The Pandigital Paradox

"Pandigital numbers" (numbers that contain every digit) might seem random, but let me show you how important and mysterious they truly are. Prepare for pandigital pandemonium, including primes, palindromes, perfect powers, and paradoxical truths about numbers. This is the longest episode on this channel yet and is full of surprises (leave a comment if you see all 3 squirrel cameos!).
[DISCLAIMER: DO NOT COPY ANY OF THE PARTS WITH FIRE. In general, never copy any physical actions you see in Combo Class, such as anything involving fire, tools, or other science experiments.]
For more details about some of these topics, check out:
My episode about factorials - • Factorial Fact Frenzy (!)
My episode about palindromic numbers - • The Perplexing Pattern...
My episode about “hyperelevens” - • Hyperelevens (and the ...
My latest two livestreams on my bonus channel:
(the one before this episode) - • Special Saturday Strea...
(the one after this episode) - • The Pandigital Paradox
Special thanks to my Patreon supporters: Tybie Fitzhugh, Henry Spencer, Mitch Harding, YbabFlow, Joseph Rissler, Plenty W, Quinn Moyer, Julius 420, Philip Rogers, Ilmori Fajt, Brandon, August Taub, Ira Sanborn, Matthew Chudleigh, Cornelis Van Der Bent, Craig Butz, Mark S, Thorbjorn M H, Mathias Ermatinger, Edward Clarke, and Christopher Masto, Joshua S, Max, Joost Doesberg, Adam, Chris Reisenbichler, Stan Seibert, Amy, Izeck, and Max.
Combo Class Links:
Patreon: / comboclass
Domotro bonus channel: / @domotro
Reddit: / comboclass
Discord: / discord
Some of the topics in this episode include: pandigital numbers and zeroless pandigital numbers, pandigital prime numbers, pandigital perfect powers such as square numbers, pandigitally “persistent” numbers, “polydivisible” numbers, palindromic numbers, hyperelevens (often referred to as “generalized repunits”), pandigital numbers in other bases like Base 2 (binary), Mersenne primes (the biggest primes ever discovered), ways to compare sizes of infinite things such as “natural density”, and some paradoxical traits about pandigital numbers that I realized when investigating them!

[@ComboClass]

This is the longest episode yet. Thanks for watching! Also check out my channel www.youtube.com/@domotro for my shorts, livestreams, and bonus videos, and consider checking out my Patreon at www.patreon.com/comboclass

[@Amipotsophspond]

No one likes shorts they are just pushed by KZbin for $. views per video time, make it look more profitable then it is. a lack of add blockers on cellphones vrs computer makes mobile look more profitable then it is. the shorts page is designed to turn a simple scroll down in to a view then a scroll up back is yet another view. comments are greatly reduced on shorts as they are hard to view and hard to make, so censorship cost are lower. like auto play the shorts screen is more prone to letting KZbin decide the next video not the user, this is useful for gas lighting. accidental clicking from pushing the shorts make it look more profitable then it is, look at how much screen real estate shorts are given. percentage of the video watched is higher, that 1 sec video it turns out 60% of viewers watched 100% of the video because they physically could not click away fast enough and 40% were bots. KZbin thinks it's profitable, so KZbin advertises and pushes shorts more so shorts get more views and it becomes profitable at scale, self fulfilling prophecy. if tiktok gets banned KZbin wants to be their to make sure no upstart alt tech gets a foot hold in the market smooth replacement. if most advertisers switched to only paying for actual sales san francisco/big tech would collapse, but ESG money says that's not going to happen.

[@harriehausenman8623]

#mayyoutubeshortsperish

[@pythx]

It can never be a Combo Class video without any falling dry erase boards

[@nio804]

And fire. Don't forget the fire.

[@Voiding210]

And a few squirrels

[@asheep7797]

And clocks.

[@a71official]

and dice

[@nio804]

I find it amazing how you somehow manage to sound bored and excited at the same time.

[@geekjokes8458]

ever since i found this channel i had some thought about domotro i didnt really know how to describe, but ive figured out that he's a "field mathematician" much like a field biologist, he likes to get dirty, burn stuff, things falling, and whatever *hands-on* things one can think of he's not exactly "finding real maths in natural things", like other people have used that term for, domotro is messier in a more realistic way, grounded, and more interesting and charming

[@otakarbeinhauer]

100% of all natural numbers include in its string a larger prime than the largest we have found.

[@ThePeterDislikeShow]

And they contain the phone number of your perfect match. 100% of them!

[@proto_carl8404]

I love this channel…. I only recently discovered it… it’s so smart, I learned so many things from it

[@ThecrackpotdadPlus]

Great Combo Class today. I’m loving the hyper-elevens and pan digital numbers. I love repeating numbers and number patters. 12345678987654321 with its square root is perfect for me.

[@CjqNslXUcM]

I noticed this as a kid when multiplying 11*11 using the technique of multiplying for each digit of the multiplicant and then adding up. 10*11=110 plus 1*11=11 gives you 121. 111*111 is 11100 plus 1110 plus 111, resulting in 12321 imagine my childlike joy when I arranged it in columns: 111111*111111: = sum of 11111100000 _1111110000 __111111000 ___11111100 ____1111110 _____111111 = 12345654321

[@BariScienceLab]

@@CjqNslXUcM Beat me to it, but this pattern is a very beautiful one! 11*11 = 121 111*111 = 12321 1111*1111 = 1234321 11111*11111 = 123454321 etc. What's funny is that going over nine digits seems to break the pattern: 1111111111^2 = 1234567900987654321 Which seems disorderly at first, but this is just like you said, 1.111111111*10^18 + 1.111111111*10^17, etc. except that 10 ones overlap, which creates a 0 and gets one carried over to the next column, which has 9 ones, making it 9+1 ones = 10 ones, so that gets another zero written down, and the one is carried over to the 8, giving 8+1 = 9, which continues the pattern from there (which is also why the 8 is skipped, because you don't have to carry the one when you have a sum of 9, so it skips over to 7, not 7+1) TL;DR 1111111111^2 breaks this nice pattern until you actually look into the multiplication and have a funny realization!

[@ThecrackpotdadPlus]

@@CjqNslXUcM so wonderful. I love it.

[@TheMagicFellow]

Camera man brutal with that spray down lmao

[@Mr.D.C.]

18:18 Squirrel!!

[@cheeseheadfiddle]

How is it possible that this exists? The most refreshing KZbin channel to explode in years. Vast appreciation.

[@evenaxin3628]

Also, in base 2 the smallest pandigital prime is 10 which means it's also the smallest pandigital and the smallest prime.

[@Arceus3251]

The flames are a metaphor for my brain being lit on fire after both understanding and not understanding this.

[@rogeraldrich2533]

"In general, never copy any physical actions you see in Combo Class" Especially chest bumping a soap bubble, who knows what could go wrong. Great episode, I've got plenty to ponder now.

[@zacharysmith4508]

All I know is this dude must be cool as fuck to have this nonsensical aesthetically for a video and its about math? Love it.

[@chrystiantodero47]

''there is not a standard notation'' is something we are familiar now. seems like everything he teaches would be like that

[@hhhsp951]

This dude has more whiteboard than I do. For now.

[@JamieJamez]

7:53 So there could be a Rayo's number - persistent number, but not a ℵ0- persistent number?

[@Voiding210]

Yes

[@LordMarcus]

18:32 Since the cardinality of the set of even integers is equal to the cardinality of the set of both even and odd integers, 100% of all integers are even. :)

[@bambam180acidpunk]

the thing you said about how there is no infinitely persistent pandigital but that you can name any arbitrarily large number of persistence and there will be a corresponding pandigital. and i can wrap my head around that any given persistence does not imply that it can scale from there but there is some other combination that reaches any given higher persistence. so its weird that you could name every possible arbitrarily large persistence and there will be a corresponding pandigital but no pandigital can scale to an infinite persistence. could you talk more about this relation in the future, i'm just going off what i think i understood, does this kind of relation have a name or anything else you know of interesting about it or other examples of similar interactions. i've seen all your episodes and i can follow but unless you explain it as clearly as you usually do i might not know what you're talking about. for instance i'm wondering if that relation i mentioned is in any way similar to godel's theorem, the godel sentence part is what i'm looking at. forgive me i'm a noob lol , i'm not asking if they are the same just if there are similarities and if you could comment on them and whatever the vocabulary is for them, or if i'm way off i'd like to hear about that to, thanks keep it up

[@iplaygames803]

How many whiteboards do you go through every year?

[@Bovineprogrammer]

If we relax the distinction between "regular" and "zeroless" minimal pandigital numbers (so the numbers contain all digits in the base exactly once, with lead zeroes allowed), are there numbers that can be minimally pandigital in multiple bases? I notice that the lowest minimally pandigital number in base 5 (1234 = 194 decimal) is lower than the highest minimally pandigital number in base 4 (3210 = 228 decimal), so there are overlapping areas where such numbers could exist.

[@majora4]

Absolutely. One such example is 3012 base 4 = [0]1243 base 5. I'm not certain if it's possible without using a leading zero, but my gut instinct says it's not.

[@Bovineprogrammer]

@@majora4 My gut says the same, which is why I included it. Nice to know there are numbers that fit the pattern - I was too lazy/busy to code something to check.

[@Voiding210]

There are 10!-9! Minimal pandigital numbers

[@tshandcannon548]

we need more tetration what about things like pi tetrated to pi, x tetrated to x and negative tetrations???????

[@sdspivey]

In ternary 201 (19 decimal) and 21 (7 in decimal), are minimal pandigital primes. No, there cannot be pandigital primes in decimal.

[@6gradosproducciones]

Most average mathematician

[@EthanBradley1231]

I object to you referring to 10 in base two as "ten." The word "ten" should unambiguously refer to the number equal to the number of 1's in 1,111,111,111.

[@dustindoesstuff374]

You’re still killin’ it, but lose the hat 😉

[@themightyripples6582]

I havent mentioned this before here but I've been thinking about it.. I really love the continuity through out these episodes - constantly referencing previous episodes. Like the evolving mess in the yard and the dirt on your jacket or the bubbles still randomly making an appearance. This is part of what makes your channel so entertaining for me and is one way you really stand out among other maths youtubers. Thank you for the effort, it's definitely appreciated :) !

[@josephrissler9847]

Exactly! This channel has such a creative component. I love the mad scientist vibe.

[@Jimorian]

I also appreciate that the "gimmick" of the deteriorating classroom is always there, and pops to the forefront at times, but it's never belabored to "stretch" the joke, and the lecture continues on with barely any interruption. That's a very tough balance to get right. :)

[@josephrissler9847]

@@Jimorian I like trying to guess which bits of chaos are staged and which are happy accidents. Entertaining either way

[@themightyripples6582]

You're all right! The creative mad scientist vibe. The bits of the chaos that are staged versus the random! Idk which is better because the act... I like the act because I often realize I'm slightly unstudied in something. Or something new altogether! So I go play with the numbers or graphs or both (usually both lol) then ill talk about it here The random chaos is almost like fireworks, it happens and you react afterward and it isnt getting in the way of the 'conversation'

[@Fire_Axus]

I dont like it

[@ShadowKestrel]

I must confess I only saw 2 squirrel cameos 😔 Great video as ever!

[@soupisfornoobs4081]

There's at least 3! Edit: not 3 factorial

[@ohmisterjeff]

AI-generated video of Jeff from American Dad as a math-genius KZbinr! Thank you, Domotro

[@swapnildas3056]

Nice! I've been waiting for this lesson!

[@anevenbluerjay]

Genuinely reassured by the fact that the camera guy has a hose in his other hand.

[@Ninja20704]

I was hoping you would mention about normal numbers, which is a number where the proportion of each digit in it is very close to 1/10 (cause we have 10 digits), and the digits are more or less evenly distributed. Because as numbers have more and more digits, it is more likely to be normal, meaning we are more likely to find specific strings of digits in it, especially into the millions of digits.

[@SirWilliamKidney]

I love the new hat! Great episode this was really fun!

[@robo3007]

32767 is my favourite zero-less number! It has no zeros at all in bases 2, 3, 4, 6, 8, 9, 10, 11 and 12.

[@Guil118]

I don't understand how the main channel has less subs than the bonus one. Both are great but that's the first time I see that.

[@ComboClass]

Mostly because of all the “shorts” I’ve posted on the bonus/Domotro channel

[@Guil118]

@@ComboClass Oh yeah! I forgot how strong shorts are these days. Thanks Domotro.

[@abe-danger]

18:18 SQUIRREL!!!!

[@JeremyOuelletteNH]

This is the best squirrel catch, it looks like he knows he's photobombin with his little head poking up lolol🤣🙈🤘

[@TrifectShow]

Great video, gotta love those number palindromes! Just a suggestion for a video. I've recently noticed there are high composite odd numbers. Much like 360 being a high composite number, which we use for circumference. "1, 3, 9, 15, 45, 105, 225, 315, .." are all high composite odd numbers. There are cool ways to use odd composite numbers to divide portions into odd amounts

[@asheep7797]

I feel like 15 and 1001 will be the horsemen of highly composite odd numbers. 15015 is probably up there.

[@TrifectShow]

@@asheep7797 I believe 15015 is

[@harriehausenman8623]

I love the frame-perfect editing and the Hollywood-style special effects!

[@timmyjefferson1815]

Hi teach

[@robertdibenedetto6445]

Awesome video! Are there any non-pandigital irrational numbers?

[@josephrissler9847]

Yes! There are infinitely many. For example, any integer multiple of Pi or e, with any digit of your choice removed

[@ComboClass]

Yep. There are even irrational numbers that (when written in base ten) just contain the digits 0 and 1 in an infinite non-repeating decimal. However to be irrational in binary, you would need both 0’s and 1’s and thus need to be pandigital in base 2

[@Voiding210]

0.101001000100001...

[@josephrissler9847]

I’m wondering if there are any irrationals that are non-pandigital in every integer base 3 or greater, or in every base above some number. I would be surprised if there are.

[@megliosolichesole]

How many pandigital numbers in an infinite base system? 0? 1? Does the question make sense?😆

[@ComboClass]

I guess you could say 0 haha, although if you used a base system with an infinite amount of possible digit symbols, the term pandigital just wouldn’t be an applicable term you could measure

[@JeremyOuelletteNH]

Uncountably infinite base! Probably nonsensical, but maybe if u were a totally lit math deity...

[@RibusPQR]

The number 111111111 and the number 12345678987654321 make me think of the odds of getting any particular number from rolling one 9-sided die, and rolling two 9-sided dice and adding them together.

[@joaomrtins]

27:21 squirrel

[@victorribera5796]

Aren't almost all binary written numbers pandigitals, and infinity-persistent?? Is this considered a trivial or exception case??

[@ComboClass]

Although the i only found info about the pandigitally persistent numbers in base ten, there’s a good chance binary follows the same pattern and doesn’t have any infinity persistent numbers by having every number eventually have a multiple that’s a non-prime mersenne number / hypereleven. For example, 101 in binary doubled is 1010 but tripled is 1111 which isn’t pandigital Edit: as the comment from Victor below reminded me, binary might actually be a small enough base to contain pandigital infinity-persistent numbers! I hadn’t analyzed that trait in binary yet but I’ll look into more details of how that trait works in other bases sometime in the future

[@victorribera5796]

@@ComboClass but for exemple, 2 (10) whatever you multiply it to would have a 0 at the end because it is even, and it will also always have a 1 since it is different to 0 therefore 2 is inifinity pandigital persistent right?? Again, I suppose this is considered a trivial case and the non existence of infinity persistent cases may be only for higher bases since binary doesn't have much to work with

[@ComboClass]

Good point! I guess pandigital infinity-persistence might work in base 2. I didn’t investigate that particular trait in binary (which I why I only mentioned base ten examples of it in this episode) but I’ll look into that more sometime in the future :)

[@WillyTheComposerOfficial]

Does this guy film these at the house he inherited from his grandma?? Cuz that’s what it looks like

[@renerpho]

I was hoping you'd touch on "pandigital formulas": Combinations of all the numbers from 1 to 9 (or from 0 to 9), together will the standard arithmetic operations (+, -, *, /, exponentiation, and decimal points). There are a few pandigital formulas that produce interesting results. My favourite is (1+0.2^9^(6*7))^5^3^84, which is a reasonably good approximation for the number e. (It is correct to 8368428989068425943817590916445001887164 decimal places.) You can leave out the 0, and write it as (1+.2^9^(6*7))^(5^3^84), if you prefer. A more famous example would be (1+9^(-4^(6*7)))^(3^2^85), which was the "star" of a Numberphile video a few years ago (then considered the record holder for the best pandigital approximation for e, but actually not nearly as good as the other example).

[@Xnoob545]

If you take Graham's number as a number string, it probably appears many many times inside the digits of TREE(3)

[@miachen2635]

Missed opportunity to share a very nice fact: the only number in base 10 (or any known base) whose square and cube are together pandigital in that base is 69.

[@smaza2]

ah, you're a programmer! this somehow explains a lot

[@emilyrln]

Watching you is like watching a juggler feign bad juggling: it looks like bumbling chaos, but everything is carefully controlled. At least, I _hope_ all these fire shenanigans are under control… 😅

[@muskyoxes]

"Almost all numbers contain all digits." Yawn "Trivial corollary: almost all numbers contain any particular string of digits you can name in that order." Mind blown

[@pepebriguglio6125]

Almost one in every 3765 one hundred digit number is not pandigital ☝️☝️☝️

[@pepebriguglio6125]

Does your squirrel appear in all of your videos? ... Please don't let your assistant mistake it for a stray flame, when he plays firefighter 🙏🙏🙏

[@jimmx2]

Just take coal chamber's "Loco".... replace Loco in the song with Domotro!

[@WillyTheComposerOfficial]

I once dated a girl who was pandigital

[@pkmnhx43_27]

Its quite interesting that despite the square numbers being able to be mapped to the integers, they have a density that approaches 0

[@josephrissler9847]

Yeah, man. Infinity gets weird.

[@carolinabullfighter574]

I cheered when he ripped a chunk off the dry erase board. Then the squirrel...what an inspiration

[@Voiding210]

5:56 squirrel

[@LordMarcus]

This is an auditory cognitohazard, and I don't mean the lecture.

[@Envy7817.]

Man the way he and his surroundings "look" is the reason i started studying maths again 😂

[@HomieSeal]

I just want to say that the square root of 12345678987654321 is honestly the coolest thing

[@a71_special]

look combo class this is about pandigital numbers.

[@bakerrboyywoo]

I absolutely love these videos thankyou so much

[@jordough4495]

Does gravity get stronger when you shoot these things?

[@aioia3885]

7:37 is the proof for that accesible? I'd like to see it

[@ra1nman_mashups]

23:12 reminds me of this game me and some friends played in middle school. You’d roll a 10 sided D&D die 10 times to get 10 random digits in base 10. Whatever number you rolled you would have to prank call!

[@asheep7797]

*rolls 0 000 000 911*

[@harriehausenman8623]

Some pseudo random engagement! 🤣

[@petros_adamopoulos]

27:21 for the squirrel.

[@tylerowens]

What is different about pandigital numbers in non-integer bases, since you specified integer bases in most of what you discussed here?

[@ComboClass]

Some irregular bases would follow many of the same patterns, but since I mentioned some of these patterns apply to “all/any bases” I wanted to be clear that in this episode i was just describing/investigating positional numeral bases with a base number of an integer 2 or greater

[@JeremyOuelletteNH]

human brain 🤯

[@evenaxin3628]

Are there any interesting properties mathematically of palindromic numbers?

[@evenaxin3628]

Based on this, I just learned about an alliterative number. 1023985674765893201 is a palindromic pandigital prime.

[@ameteuraspirant]

could you consider a "zeroless" pandigital number to just have the restriction that any zero exists at the beginning of the number in the highest place(s) and therefore to be a subset of all pandigital numbers only with an extra restriction?

[@ComboClass]

Yeah I hinted at that when counting the minimal amount of each here, but didn’t go into it much so to clarify more here: if you were to include the infinite amount of leading zeros to the left of a number (which numbers technically have but usually aren’t included when analyzing the number’s digits), then the zeroless pandigitals would basically be “the pandigitals whose only zeros are of the leading type” and general pandigitals would have to be redefined as “pandigitals with every digit including a non-leading zero”

[@ToanPham-wr7xe]

😮

[@NoOffenseAnimation]

i always thought that 'domotro' was a very steampunk name, but ig it makes sense, cuz you have a lotta clocks

[@renkothebird]

The clearly superior pandigital number is 1 in base 1

[@steepslopesmm2]

don't you mean 0?

[@monksandmonets]

Well the algo just recommended this to me and it's dope. Congrats on your new fame.

[@Tletna]

So, I was wondering with regard to your earlier writing numbers in various bases videos. Has anyone ever written out how each value on the line y = x would look as represented in base x, same thing for y = mx + b, or y = mx^2 + nx + b, or other functions y of something x. Since that includes so many irrationals both algebraic and transcendental, then, at first, let's restrict things to integer values of x to begin with. So, base zero at y = x = 0 would be difficult to express. However y = x = 1 would just be 1 or a dash just like counting. y = x = 2 would be like binary so 10, y = x = 3 would be 10 in ternary, so all values express in base = x would be 10 for positive integers. But what about other functions? So now lets do y = 2x. For y = 2x = 2*1 = 2 we have 2 dashes or 1 1, for y = 2x = 2*2 = 4 we have 4 in binary or 100, for y = 2x = 2*3 = 6, we have 6 but in ternary which is 20. For y = 2x = 2*4 = 8 or in 20 in quarternary as we'll see this pattern will continue. What about y = x^2? 0 again doesn't mean much, 1 is 1*1 = 1 or a dash. y = x^2 = 2^2 = 4 is 100 in binary. y = x^2 = 3^2 = 9 is 100 in ternary. y = x^2 = 4^2 = 16 = 100 in quarternary. y = x^2 = 5^2 = 25 = 100 in quinary. So, again a pretty boring pattern. Now, I only did positive integers for x, and only did y = x, or y = 2x, or y = x^2 so far. So, I guess, I'd ask, are there more interesting functions y(x) when we represent the answer y in base x both restricting x to positive integers and when not? More important questions: is there a special base for counting squirrel cameos? And, have you considered a sponsor who provides insurance or sells fire extinguishers?

[@ComboClass]

That sounds interesting. I’ve actually been planning a future episode about a connection that I realized bases have to polynomial equations, although with a different approach than the one you mentioned :)

[@justRD1]

Squirrel!

[@infectedrainbow]

He has a schnitzel.