Secrets of the lost number walls

This video is about number walls a very beautiful corner of mathematics that hardly anybody seems to be aware of. Time for a thorough Mathologerization :) Overall a very natural follow-on to the very popular video on difference tables from a couple of months ago ("Why don't they teach Newton's calculus of 'What comes next?'")
00:00 Intro
01:02 Chapter 1: What's in a wall
03:35 Chapter 2: Number wall oracle
14:31 Chapter 3: Walls have windows
16:34 Animations of Pagoda sequence
18:13 Chapter 4: Zero problems
25:31 Chapter 5: Determinants
32:49 Animation sequence with music
35:22 Thank you :)
References for number walls
The main reference for number walls is Fred Lunnon's article "The number-wall algorithm: an LFSR cookbook", Journal of Integer Sequences 4 (2001), no. 1, 01.1.1.
cs.uwaterloo.ca/journals/JIS/...
Also check out Fred's article "The Pagoda sequence: a ramble through linear complexity, number walls, D0L sequences, finite state automata, and aperiodic tilings", Electronic Proceedings in Theoretical Computer Science 1 (2009), 130-148. arxiv.org/abs/0906.3286. Among many other things this one features lots of pretty pictures :)
Conway and Guy's famous "The book of numbers" has a chapter dedicated to number walls. This is where I first learned about number walls. Sadly, Figure 3.24 on page 88 which describes the horse shoe rule is full of typos. Careful:
1. (formulae on right) Negate signs attached to w_l/w and e_l/e ;
2. (diagram on left) Leftward arrow missing from edge marked w_2 ;
3. The last row of arrows bears labels " s_3 " ... " s_2 " ... " s_1 " , which should instead read " s_1 " ... " s_2 " ... " s_3 " .
More articles/books to check out if you are really keen:
tinyurl.com/bdhyzscw
core.ac.uk/download/pdf/82737...
Jacek Gilewicz, Approximants de Padé, Springer Lecture Notes in Mathematics 667 (1978).
The Wiki page on linear recurrence with constant coefficients is a good resource for finding out about how the characteristic polynomial of a sequence translates into a "function rule"
en.wikipedia.org/wiki/Linear_...
Coding challenge
Create an online implementation of the number wall algorithm using determinants or, ideally, using the cross and horseshoe rules and do a couple of fun things with your program. Here are some possible ideas you could play with: 1. generate pictures of even number (or, more generally, mod p) windows of random integer sequences or of sequences grabbed from here oeis.org/ . 2. Explore the Pagoda sequence number wall, again mod various prime numbers. Here is the entry for this sequence in the on-line Encyclopaedia of integer sequences tinyurl.com/yc45cfvf 3. Be inspired by the examples in this article arxiv.org/abs/0906.3286 Send me a link to your app before the next Mathologer video comes out and I'll enter you in the draw for a copy of Marty and my book Putting two and two together :)
Research challenge
Prove the Pagoda sequence wall conjecture or find a counterexample.
Bug report
In the video I say that figuring out the factor rule is easy. This is only true for windows of 0s of even dimensions. Showing that the factor rule has a -1 on the right side for windows of odd dimensions is actually somewhat tricky. Details in the first article by Fred Lunnon listed above.
Today's music: Asturias by Isaac Albeniz performed by Guitar Classics and Taiyo (Sun) by Yuhi (Evening Sun)
Today's t-shirt: Yes, I am always right. If you are interested in getting one just google "Yes, I am always right math t-shirt" and pick the version you like best.
Enjoy!
Burkard

[@Stekey21]

This is mad! I am a PhD student who is actively studying number walls! Specifically, I have written a program to generate the number wall from a sequence and I have some nice formulas for the number of sequences that have a given window in their number wall. They have some beautiful relationships to diophantine approximation over function fields and linear complexity! It's so excititing to see them be popularised more.

[@briandublidi4708]

hehe is your code open sorced? :D i will definitely program the number wall to solve sequences for so i can make a "..." function :)

[@hexeddecimals]

Is you code available anywhere? I could definitely use it to help find some recurrence relations in my mathematical pet projects!

[@gamekiller0123]

Do the cross and horseshoe formulas hold for any commutative ring?

[@Stekey21]

I'm a little hesitant to share my code as it is a valid solution to this videos "Coding Challenge". Sorry for any disappointment! Just to prove I'm not talking total rubbish though, here are some cool number walls created by my code. drive.google.com/drive/folders/15GEhc5FCTcPDSZoEZxBpDcOSfobyoxjX?usp=sharing

[@Stekey21]

@@gamekiller0123 to the best of my understanding they do! Though I work over finite fields and usually take these formulas for granted.

[@johnchessant3012]

Yet another awesome topic that has me saying "how is this the first time I'm hearing about this?"! The mystery of why the number walls will always have integer entries would've been enough to hold my attention, let alone all the other cool properties! Also, fun fact: The characteristic polynomial is the denominator of the generating function of the sequence! The numerator will be a polynomial of degree less than the denominator, determined by the initial terms of the sequence.

[@alexanderlevakin9001]

May be you know the answer. Supposr we have f(n) = a_1*f(n-1)+...+a_k*f(n-k) where k > 5. We want to express f(n) as function of n. How can we do it in case we can not solve the characteristics equation?

[@gammano0b858]

The recursive formula for the squares translates to: x^2 = 3(x-1)^2 - 3(x-2)^2 + (x-3)^2 Expanding the right hand side gives us: (3x^2 - 6x + 3) - (3x^2 - 12x + 12) + (x^2 - 6x + 9) = 3x^2 - 3x^2 + x^2 - 6x +12x - 6x + 3 - 12 + 9 = x^2 The formula works :) The coefficients of the formulas look like the binomial coefficients, they are matching with the rows in pascals triangle. You can generate the formla by expanding the right side of the equation 0 = (x - 1)^n and replacing every x^k by the k-th coefficient c_k. Then solve for the highest power (x^n) and you have the formula for the sums of x^(n-1). Notice, the formula for the squares uses the coefficients of the 3rd row, expanding (x-1)^3 not (x-1)^2.

[@TheEternalVortex42]

The polynomial you get for the sequence n^k is just (x-1)^(k+1) (hence the coefficients in the expansion are binomial coefficients). The general solution of such a recurrence is a_0 + a_1 n + a_2 n^2 + ... + a_k n_k (a *very* special case of the general formula, since all the roots are 1) To get the a_i you can just plug in the initial values and solve the linear system. It's a bit tedius for larger k, but say for k = 2 we get a_0 + a_1 n + a_2 n^2 at n=0 -> a_0 = 0 a_1 n + a_2 n^2 at n=1 -> a_1 + a_2 = 1 a_1 n + a_2 n^2 at n=2 -> 2 a_1 + 4 a_2 = 4 So 2(1-a_2) + 4 a_2 = 4 -> a_2 =1 and thus a_1 = 0 so we get n^2 as the solution as we expected

[@moskthinks9801]

This really reminds me of solving differential equations, especially y'''=3y''-3y'+1 The characteristic equation is r^3-3r^2+3r-1=0 or (r-1)^3=0 We get r=1 with a multiplicity of 3. For d.eq, it meant that the solutions are of form Ax^2 e^x + Bx e^x + C e^x But when it comes to solving functions, the e^(1x) is replaced with 1^n. If it were e^(2x), it would become 2^n. This means that the solutions are now of form A*n^2 * 1^n + B*n * 1^n + C * 1^n = An^2+Bn+C, which is just a quadratic! Note because Sn = 3Sn-1 - 3Sn-2 + Sn-3 is a recurrence relation with three variables on the right, we need three initial values, just like how the fibonacci sequence has F1 = 1 and F2 = 1. Because we are dealing with the square numbers, the three initial values are S0 = 0, S1 = 1, and S2 = 4. This means that C = 0, A+B = 1, and 4A+2B = 4. We get B = 0 and A = 1, giving us Sn = n^2, as it was desired to show. QED.

[@user-ri8ps6cl4w]

No guys you dont understand because 35((818%*+=+*))10^83

[@marekmatusiak8397]

The equation works - but how well it works ... x ^ 2 = 3 (x-1) ^ 2-3 (x-2) ^ 2 + (x-3) ^ 2 This equation also works: for any k x ^ 2 = 3 (x-k) ^ 2-3 (x-2k) ^ 2 + (x-3k) ^ 2 It is not the end. This also works: f (x) = ax ^ 2 + bx + c and any k f (x) = 3f (x-k) -3f (x-2k) + f (x-3k) Also for higher power exponents

[@adamsniffen5187]

​@@marekmatusiak8397 x^m=\sum_{i=0}^{m}=(-1)^i\binom{m+1}{i+1}(x-i-1)^m If you want to shift by k units, here it it shifted by 1 (x+1)^m=\sum_{i=0}^{m}=(-1)^i\binom{m+1}{i+1}(x-i)^m also, = \sum_{i=0}^{m}\binom{m}{i}x^i :) eg. A quintic integer x in terms starting at (x-5): (x+5)^5=\sum_{i=0}^{5}=(-1)^i\binom{6}{i+1}(x-i+4)^5=3125+3125x+1250x^2+250x^3+25x^4+x^5

[@maxwellgrossman]

I love how every time mathologer posts he just shows up with a new video about something I've never heard of and then suddenly I have something new to explore for the next month or so.

[@cynthiabinder3730]

Top secret stuff .... been around for 50 plus years 4sure. Like lock combinations that don't have keys. ... its great for art complexities even in cloth manufacturers trades secrets....

[@canaDavid1]

This seems very similar to second order automata - each cell's value is determined by the three cells over it, and the cell 2 over it.

[@Mathologer]

If you check out some of the references given in the description you'll find that there is indeed a connection ;)

[@kees-janhermans910]

It's a bit like Conway's game of life. Albeit one that has an evaluation direction.

[@sschmachtel8963]

Yeah and another thing that immediately came to my mind is the method of finite differences and now I wonder if there is any usable connection between pdes(/(i)bvps?) and those sequences. Meaning e.g. use properties of pdes to find something about the properties of these sequences, or vice versa use properties of these sequences to find out something about pdes(/(i)bvps?) Seems anyway like a not really fitting match up, because you have only one boundary to start with ... but who knows if there isnt any usable stuff on the interior???? Particularly interesting seems to me, if you have such a "stencil" or method of calculation, can you actually show that some pdes are aphysical, meaning end somewhere else than on the boundary? Sure those are only approximations .... and in the end does that mean these approximations break down because you cannot zoom in, or is it that you'd need to also change the starting row if you zoom in...

[@wyattstevens8574]

​@@kees-janhermans910Maybe Wolfram rules?

[@matthewgiallourakis7645]

I love the feeling of having just about finished watching another great math video, and then I remember this is Matholologer and I've only just finished chapter 2 of 5! You spoil us with knowledge!

[@joseville]

What was chapter 1?

[@ardlight172]

I really enjoy watching your videos Burkard, thanks to you I've rediscovered the thrive and passion to take pen and paper and try to prove things. Thank you so much.

[@theMichaelsLi]

For years people have discussed the level of mathematics that can be found within art. Number Walls are proof that a hidden masterpiece can exist if you recognize the pattern of beauty. In the past I watched your videos for the lessons and enthusiasm, given the level of excitement you've displayed concerning Number Walls - I'm afraid you'll forever be the Paint by Number Guy in my book. It's really something to see a scientist morph into an artist in a video. Good job!

[@trizgo_]

i can't help but smile whenever self-similarity pops up out of seemingly nowhere!

[@elderpastitsios1569]

Self similarity must be linked to lack of information, in some way that escapes my capacity, but I wish I knew how to evalué and link the information with the self similarity.

[@chriszachtian]

How on earth could you resist to check what happens with the sequence of primes??? Thanks for the video.

[@MrPiks0u]

I spent the whole video wondering what it would look like with the sequence of primes as a second row... guess I'll have to test it myself some day.

[@iggusify]

Maybe it will become a mandelbrot fractal?

[@petrospaulos7736]

If the second row is the primes, then third row is oeis A056221

[@petrospaulos7736]

The Prime Wall goes like.... - - - 1 1 1 1 1 1 1 1 1 1 1 1 1 1 - - - - - - 7 11 13 17 19 23 29 31 37 41 43 47 53 59 - - - - - - -6 30 -18 42 -30 -22 128 -112 98 90 -78 -70 36 248 - - - - - - -12 72 -72 72 96 188 480 0 532 384 288 164 352 1184 - - - - - - 0 144 0 288 144 488 1800 2280 2888 -64 -256 1064 -1952 5312 - - - - - - 288 576 1152 -1248 -112 4432 -1552 15952 1936 464 3856 -5232 30928 - - - . . .

[@chriszachtian]

@@petrospaulos7736 great hint! And sadly, no zeros in sight?

[@darrell1766]

Fascinating as always. Thank you Burkard! Lately I've become obsessed with spiral mathematics. The obsession was spurred by a game called Idle Spiral that was released about 4 months ago. I'm no mathematician so figuring out where the game-play mechanics and real math diverge is a challenge for me. But it's fun to try!

[@SuperYoonHo]

Woah! Another CRAZY video!!! These videos i cannot resist to watch:) So beautifully made are they.

[@ritchards]

Immediate thought: what does the number wall of the prime numbers look like?

[@richardgratton7557]

Interesting…probably no rows of zeros?

[@HighFiveTheTodd]

That was my first thought, followed by "I'm sure this would be used to predict primes if there was a recursive formula you could derive from this method...but there isn't."

[@cynthiabinder3730]

Depends on the wall created to open() gate() or creating a wall. Does the numbers match????()

[@publiconions6313]

I was thinking a number wall of a non-polynomial number... like the digits of e or pi

[@petrospaulos7736]

Here is the number wall of PRIME NUMBERS! - - - 1 1 1 1 1 1 1 1 1 1 1 1 1 1 - - - - - - 7 11 13 17 19 23 29 31 37 41 43 47 53 59 - - - - - - -6 30 -18 42 -30 -22 128 -112 98 90 -78 -70 36 248 - - - - - - -12 72 -72 72 96 188 480 0 532 384 288 164 352 1184 - - - - - - 0 144 0 288 144 488 1800 2280 2888 -64 -256 1064 -1952 5312 - - - - - - 288 576 1152 -1248 -112 4432 -1552 15952 1936 464 3856 -5232 30928 - - - . . .

[@gideonk123]

Fantastic video! A more computationally-efficient method than the determinant method, which also easily and transparently deals with single and multiple zeros: Step 1: After using k delayed copies of the original sequence, instead of repeatedly computing determinants (as in the video), take a rectangular “snapshot” - this is a rectangular matrix of size k-by-n, with n > k. Step 2: Transpose this matrix (so now it’s n-by-k with n > k) and compute its “economy” SVD (singular value decomposition). There are many existing software libraries to do this. This step is the only computationally-intensive step, whose complexity is O(n k^2), which is much less than the determinant method as stated in the video, which is O(n k^3), or even more if care is not taken… Step 3: Now, look at the singular values: If one or more of the smallest singular values is/are zero, then we’re sure this is what Burkard calls “Fibonacci-like sequence”, something usually called “linear recurrence with constant coefficients” (which he mentioned in the video). However, if none of the singular values is zero, then repeat with larger k, i.e., more delayed copies of the original sequence. Step 4: The linear recurrence with constant coefficients is easily determined from the column of the SVD result corresponding to the zero singular value (because this is the vector which can zero-out any set of k consecutive elements of the sequence). Specifically, if the “economy” SVD result of the original n-by-k matrix is U, s, Vt, then the last row of the square k-by-k matrix Vt will be the desired set of linear constant recurrence values. This last row has size 1-by-k, of course. These values will probably need to be scaled by a common multiple if nice integer values are desired! Move the 1st element of this set of k values to the other side of the equation to use as a prediction recurrence equation, i.e., the next value is determined linearly from the previous k-1 values. Remark: The idea of a singular value decomposition (SVD) is mathematically very closely related to a determinant, because it essentially also determines linear combinations of columns (and/or rows). But it is more computationally-efficient in this case, because it can be computed for a rectangular set of values simultaneously, rather than small squares of values one after the other. The second advantage of the SVD is that it can also give us the linear recurrence with constant coefficients “for free” (from the same computational result).

[@Mathologer]

Very nice :)

[@gideonk123]

@@Mathologer Thanks :)

[@hybmnzz2658]

If discrete calculus leads to calculus, I wonder if there is a continuous analog to number walls.

[@PeterBarnes2]

That would sort of imply a sort of continuous form of the matrix and determinant, I think, by what's said in the video.

[@theodorealenas3171]

Yeah. And, if you zoomed out fat enough on one of them that look like zoomed out versions of themselves? Wouldn't you be able to define a limiting, fractal version of the thing

[@f0rdgamer]

@@theodorealenas3171 you would need to define what it means to be between cells or what it means to be an aggregate of nearby values, neither of which are clear.

[@alokaggarwal6859]

Thank you the video was fantastic!! I never heard of number walls before, and the graphical presentation was very informative. I look forward to learning more from the links.

[@Mathologer]

Thank you :)

[@mathyland4632]

Well, you’ve set me down a rabbit hole again for at least the next week. Amazing video as always!

[@cynthiabinder3730]

A like the 🐇 🐰 🐇 🐰 hole.hop hop hop hop hop not lost not lost not lost.♾♾♾♾♾♾♾♾

[@ApteraPioneer]

More (fabric designers) should partner with mathematicians. Gorgeous animations!

[@richardgratton7557]

Yes!! I was thinking of linoleum floor patterns myself.😊👍

[@cynthiabinder3730]

Did done been there secrets out

[@cynthiabinder3730]

@@richardgratton7557 right on Richard.

[@michaelharrison1093]

My mother was a commercial artist and back in the 1980s she was designing patterns for carpets that were being manufactured using a Wilton loom process. The Wilton loom could accommodate a multi-colored pattern however there were a set of physical rules limiting the transposing of colors in a given row and also a different set of rules determining the transposing of colors from one row to the next. These rules came from the physical limitations of the mechanical loom design. The loom itself was driven from punch cards and this is the earliest use of a mechanical punch cards. I worked on a computer program that could replicate the rules imposed by the loom and could allow for different new patterns to be generated from a few adjustable parameters, such that each of these designs met the loom manufacturing rules.

[@johnchessant3012]

The closed formula for the mortal enemy's sequence is: E_n = 0.8469 (1.5747)^n + 0.2636 (1.3802)^n cos(1.7805n + 0.9507). All the decimals are rounded so this won't be very accurate past n = 15 or so, but we can use this to get the asymptotic formula for E_n. Since 1.3802^n

[@cynthiabinder3730]

Gums drops in a gum disperser..... or atoms ⚛ 🤔 🤣 😏 🙄 electrons protons in the power 🔋 plants??◇operational ◇

[@lucasvignolireis8181]

Loved that you used Asturias played by Andres Segovia for this .... Perfect match

[@michaelmam1490]

This is very useful for a problem I'm working on right now. Thanks you so much for making this!

[@FloydMaxwell]

Another amazing video. Great subject to revisit. Math dazzles like a super nova in Mathologer videos.

[@SaturnCanuck]

I always love your videos, a great way to relax on a Sunday afternoon

[@jneal4154]

This and the difference table video have been my favorite videos from you so far. Excellent work. Love it. I swear, all of math is just Pascal's Triangle and the Pythagorean Theorem in disguise.

[@binathiessen4920]

You can get the non-recursive function from a recursive function using the z-transform. This is used often in Digital Signal Processing problems.

[@sschmachtel8963]

interesting aspect I must say ... 🙂 Maybe indeed useful to try to apply if a problem seems to be of similar kind

[@karolakkolo123]

Wait so what is the difference between the z-transform and using generating functions to solve the recurrence relation?

[@gideonk123]

@@karolakkolo123 z-transforms are a different name for a sub-type of generating functions

[@cmilkau]

A nice brief explanation of the determinant is that it is 1 for the identity matrix and additive as function of each is its rows. May sound complicated but if you know these terms you know they can be quickly demonstrated on elementary level. They also easily explain why it's zeroed by linear dependence with a little work

[@scottmiller2591]

Fascinating. I've been working with linear sequence recurrence for decades, and never heard of this. Once I saw the initial construction, I began thinking about determinants, due to the small cross formula being related to the 2x2 determinant. Computing determinants by the efficient sharing of all previous subdeterminants - we all have seen how to decompose determinants into subdeterminants, but it quickly becomes impractical to compute, even on a computer, since determinants are based on the combination of all possible combinations... This is very nice. Usually to solve for linear filters you need to trot out at least Cholesky decomposition. This is just a simple rule (plus a couple of complicated rules, the big cross and horseshoe).

[@EulerD]

Thank you for bringing this back. Determinants can be deterring. Ahaha Meanwhile ... Store Associate: "Which gift wrapping paper would you like?" Me: "Please give me a roll of ternary, mod 3 pattern with the maximal zero-free diamond region." Store Associate: "What colour scheme?"

[@ovidiufarcau243]

Man if I had a math teacher like you , I would be a rocket scientist by now ! I was in high school in an elite math class but switched to Biology going to university cause of losing interest in math, but i understood 99 % of your video and really got me excited again by math.

[@Mutual_Information]

Very original video. I watch a ton of math edu content.. and I’ve never heard of math walls. Well done!

[@cynthiabinder3730]

Laughing 😆 👽👾😆 🎃😂🤣😅😇🤓😏the spy who loved me....ta da super hero📓

[@aweebthatlovesmath4220]

It's always a good day when mathologer post a video!

[@agostinhooliveira5781]

This is awesome. Thank you!

[@peterflom6878]

Great stuff as usual!

[@kayacanoedeloin8703]

this video is just amazing , thank you !!! (and I think you can be prod of your job when Notch is supporting you for doing this amazing stuff !)

[@KipIngram]

This is absolutely amazing. Numbers are just way cool. I really need to go study this stuff more.

[@christheone248]

Math is just amazing! First time that I've seen or heard of the number wall, very interestting

[@TheJoaovascorodrigue]

What a wonderful discovery these walls! Great music choice

[@user-vf5di9nz4s]

Thanks for such a masterpiece. love your shirts as well.

[@jimmy685]

So, could the infinite row of zeros at the top be considered to be infinitely high moving upwards too? … thus also being a square window ….. which, for want of a better term, I think should be called the “sky” 😀

[@uriel578]

Sooo the sky's the limit? 😅

[@killerbee.13]

I think that actually, the row of zeroes at the top wouldn't extend to an infinite series of rows of zeroes. Since the number wall rule is symmetrical, I believe that above the wall would have to be either symmetric or antisymmetric (all values negated) with the part of the wall below the line. I haven't checked exactly what happens though.

[@__ocram__]

@@killerbee.13 yeah, like the extension of the Pascal's triangle

[@aaronr.9644]

determinants always seemed rather mysterious to me and it seems like they never fail to amaze me :)

[@ckq]

This is crazy. I was just exploring recursive sequences and the characteristic polynomials to generate a function that's the sum/product of exponentials and polynomials. I was doing it to generate coefficients to best predict a time series with noise.

[@donwood7541]

Came across this from a Fred Lunnon paper about Pagoda Sequence... John Conway, who took an early interest in this topic, christened the zero regions windows, and the table a wall of numbers, [2]. Apparently, on first encountering these results, he transcribed them for safe keeping onto his bathroom wall (the way one does); but having moved house by the time the book came to be written, was obliged to rely on memory, and as a result (to his evident embarrassment) committed two separate typographical errors in restating them.]

[@waheisel]

Thank you so much for "I definitely don't expect you to have understood everything"! (24:38) And thanks for the awesome video!

[@gobyg-major2057]

I also see Pascal’s triangle in the Fibonacci, square and cube formulas at 9:47 I.e. 1,1…1,3,3,1…1,4,6,4,1…etc.

[@wahyuadi35]

Wow... Me too. Seems so obvious.

[@wyattstevens8574]

It's just row n+1 (the one starting 1, n) for x^n (n=0 is trivial!)

[@marcozarantonello2180]

You are by far the best math channel on KZbin

[@Mathologer]

Glad you think so and thank you for saying so :)

[@accountname1047]

You're a pedagogical genius!

[@namespace17]

Wow! This definitely felt like magic!!!

[@NonTwinBrothers]

This was a fun coding challenge, it's very satisfying to have the computer do all the work for you. plus you get pretty patterns so I consider that a win-win :D

[@InfluxDecline]

Thanks for using Albeniz's beautiful "Asturias (Leyenda)" at 1:00, and of course for the video as well. :)

[@mehdimabed4125]

Always amazing how you make us discover new, interresting, beautiful, "simple", ... maths !! And from what you said in this video, couldn't this number wall be used to calculate determinant faster ?

[@heliy_25]

In diagonal recording, this is even more amazing.

[@heliy_25]

Why do I need a cart? I really like what Master does on KZbin. The Soviet school of mathematics was very specialized. Now I'm interested to know the context of the work of great mathematicians :). However, it may not matter soon :(. Bureaucracy and ignorance will kill this world, we probably deserve it. The danger of making stupid decisions is very high (sorry, I'm from Russia, Bleak House is waiting for wise words and decisions;).

[@yahccs1]

Amazing thank you! More patterns from numbers! Those recurring patterns with the 0 squares are so much like fractal patterns or that unpronouncable triangle I keep forgetting the name of. Yes they might make good designs for t-shirts! It was just a little confusing how the 'x' you introduced relates to the numbers along the original sequence to convert it to Sn-1 etc.

[@Veptis]

I got a mystery sequence in my head which might not even have a formula. And it's base 10 dependant: 6, 2, 5, 5, 4, 5, 6, 3, 7, 6, 8 I should check the Oeis to see if they have a neater formula and no there wasn't. The self similarity might look more beautiful if you zoom out instead of in, since the resolution increases and not decreases.

[@AdrianCruz_]

How interesting, when I have some spare time I know what I’ll be looking through/trying out. On a slightly off topic note Wonderwall was stuck in my head while watching this

[@wyattstevens8574]

"'Cause after all... that's its number wall..." ("it" being any certain number sequence)

[@johnb3513]

Almost as excited to see the t-shirt as I am to hear the math!

[@Nethershaw]

Unexpected symmetries are always delightful.

[@ozradek1]

So I see a pattern. About 1/2 way through every Mathologer video I comment that this is simply the best and most amazing video yet. And again. No number wall needed to decode the pattern. You are simply the best. And getting better.

[@lorenpearson1230]

We hit on these as recreational math problems one course in university. They were advertised the same as Collatz - dive in and get lost forever in the patterns. I suspect it came up because that was around when computer networks were starting to be used and the horseshoe was published.

[@AndrewMakesPuns]

Mind-blowing

[@wmafyouni]

I just want to share some love and say I always enjoy your musical choices but this time Asturias was a very good fit for a less-known topic.

[@johngreen9769]

When I saw the cross rule it immediately reminded me of matrix multiplication and lo and behold we get into linear algebra and determinants, it would have been nice to have some history of number walls though. The patterns that come from the integers is amazing, I can only think they somehow describe the universe in ways we still do not see.

[@ethanjensen7967]

You can do this with symmetric polynomials. Hn(eigenvalues) are xn and En(eigenvalues) are the char poly coefficients

[@bloomp7999]

love this kind of intricate pattern appearing out of logic

[@eriktempelman2097]

Math fan slash guitarist here. Starting music is the (in)famous "Asturias", which was inspired IIRC by a legend from Asturia (= Spanish province) about a group of miners that got stuck when then mine collapsed. So, in this "Asturias Leyenda" they kind of hit a wall. Isaac Albeniz was the composer, by the way.

[@eekee6034]

17:28 The 7 wall contains diagrams of radio telescopes. ;) Look for nearby dots making parabolas with 2 diagonal lines and 1 partial vertical or horizontal. I see one facing up on the left side and one facing to the right somewhat left of the middle. Beautiful video! :)

[@crossiqu]

Wow! Pagoda number wall also contains an optical illusion: little squares seem to be rotated to the right or to the left depending on the sorrunding squares!!!

[@SherlockSage]

I just spent twenty or so minutes trying to make a number-wall for the partition function by hand. I may have given myself a mathematical concussion in the process😵Great stuff! Maybe next time I'll start with a simpler sequence though 😂😭

[@SherlockSage]

Actually I could use some math triage here. I got a fraction in my counting and I'm not sure where I went wrong Here is my number-wall so far * 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 * * 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 * * 0 0 1 1 2 3 5 7 11 15 22 30 42 56 77 * * 0 0 1 -1 1 -1 4 -6 16 -17 34 -24 84 -98 * * * 0 0 1 0 0 -1 2 -4 14 -17 34 -76 112 * * * * 0 0 1 0 0 -1 0 2 8 11 -4 82 * * * * * 0 0 1 -1 1 -1 1 -1 3 -9 27 * * * * * * 0 0 1 0 1 0 2 -1 0 0 * * * * * * * 0 0 1 -1 1 -2 4 -1 0 * * * * * * * * 0 0 1 0 1 -3 7 -1 * * * * * * * * * 0 0 1 1 1 1 (11.5) 11.5? Did I miscalculate somewhere?

[@stighemmer]

The computer is your friend.

[@adsoyad2607]

The partition function was the first function I thought of as well!

[@NoNameAtAll2]

what's partition function?

[@wyattstevens8574]

@@NoNameAtAll2 The "hardest what-comes-next" video can show you. It's the number of ways to break the index up into equal or smaller integer parts.

[@MooImABunny]

Haven't watched yet, just wanna say the name reminds me of 6! 3.. no 4! 11? That's Numberwang! Edit: now that I've watched the video, hot diggity damn that was wonderful. The whole process was pretty mysterious, the cross relation vaguely reminded me of the 2×2 determinant but I didn't take this thought far enough. The fact that these algorithms can give you these shifted sequence determinants and thus the recursion formula for any sequence with one is pretty amazing

[@rasowa2958]

@22:36 Theoretically we don't need other factors to calculate factor at the bottom edge. Since the row has exponential growth and we know first and last term: 992436543 = -243 * x^5 (5 is a number of steps to get from -243 to 992436543) thus: x = (992436543/-243)^(1/5) = -21 this factor from left to right (-243) * (-21) = 5103; 5130 * (-21) = -107163 and so on. But I grant you that the equation with other factors may be easier to use manually, or to get result as a fraction rather than decimal.

[@michelmarin4641]

Great video , really thanks

[@snowfloofcathug]

How does it feel to never be able to make a video about something “barely anyone knows about” when you will inevitably popularise it? I think it’s CGP Grey that has to deal with people commenting on his “common misconceptions” videos saying how “everyone knows it” but the only reason the correct knowledge is so well known is partially *because* of said video

[@Mathologer]

Yes, in a couple of days from now what I say at the beginning of the video will no longer be true :)

[@a.osethkin55]

So strange and amazing!

[@asaadalsharif7832]

Really amazing I must play with this algorithm

[@AndreasEisele]

Truly fascinating stuff, and presented in such a wonderful way! Thanks so much, also for all the other videos on this channel! I was curious as to how a number wall for a sequence like 1 1/2 1/3 1/4 1/5 ... would look like, but the fractions grow out of bound and I could not find any meaningful pattern. Is there any way to generalize this idea to fractions?

[@mathsnewdiscoveries]

I like your teachings Sir.

[@senthilkumaran5255]

The zero windows seem eerily similar to black holes! Especially with the square rule breaking down at the event horizon (the borders of the windows) and requiring us to use special rules

[@sphakamisozondi]

Mathologer, your shirt-gamecis criminally underrated!

[@rg8766]

I wish you had structured this video to start with the explanation of a number wall as the result of determinant operations instead of using up 30 minutes of my life confusingly going on about unexplained and "apparently arbitrary" cross and horseshoe games. Then you could talk about how the computationally quicker cross and horseshoe rules are derived from the original determinant operations and then how they lead to fibonacci type solutions of sequences. It would have made MUCH more sense.

[@LoganStargazer]

Your videos are always so entertaining and edjamacational. Every Calculus video I have ever see only mentions what it's used for (For example: to calculate the area of a random blob), but they never use it in the video. It's all talk and no do. Would you please make a Calculus video where an answer is *actually calculated*?

[@SimonBuchanNz]

The thing is, calculus is mostly "algebra auto pilot" once you start actually using it. You might like channels like blackpenredpen, which are largely focused on the working out bit.

[@CM63_France]

Hi, Wonderful! The guitar piece is called "Asturias" and is from Isaac Albéniz.

[@SeanCMonahan]

An extra credit question on my linear algebra final back in the day involved "look-and-say sequences" (A005150 on OEIS is the simplest of them). I wish I could remember how it worked because I feel like it would tie into this number wall idea somehow. The first number in the sequence is 1. Read aloud, that is "one '1'," (11). Read aloud, that is "two '1's" (21). Then "one '2', one '1'" (1211), etc.

[@sofia.eris.bauhaus]

as you explained it i straight away had to implement the number wall formula in a spreadsheet and put in the Thue-Morse sequence. aaand get div/0 errors in my fourth row 😅.

[@nathanielreichert4638]

Your ironic t-shirts are the second best part of your videos

[@brianerisman3192]

There was mention of number walls for both finite and infinite sequences. On the finite side, I am curious if there’s any pattern that would emerge from any of various starting points in the Collatz sequence - e.g., the 100-ish-digit sequence starting with 27.

[@yinq5384]

Amazing video as always! 15:52 Those "divisible-by-prime squares" can be viewed as "p-adic 0-squares" right?

[@colinpountney333]

The number wall for the Catalan series is crazy.

[@rocknrolladube]

The fact the square numbers have a formal that we take for granted suggests we might want to use other polygonal numbers when trying to work with irrationals. There could be very simplistic algebraic expressions if we adopted a small super script triangle or pentagon, etc, in our equations.

[@AvanaVana]

This is mind blowing and awesome. Always good here. Edit: one thing I wasn’t clear about: do all number walls with windows greater than 1 in dimension have geometric sequences bounding the windows?

[@wyattstevens8574]

I think that's what he said.

[@joseville]

I think the long cross rule can be used to compute the two numbers directly below a 2x2 square, but then to compute the two numbers below those 2, you still have to use the horseshoe rule.

[@ignacioetcheverry232]

Gracias!!!

[@EthanLeeKonKeong]

Thanks!

[@HYEOL]

The end Animation made it worth it

[@miriclamier2875]

Top notch

[@dodisboy]

I love the music from my country: Asturias by Isaac Albeniz performed by Guitar Classics and Taiyo (Sun) by Yuhi (Evening Sun). I was surprised when it started playing. :)

[@asdfgasdfgadsfgadsfg]

very good sir !

[@at7388]

I always give u a LIKE before I look your video.

[@josebarcena50]

Great content to have breakfast with :). Would be great to see the prime numbers number wall :0

[@tim40gabby25]

I'm guessing that 99.99% of professional mathematicians not knowing about number walls is a poor estimate :) - and what a wonderfully taught video. Marco Pizzato is such a Pal you named him twice :)

[@rodolfojoseespino6729]

You are "capo"😁... thank for bring it (too hard work)so beautiful and enjoyable chores ...👍🇦🇷🌎

[@MasterHigure]

Anyway, here's numberwall. Also, at 22:40, I would think it much easier to remember how to do it if you just walk all four sides clockwise and multiply the ratios.