343867 and Tetrahedral Numbers - Numberphile 

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

The stop motion is georgious. Appreciate the effort

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

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!

Very happy to see Dr Grime back on numberphile!

Whoever animated this episode, you earned your paycheck.

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.

6:34 The Fermat-Haran Conjecture 😀

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.

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

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

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

James is really Mr. Numberphile =D

The animation / stop-motion is looking smooth as heck

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

Always waiting James' videos❤

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

GRIIIIIIME I MISSED YOU, MAN welcome back, singingbanana!

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

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~

Good to see James Grimes again!🌞

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. 🤞🏼

I love maths! James adores it.

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

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.

Nice! I love (that) stop motion!

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

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.

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

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

Glad to see James Grime again!

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

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

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

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, ...

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

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

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.

Numberphile is extra special with Dr. James.

I love James' little speech at the end of this

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!

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

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

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.

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

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

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

James is the best

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

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.

I like Brady's proof by pronouncement.🎉

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!

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?

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)

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

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.

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

That stop motion was pretty sweet!

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

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

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!

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.

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

thank you so much for your kindness and information

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

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

Ive never been this early to a numberphile

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.

What a coincidence, one of yesterday's videos on friends of the channel Cracking The Cryptic involved a puzzle where the solution path touched on tetrahedral numbers!

Love that they still used the brown paper

5:20 also that was my conjecture :)

What a beautiful video. Thank you.

The stacking sound effect is adorable

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

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

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

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

The Katamari speaking sound effects are perfect

James is back!!!

Now do it for all 4-dimensional pyramid-numbers 😄

Classic numberphile!

Cracking the cryptic mentioned a few days ago the concept of tetrahedron numbers. Nice coincidence 😃

I love the episodes with connection to Geometry.

Oh Cauchy, always ruining my life by being a better mathematician than I could ever dream of aspiring to be

I am new to this problem. What I see is that any way you attempt it, you will require 5 separate sequential logical axioms to describe the full body of any tetrahedron.

my favorite presenter on Numberphile 👍👍 1:24

Great animation

That stop go animation must have taken ages to do. Good job Brady and his elves

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?

there's nothing like James Grime in a Numberphile video

This all makes sense from very basic number theory, or more clearly, from the construction of the integers themselves. Normally the integers are constructed by incrementing them by some unit size, from some initial value. Using that operation, and the unit size of 1, and the initial value of 0, the positive integers can be constructed. Clearly if this process were altered, the resulting sequence would vary from that of the sequential positive whole numbers in some way. Like by changing the unit size, you can produce multiples of some integer, or you can produce a cyclical pattern of ratios, or you can produce a sequence of the harmonics of some irrational or transcendental number. Another way to change that process would be to adjust the algorithm, like by altering the unit size according to the step number. By incrementing the unit size by the step number, you get the triangular numbers. By incrementing the unit size by two times the step number, you get the square numbers. Likewise, by incrementing it by 3n, where n is the step number, you get the pentagonal numbers. In general, for polygonal number P with m sides and step count n, and initial value of 0, P(n) + 1 + n(m-2) = P(n+1) ... for triangular numbers, this produces the following: 0+1+0*1 = 1 1+1+1*1 = 3 3+1+2*1 = 6 6+1+3*1 = 10 ... and for squares it produces: 0+1+0*2 = 1 1+1+1*2 = 4 4+1+2*2 = 9 9+1+3*2 = 16 ... So, having said all that, since these are variations on the basic constructive sequence that builds all positive integers, it follows that they are going to have one of those altered sequences as a result, with characteristic gaps in the integers they produce. These gaps get larger, so they aren't some cyclical pattern produced by the beat ratio of some rational number. Also they still consist only of whole numbers, and so it doesn't produce the patterns made by transcendental or irrational numbers. So then, these are transformations applied to the basic integer construction operation that are both aperiodic, and that produce nothing but integers. Since they are aperiodic, and only produce integers, the outputs up to and including some step can all be combined in various ways to produce different sums. And since they themselves are comprised of elements that are aperiodic in origin (as opposed to a list of elements where each is the same distance apart), the number of possible sums from combining them grows nontrivially (if they were equidistant, there would be far fewer unique results, as more combinations would be redundant). I lack the rigor to connect the dots the rest of the way, it seems very intuitive that, because the gaps grow at the rate they grow, and because the list of unique elements from which to make sums also grows at a rate influenced by the nonperiodic growth of those gaps, those two would grow in balance, and that the gaps created by the sequence would never grow fast enough such that they could not be filled in using the same number of elements or less.

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.

Hmm continuing this pattern, I conjecture that for every Platonic solid, you can write every natural number as a sum with at most n+1 terms, where n is the number of vertices of the Platonic solid. So for example, every natural number can be written as the sum of 21 dodecahedral numbers.

For the longest time I thought Grimey was the host of Numberphile as he was in so many videos, until I saw Brady!

nice physical animations :D

Nth Tetrahedral number is the sum of Triangular numbers from 1to N !!!!

1:25 Hello. I'm the number 2. Pleased to meet you.

James is great. :)

Give that man a wider margin!

Every number can be written as the sum of at most one 2-gonal number (every number is a 2-gonal number).