!

[@ChrisTian-uw9tq]

can anyone explain this differently? "There exists a non-self-intersecting path starting from this node where N is the sum of the weights of the edges on that path. Multiple numbers indicate multiple paths that may overlap." Not quite catching how it relates to the numbers in the graph

[@Artaxo]

@@ChrisTian-uw9tq You can follow any path and choose when to stop. The edge weights you pass (not the nodes) need to sum to the number (or one of the numbers) of the dark green node.

[@ChrisTian-uw9tq]

@@Artaxo Then how is the pre-populated 31 meant to have its following edge filled to sum to 31 if max number allowed is 24?

[@woody442]

The stop motion is georgious. Appreciate the effort

[@numberphile]

By our man Pete 👍🏻

[@woody442]

@@numberphile Thanks Pete! :)

[@harriehausenman8623]

@@numberphile Thanks Pete! :)

[@brianbrianbification]

Pete ftw

[@stephenbeck7222]

Wait you didn’t just put an overhead camera on top of James’ paper and let him slowly move all the dots around then edited out the hands?

[@brouquier7172]

I've come to the comments section to write how happy I am to see Dr James Grime again on Numberphile and how much he's been missed, but I see everyone's done the same thing already!

[@CallousCoder]

A big applause for all the stop motion inserts and the clay balls and the discs! Wow ❤ I adore the clay Bollocks run Pollocks 😅

[@sergio_henrique]

I wonder if it's actually stop motion or if it was just made to look like stop motion (like the Lego movie).

[@pmcpartlan]

​@@sergio_henriqueall real, moving little things around and taking photos

[@burnttoast6924]

Very happy to see Dr Grime back on numberphile!

[@allasar]

Whoever animated this episode, you earned your paycheck.

[@ClayGordon]

Reminded me of an episode of Gumby.

[@forthrightgambitia1032]

For reference Lagrange actually proved any number is the sum of four squares. Which is why it is usually called Lagrange's four-square theorem.

[@sethpeck7179]

I loved that game when I was in grade school

[@smylesg]

6:34 The Fermat-Haran Conjecture 😀

[@respitesage]

I always remember triangular and tetrahedral numbers because of the song 12 Days of Christmas. If you interpret the lyrics as listing all gifts up to that point (including previous days), then the running total of gifts is the first twelve triangular numbers. If instead you interpret it as listing the gifts for only that day (i.e. the gifts from all previous days are given again, leading to, e.g., 12 partridges in 12 pear trees) the running total of gifts is the first 12 tetrahedral numbers.

[@hughcaldwell1034]

I think having that damn song stuck in my head in class was the reason I worked out the tetrahedral formula.

[@GaryFerrao]

8:34 “I said «Pollock’s», you’ve heard me quite distinctly.” 😂

[@GeorgePlaten]

The only mathematician owned by a dog

[@flickingbollocks5542]

Sounds like ☝️

[@shruggzdastr8-facedclown]

Was he trying to make sure that people weren't mishearing him as saying "bollocks"?

[@ericherde1]

@@shruggzdastr8-facedclown I think so. It took my a while to realize that since it isn’t used as profanity (or really at all) in my dialect of English.

[@talastra]

Pollock's conjecture is bollocks. Or, alternatively,, Pollock's conjecture is the dog's bollocks.

[@SherlockSage]

EYPHKA! Delightful historical coincidence that you can still write this Greek word with Latin characters

[@aftertwentea]

ЕВРИКА

[@jlljlj6991]

ΕΥΡΗΚΑ is not EYPHKA 🙂

[@zmaj12321]

@@jlljlj6991 I see what you did there

[@drenzine]

​@@jlljlj6991 oh don't go splitting hairs

[@WillBinge]

@@jlljlj6991I can’t tell the difference

[@alansmithee419]

Gaus and Euler, the people who took a look at mathematics and went "that s***'s boring, but I can fix it."

[@JamesDavy2009]

The latter being the guy who gave us the base of the natural logarithm and the formula: e^πi + 1 = 0.

[@alansmithee419]

@@JamesDavy2009 Honestly the two were so important that listing any one thing they did as an example feels like it can only ever understate their contribution. Even that formula is just one example of an expression that drops out of what is an entire mathematical framework that Euler pretty much constructed from scratch, and that entire framework is just scratching the surface of his contributions to mathematics.

[@akshaj7011]

@@JamesDavy2009 Many things in math are named after the second person who discovered them, because the first person was always Euler.

[@marvindet3775]

James is really Mr. Numberphile =D

[@YuriFurtado]

The animation / stop-motion is looking smooth as heck

[@onecupofconsciousnessplease]

I haven't watched the video yet, but I'm very excited about the combination of Numberphile, James Grime, and a specific large number.

[@numberphile]

A perfect storm

[@harriehausenman8623]

@@numberphile Superior highly perfect storm 😉

[@АністратенкоАнтонінаВіталіївна]

Always waiting James' videos❤

[@nicolasfpauly]

Same 😅❤

[@benjamingarrido5494]

I watch your videos, I don't understand anything about numbers, but I like your enthusiasm and your healthy joy, greetings from Chile

[@harriehausenman8623]

What wonderful video! As usual, perfect presentation by Mr. Grime and a generally very interesting topic 🤗 Thanks so much. 🙏

[@numberphile]

Glad you enjoyed it! Cheers.

[@palestinianperspective]

I love maths! James adores it.

[@jacksonstarky8288]

James' closing comments are spot on. I was in high school (late 1980s for me; my brain is very middle-aged now) when I found the pattern of adding consecutive odd numbers to generate the square numbers, and then I figured out that the Nth level difference between consecutive N-dimensional numbers was N! (N factorial)... it's easiest to see this with the square/odd numbers, in which adding 2! starting at 1 generates the odd numbers. I found some hiccups in the first few iterations at each new power, but in general the pattern normalized at N^N.

[@azlhiacneg]

Fun fact: 2024's the only tetrahedral year all our lives~ And there's a book all about triangles coming out later this year! Seems like a triangle-y type of year~

[@stickfiftyfive]

True, but 2024 is also the only year we'll live through that is also a dodecahedral number and the first one since 1330. Every (3n + 1)th triangular number is the nth dodecahedral number.

[@528Circle]

That IS a fun fact!

[@zawbones5198]

For anyone curious 1771 was the last one and 2300 will be the next!

[@stickfiftyfive]

2024 is also the only dodecahedral number year we'll live through.

[@neildegrassebison]

Great to have you back on Numberphile, James, and thanks for the video! And congrats on the ring 😉

[@derekhasabrain]

It’s incredible that to this day, every episode gets its own special animation to make visualize the lesson in a delightful way. Stop motion!! Brady you animate so well!

[@deliciousrose]

Classic Numberphile with the OG presenter! ❤❤❤ I'm happy to see James again, being guest in other channels. Hopefully he'll upload new video in his own. 🤞🏼

[@black_platypus]

Loving the sound effects! Has a very 70s animation vibe (or thereabouts) ✨

[@mojeogame]

I really appreciate the precision with saying (every time) that any POSITIVE WHOLE number :)

[@phonomancer_thepossum6279]

This guy makes maths ALOT more fun than when I was in school.

[@PapayaJordane]

11:33 this is exactly why I started working on the Collatz conjecture. I knew I'd learn a lot by thinking about the numbers and how they connect, and I was right.

[@michaeld5555]

I don't know exactly why but this is the most beautiful fundamental proof I've stumbled upon in Mathematics thus far. Thanks so much for making this video!

[@NoNameAtAll2]

GRIIIIIIME I MISSED YOU, MAN welcome back, singingbanana!

[@agrajyadav2951]

Fr

[@MrsAjergirl]

That joke about Fermat's margins is so granular, and I'm 100% here for it

[@davidiverson5928]

Parker squares are the 21st-century version of Fermat margins.

[@cordial001]

I love James' little speech at the end of this

[@danix30001]

Amazing video as always, I’m glad with the stop-motion, can’t imagine how much work it took to make

[@maxaafbackname5562]

Nice! I love (that) stop motion!

[@vsm1456]

today I was reminded about figurate numbers and went to read more about them. and now you release a video :D love this coincidence!

[@sadaharu5870]

Glad to see James Grime again!

[@fwekker]

2:54 'try and go even further' sounds a lot like 'triangle even further' lol. was that intentional?

[@johnrichardson7629]

My favorite tetrahedral number fact: The numbers along the finite diagonals of the multiplication table sum to the tetrahedral numbers. 1, 2+2, 3+4+3,4+6+6+4, ...

[@rosiefay7283]

4:19 Funny: The first way that occurred to me was one you didn't mention. Seeing as 4|28, I divided it by 4, getting 7=4+1+1+1, then enlarged, getting 28=16+4+4+4.

[@Matthew-bu7fg]

I love how we can shine a light on an arbitrary number like 343,867 with this channel Also always great seeing James in a video!

[@richardlynch5745]

my favorite presenter on Numberphile 👍👍 1:24

[@joelproko]

Given that you seem to need at most 5 tetrahedral numbers to construct any number and at most nine cubes, it would seem that one would in general need at most n+1 3D-numbers to construct any number, where n is the number of vertexes the 3D-number has.

[@bfmdsm2020]

Fun fact that i just found out: If you, let's say, color tetrahedral numbers in Pascal's triangle, you'll notice a straight line of colored numbers! This also works with triangular numbers and tentatopal numbers (numbers that can be arranged as a hupertetrahedron, though I'm not sure tentatopal is the correct name)

[@maynardtrendle820]

Good to see James Grimes again!🌞

[@harriehausenman8623]

My favourite banana! 🍌

[@spaceyraygun]

i've used triangular numbers to verify if a group of unique integers (in any order) was a gapless sequence or not. i was goofing around with some very basic arithmetic and i kept getting results that were oddly familiar. they turned out to be triangular numbers! around this time i had just been introduced to triangular numbers from numberphile! my specific use case was to determine if a set of years had gaps in it. turned out that there were much easier ways for me to do this programmatically with code, but i'm still proud of having such an epiphany as a non-mathematician. i have a working demo and explanation that i can link to, but i don't want this comment to go to spam jail! basically, the formula is this: `(max(set) * length(set)) - sum(set) = T(length(set) - 1)` where `T(n) = (n * (n + 1)) / 2`. `length` is the amount of entries in the `set` of unique integers.

[@benjaminpedersen9548]

It is a cool find and definitely works assuming the integers are unique, however, if you know the maximum you probably also know the minimum and thus max(set) - min(set) = length(set) - 1 is likely easier to check.

[@spaceyraygun]

@@benjaminpedersen9548 lol of course i was overthinking it! it's funny because i did think of something like this but i must've forgotten to -1 from the length before i derailed and went on this magical journey. also, i almost immediately found another way to do this leveraging the native features of the programming language i was using. i ended up not using my original idea at all. but i won't let that take away the epiphany i got from this "discovery", however useless it may be. 🤣 thank you for the simplification!

[@IamGod13th]

1. So if we name triangle-, square-, pentagonal- etc numbers as "plane" numbers; 2. And we have proof that we can write any whole number as sum of 1n of n-numbers for "plane" numbers; 3. Also we can name tetrahedral-, cube-, dodecahedral- etc numbers as "volume" numbers; Could there be relation between shape of plane and quantity of planes to describe how many "volume" numbers we need for different shape of volumes? Or something further beyond: relation between quantity of planes and volumes, and shape of these planes and volumes for description of "hyperspace" numbers?

[@twt2718]

I love that Gauss uses the same asterisk I his writings that I overuse today.

[@harriehausenman8623]

✺✺✺ I switched to the Sixteen pointed asterisk 😄 ✺✺✺

[@danielw.4876]

My favorite tetrahedral number is 4060. It is the 28th, and it is exactly 10 times bigger than the 28th triangular number which is 406. And 28 itself is a triangular number

[@danielw.4876]

Also, the digits of all these numbers each add up to 10

[@Sillu129]

I have encountered a lot of content on this channel where people have checked a conjecture up to a very large number but with no proof, i think it would be rather more useful to learn about all of the anomalies unproven conjectures which even after checking it up to very high numbers would eventually show something unexpected. Knowing about all of the anomalous unexpectancies would give one a good head start approaching any new theories.

[@ophthojooeileyecirclehisha4917]

thank you so much for your kindness and information

[@publiconions6313]

Numberphile's vid editor is probably my favorite person in the world that I don't know

[@bigpopakap]

I think it makes sense to me that it doesn't require more than n n-gonal numbers. Here's my hand wavy intuition/psuedo-proof: Lemma: any sequence of n-gonal numbers starts as "1, n, ...". This is almost by definition: you start with 1, then add as many red checkers as it takes to make n sides. Of course, that's n checkers total. So the second number in the sequence is n. So now let's just keep adding checkers (start with 1, then 2, etc.) to see how to arrange them into at most n n-gonal numbers. If we add 1 checker, it might take 1 more n-gonal number. If we add 2, it might take 2 more n-gonal numbers (a 1 and another separate 1). Once we get to adding n more checkers, then it only needs 1 more n-gonal number, because those extra n checkers can be arranged into 1 "pile" (the lemma). So this shows that every n new checkers we add, it sort of collapses back down to one extra pile. Of course, that alone doesn't necessarily mean the "collapsing" keeps it under n piles *forever*, but it's some sort of intuition. I wonder how close this is to the real proof, if at all

[@WRSomsky]

I was wondering if "Any number can be written as N N-gonal numbers" is optimal? IE, for all N, do there exist numbers (for that N) such that *require* N N-gonal numbers? Or are there some N for which you can do better than N N-gonal numbers?

[@WAMTAT]

James is the best

[@scottabroughton]

This video, more than any other, reminded me of a Sesame Street episode brought to us by the number 343867.

[@courtney-ray]

How did I miss a James Grime vid! First things first: Click like! Now let’s watch what this video is about…

[@qdphi]

Wow, I just noticed that for the square numbers you used square waves and so on. Pretty nice touch!!

[@charlesmurray3255]

I noticed that but i forgot what they were called :)

[@Essin62]

Why why WHY is this so fascinating? It should be complicated, abstract and boring but it's interesting as heck and I don't know why

[@aminramazanifar9743]

Numberphile is extra special with Dr. James.

[@brumd]

It might not be the main point of the video, but, I am really enjoying the sounds in the animations. Assuming these where created by the animator, this is really classy sound design, very buchla-esque / synthi etc. It really adds to a great video; always good to see James Grime. Like +1

[@Sci-Marvels]

Your method and solution are so intresting!! Wish you the best 🙂

[@o_enamuel]

2:55 Can we triangle a little further?

[@ExplicableCashew]

Getting a new Singingbanana and a new Engineerguy video in one day, nay, within an hour of each other is *crazy*

[@leovanwinkle8812]

That stop motion was pretty sweet!

[@The_Commandblock]

Fun Fact: 2024 is also a tetrahedron number. I think the side is 22

[@TheSireverard]

Tetrahedral numbers are the running partial sums of the sequence of triangular numbers. Somebody make something of that.

[@zxuiji]

I imagine the way to prove the conjectures is through the jumps between singles. So for example with the triangle ones the jump from 1 to 3 is 2, 3 to 6 is 3, 6 to 10 is 4, 5 the next, 6 the next, you get the picture. Presumably the numbers between will only refer the the Ngonals that came before.

[@keyaanmatin4804]

Love that they still used the brown paper

[@maynardtrendle820]

I like Brady's proof by pronouncement.🎉

[@FloydMaxwell]

Great animation. The kind of thing that hooks the kids.

[@Zambicus]

The animations are great, but the synth effects i liked even more. Reminded me of those VHSes math teachers might put on in the 90s showing weird math ideas.

[@muhammetboran8782]

5:20 also that was my conjecture :)

[@somebody9232]

The difference between the same (in order like the 5th pentagonal and the 5th hexagonal) pentagonal and hexagonal number is a triangular number and then the difference between the next pentagonal and hexagonal numbers is the next triangular number Same goes for square and pentagonal Triangular and square etc Very interesting

[@chris_dixon]

What a beautiful video. Thank you.

[@acdenh]

Worth noting with the new year 2025 coming up that (1) 2024 is the 22nd tetrahedral number and (2) 2025 is the sum of the first nine cubes (related to 10:14)

[@jareknowak8712]

I love the episodes with connection to Geometry.

[@fwiffo]

So, triangular numbers t(n) are defined as sum of natural numbers from 0 to n. Tetrahedral numbers t3(n) are the sum of the triangular numbers from t(0) to t(n). What if we go 4 dimensional there? 4-tetrahedral numbers t4(n) as a sum of tetrahedral numbers? What do we know about those? For 3D analogs for squares, pentagons, etc seem to be pyramids in this context. And what if we diagonalize? Sum of the triangle(3), square(4), pentagon(5) ... n-gon(n)?

[@robinbrowne5419]

Just when we thought we had seen everything, Numberphile comes up with yet another 👍

[@danielbickford3458]

If all real whole numbers can be made up of n n-gonal numbers what percentage of them are made out each number per n-gon? Ei: what is the ratio for one triangle composite numbers, two triangle composite numbers and 3 triangle composite numbers?

[@brentjeanneret]

Love this. Two questions... 1) is there any rule known for how often the maximum number of n-gonal numbers are needed? Is there any regularity? 2) (my young son's question) for square numbers, the length of each side is, obviously, the square root. For other n-gonal numbers, is there a name for the operation that returns the side length?

[@adityapotukuchi4043]

Lovely video that reminds us all why we love math! Also, please come to Toronto when you can, there's pretty fun math happening here :)

[@zakmaniscool]

0:40 "Do you know who else loves triangles? Matt Parker, because they're not squares"

[@aaronloach]

"Every triangle is a love triangle if you love triangles" - Pythagoras...probably

[@jpdemer5]

They're Parker squares . . . only off by one vertex.

[@emre42960]

what is the name of the music at 9:40?

[@Chompingbits]

The stacking sound effect is adorable

[@joaquinvigara1356]

I’m a simple man, I see james, I click 🙌🏻😹

[@duncanhill4434]

As the number of people mentioning they are happy to see Dr Grime back approaches TREE(3), I'll just add my contribution!

[@renerpho]

This is closely related to Waring's problem: What's the smallest number k such that every positive integer can be written as the sum of at most k n-th powers? For square numbers (n=2), the answer is 4. For cubes (n=3), it is 9, although 4 are enough for sufficiently large numbers. For n=4, you may need 19, but 16 are enough for large numbers. For n=5, it is 37, and it is conjectured that 6 may always be sufficient if the number is large enough. The general case (dimensions larger than 4) remains unsolved.

[@legamerfandesience2987]

2:36 Can you put the link of the proof please. I'll try to understand that thing.

[@fahrenheit2101]

James is back!!!

[@dejavu5838]

there's nothing like James Grime in a Numberphile video

[@androidlogin3065]

5:46 Why that internal red dots are not removed to create the next one? Three dots less. Why are not added more internal red dots to create the next one? Four dots more. i mean why not hollow (only external part) or full fill (add all inside).

[@onebronx]

I wonder what happens with hyperdimensional n-topic figurative numbers, like 4d hypercube etc? For 2 dimensional the rule is "at most p, where p is the first n-gonal number after 1". For 3-dimensional the rule seems to be "at most p+1, where p is the first n-hedral number after 1". Will it continue like "at most p+d-2, where p is the first d-dimensional n-topic number after 1"?

[@JeffKaylin-ft5cx]

If 3 points can define whether they are at the vertices of the square, does that mean 3 points define a square? So, in puzzles can we technically find more squares in the pattern of dots? (the trick being finding rotated squares too).

[@aliasmask]

Cool. I solved the bug byte puzzle. Took me about 2 hours, but it was fun.

[@IvanToshkov]

Did you use a computer?

[@jacksonstarky8288]

So... any positive integer is the sum of at most three triangular numbers, and at most four square numbers... and 3 is a triangular number, and 4 is a square number. And from Fermat's conjecture, that pattern continues, as 5 is a pentagonal number and 6 is a hexagonal number. I'll be interested in seeing how this extends to the third dimension... continuing the video now.

[@WAMTAT]

Ive never been this early to a numberphile

[@swordfishxd-]

me neither

[@numberphile]

Welcome to the party

[@lyrimetacurl0]

same

[@gabest4]

Why are following rows in triangle numbers shifted by half and not in squares? Should it be more like 1, 4, 9? If we allow this offset, 5 kinda looks like a square number, too. 5 as it is printed on a dice.

[@oncedidactic]

The Katamari speaking sound effects are perfect

[@SgtSupaman]

0:34 It doesn't look like it simply because you are using the wrong shape of token. 1 is also a square number, but that didn't look like a square either.

[@IceMetalPunk]

Kind of interesting; I wonder where 5 comes from? I can understand the n terms for n-gonal numbers, but then I'd expect a similar pattern to hold, where polyhedrons would be some exponent or multiple of that sequence. But 5?