What's special about 288? - Numberphile

Featuring Sophie Maclean and superfactorials. Try the Halfsies challenge at brilliant.org/challenge/numbe... (and tell us your score!) or go to Brilliant's courses and start for free at brilliant.org/numberphile/ (episode sponsor)
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[@ceegers]

For the notation, we could go in a spanish direction and do ¡4! for superfactorial of 4.

[@BlahBlahTALMUD]

you have my vote

[@ericpowell96]

Great idea

[@christopherellis2663]

¡Bravo! 😅

[@ilplolthereturn7525]

As a Spanish person, I will sign this petition

[@neonaeon]

Very cool

[@yoru0121]

It's even cooler when you realise sf(4) can be written as 4^1 x 3^2 x 2^3 x 1^4. So sf(4) = 4^1 x 3^2 x 2^3 x 1^4 = 288 = 4^4 + 3^3 + 2^2 + 1^1

[@KalikiDoom]

Cool!

[@Fragaut]

So, 288 is multiplicatively boustrophedonic. That's cool.

[@adityakhanna113]

Ah that's cool

[@johnjeffreys6440]

isn't factorial and 'to the power of' the same thing?

[@cheeseburgermonkey7104]

@@johnjeffreys6440 Factorials and exponentiation are very different. n! is the product of every positive integer up to n, and x^n is x multiplied by itself n times

[@xcheese1]

You're doing a Numberphile video about 288? I think it's two gross.

[@ilplolthereturn7525]

💀

[@nottmjas]

Ever since the English speaking world started to move over to metric, that statement will be lost on most native speakers nowadays.

[@heisen-bones]

@@nottmjas don't worry it's obvious what they mean

[@joshmyer9]

​@@nottmjasAh, yes, the metric dozen is ruining everything.

[@NoNameAtAll2]

​@@joshmyer9demolition, not ruin

[@fechtbrandon]

Notation suggestion: Using an exclamation point after but instead as a superscript. This makes sense (to me, at least) since superfactorials are repeated factorialization in a similar way that exponentiation is repeated multiplication. And "superscript" uses the word super! e.g. 4^! = 288

[@dylanrambow2704]

I was going to suggest this too, for pretty much the same reason.

[@PhilBagels]

That's what I was going to suggest. Either that or putting the exclamation point directly above the number. Like this: ! 4 = 288 (Hard to depict in this format. We need new symbols, like the interrobang, but they're not on the keyboard. Yes, I know we can put in some kind of code to call up these "special" characters, but that's a PITA. People who make keyboards need to get with the program and update them for new characters!)

[@stephenbeck7222]

The exclamation mark as superscript is really easy to confuse with just a regular exclamation mark.

[@PhilBagels]

@@stephenbeck7222 Right. Which is why I suggest putting it above the number.

[@Nethershaw]

@@PhilBagels That's not how typography works.

[@methethpropbut8519]

As a long-time viewer of Numberphile (about 8 years and counting!) it's quite surreal seeing someone I used to see in uni lectures make it famous on here. Thank you for continuing to spread the love of mathematics to a new generation.

[@Luper1billion]

I feel like the favorite number videos is closest to the spirit of Numberphile. You get some much passion from their explanation

[@seanLee-sk2mi]

girls are not known for their brain, that's a fact. i wonder why so many girls in this channel. someone explain to me.

[@tttITA10]

Mix the dollar sign with an exclamation point! So it has an "s" for "super" and a "!" for factorials.

[@tenelitebrains]

Came here to say exactly the same $!

[@Verlisify]

Does a mathematician ever hear a new number fact and not make it their new favorite number

[@OrangeC7]

This Numberphile video in particular has a playfulness to it that I think is absolutely awesome, one of my favourites already!

[@tookiecar1]

Hi

[@johnchessant3012]

That's pretty cool! And I think the sf notation is perfectly fine, since in music sf means "sforzando", which tells you to play a note with a sudden emphasis!

[@MichaelWarman]

This was good, more of Sophie please

[@Wolforce]

Suggestion for notation. Since it's supposed to be super, it should be a buffed up exclamation mark. That could be a triangle (inverted delta) with a bold dot at the bottom. Like a comics BAM exclamation mark

[@Matthew-bu7fg]

If maths doesn't work out for Sophie then there's always scriptwriting - there were so many twists in this video before the crescendo! 😆in all seriousness, I loved the enthusiasm! Also love that we've got what I like to think as a "classic numberphile" video. A fairly run of the mill, bog standard base ten number that we never think about but - when unpicked - turns into something beautiful!

[@joseftrogl6565]

Simply numberphile from latin phileo = loving. Numberphile = loving numbers.

[@gabor6259]

Thank you, very cool! Fun fact: 288 is twice 144 which is 12 squared and is the 12th Fibonacci number.

[@jasonremy1627]

It's been a while since we've had an OG Numberphile video about an actual number! Love it!

[@joshuazelinsky5213]

288 has some other nice properties also. It is a refactorable number, which is a number whose total number of positive divisors is itself a divisor of the number. (A smaller example is 12 which has 6 divisors one of which is 6). 288 also is one of the rare numbers which are both twice perfect square and one less than a perfect square. There are infinitely many of these and they connect to what is known as Pell's Equation.

[@hughcaldwell1034]

I was aware of the second property but not the first. Thanks!

[@backwashjoe7864]

Not sure how "rare numbers" and "infinitely many" can both be true. I'll have to think about this.

[@hughcaldwell1034]

@@backwashjoe7864 It's about how frequent a type of number is, rather than the sheer number of them. Take powers of ten, for example. There are two in the first ten positive integers (1 and 10), so at that point it looks like 20% of positive integers are powers of ten. If you sample the first hundred, though, you only get three powers of ten, so adjust that estimate to 3%. Sample the first thousand, and we've only got four powers of ten, so adjust again to 0.4%. This trend toward zero continues rather rapidly. If you generate a number at random, the probability of it being a power of ten is 0%. On the other hand, if we do this same repeated sampling of positive integers but count even numbers, we get a trend toward 50%, and if we generate a number at random, it's got a 50% chance of being even. So powers of ten are rarer than even numbers, even though there's a countably infinite number of each.

[@scudlee]

I can always remember that second property coming up in a question I faced when I was a kid, which was along the lines of "If you add the numbers from 1 to 8, you get a square number. What's the next EVEN number for which that's true?" The answer was 288. If you think in terms of the formula for triangle numbers you can see why that property is important. Of course, there is an ODD number that works before 288, but that has slightly different requirements.

[@rosiefay7283]

@@scudlee Neat. Let our even number be 2n. T(2n)=n(2n+1) must be a square. n and 2n+1 are coprime (i.e. have no prime factor in common) and so must separately be squares: n=y^2, 2n+1=x^2, therefore x^2=2y^2+1, a Pell equation.

[@goodboi650]

Is it not confusing to use n!! notation as given in the video as opposed to taking a factorial twice (as in, factorial of n!)? The second one seems like a more obvious way to interpret the double exclamation.

[@KeimoKissa]

Maybe, but if you're gonna use two exclamation marks, this choice is obvious, since taking factorials multiple times can be expressed with parentheses: (n!)!

[@radeklew1]

It is confusing, but it's also standard 🙃

[@88porpoise]

​@@KeimoKissaExcept (n!)! would be taking the factorial of n!. For example (3!)! would be 6! not the superfactorial of 3. I would say n!! makes more sense for superfactorial, but that notation is already taken.

[@Henrix1998]

The fact that 5!! and (5!)! are different numbers is kinda confusing

[@filipsperl]

​@@Henrix1998 it's pretty common that you need parenthesis to make a distinction in maths. For example: 4^3^2 =/= (4^3)^2

[@winstonnn9863]

Not even ashamed to say I'm simping for Sophie

[@johnsecunde3321]

Classic numberphile! Love it!!

[@ryanhinojosa1205]

I used this to generalize further extensions of superfactorials. By using my modular reduction technique i`ve developed this pattern holds for factorials, superfactorials, and my generalized extension, yielding the same 6 numbers of cyclic permutation group

[@stephenaustin3026]

I've lived and worked on both sides of the Atlantic, and I don't think I've ever heard anyone say "n take 2" for n-2, for example. I have heard people say "n take k" to refer to C(n, k), though it's more common to say "n choose k". In elementary/primary school, some teachers might say "n take away 2" but "n take 2" on its own sounds odd to me.

[@SgtSupaman]

Yeah, it feels like a case of someone wanting to sound like they are talking about something more complicated than they really are. I mean, how hard is it to refer to a basic function that already has a name (minus) by that name?

[@curtiswfranks]

I dislike the notation "n $" specifically 𝘣𝘦𝘤𝘢𝘶𝘴𝘦 I write monetary values with the monetary unit symbol (such as "$") exclusively after the numerical part. That notation is consistent with other units, allows for prefixes, and does not interfere with negative signs. On the other hand, univariate functions should typically prefix their inputs (which should be in parentheses), the negative sign being the major exception. So, I would use "!(n)" for the factorial of n and "$(n)" for the superfactorial of n. Also, I prefer to put a dot under the single vertical stroke of "$", so that it looks like an "S" superimposed on "!", for the superfactorial. My monetary unit symbol "$" has two vertical strokes.

[@veqv]

Calling it now. This woman is the next Hannah Fry. Wonderful exposition - anticipates the reader, has fun quips and keeps folks engaged.

[@sicapanjesis3987]

Most of the video was just muttering 1st grade math, and then an one-liner fact. Just like those 'keep watching' shorts.

[@ferretyluv]

@@sicapanjesis3987She’s clearly in number theory, which is all about first grade math and making it super complicated.

[@liliwheeler2204]

All that AND an affinity for the interrobang? Definitely the next Hannah Fry

[@SamuelEstenlund]

Exactly my thought!

[@chrispi314]

For Super Factorial, as an ease of use, I would write 5^! (in computer form) or just add a small "!" as an exponent on the number.

[@filipsperl]

That could suggest that superfactorial is defined: n^!= n^(n-1)^(n-2)^...^2^1. At least I thought that was going to be what superfactorial is, but it's probably something totally diferent

[@chrispi314]

@@filipsperl well, in some specialized field, to put an exponent it is often referred as "sup", so putting the exponent "!" would quite literally translate to "sup factorial" And I used the notation x^! because "^" is often use in computing to express a "sup exponent"

[@michaels4340]

6:20 It's square because it's an even power, not because it's a power of two.

[@FujiSkunk]

I don't know that this has been done before and I'm just now noticing it, but recreating the on-screen graphics using the same writing from the actual brown papers is a nice touch!

[@I.____.....__...__]

- So if a super-factorial is a the product of the factorials (S5=5!×4!×3!×2!×1!), then would a hyper-factorial be the toootally useful factorial of the factorial of a number (H5=120!)? 🤔 - 2:58 "You never put a dollar-sign after a number, so it feels so wrong" - Just making it clear that Sophie has never been to France or Québec or various other places.

[@unvergebeneid]

"My favourite number changes a lot." I like her already 😄

[@Pffffffffffffffffffffff]

My favorite mathematician changes a lot, but for now it is Sophie Maclean.

[@Dheeraj5373]

288 is special because it's my room number today. What a coincidence 🤯

[@Arc125]

Superfactorial notation should be a fat exclamation point, so instead of a line and a dot, it's a rectangle and a circle. Kind of like the blackboard notation for number systems - integers (Z), rationals (Q), complex (C), etc.

[@D0ct0rD4RK]

Sophie is beautiful in so many ways.

[@colonialgandalf]

A lovely video. Got me hooked halfway through. Fun to see the mathematicians of the future show their passion for maths and creative spirit. Good luck girl!

[@fedor-kokosik]

Sophie is very nice! Need more videos with her

[@Heinz-bx8sd]

Simp

[@UnimatrixOne]

@@wernergamper6200 agree

[@ForAnAngel]

@@wernergamper6200Why not?

[@wernergamper6200]

@@ForAnAngel I don't know. Maybe it's the uninteresting topic

[@johntaylor6211]

In a bikini would be interesting ! 🎉

[@vigilantcosmicpenguin8721]

I think the interrobang works really well because it also represents how I felt to find out about the superfactorial.

[@ZainAK283]

She's brilliant - more of her please!!

[@dameanvil]

0:00 🧮 Introduction to the fascination with the number 288. 0:22 🌀 Factorials: Definition and variations like double and super factorials. 1:26 🔍 Super factorial definition: Product of n! * n-1! * ... * 2! * 1!. 1:50 🔢 Examples of factorials: 1!, 2!, 3!, 4!, 5! and corresponding super factorials. 2:41 🤔 Proposals for a symbol for super factorial: Interabang or 'super' text. 4:07 🧩 Interesting property: 4k super factorial divided by 2k factorial yields a square number. 6:35 💡 Fascination with 288 due to its unique expression as 4! * 3! * 2! * 1! and through powers sum. 8:04 🧠 Advertisement for the Halfsies game by Brilliant, showcasing a puzzle involving visual estimation skills. 8:52 🔍 Mention of a recent discovery of a shorter super permutation than previously known.

[@MindstabThrull]

I'd love to see how many favourite numbers Sophie and Matt Parker end up going through because I feel like it's a lot :D

[@bonanzaguy04]

I propose as the notation for super factorial an exclamation point overlayed over an S. So similar to a dollar sign, but instead of a simple straight vertical line, it's an exclamation point!

[@gugus8081]

For sure 288 is an absolute beautiful number, it's the number of blocks available in Trackmania Nations Forever

[@amirharoush5210]

for the super factorial I suggest !_2 (where _2 is sub 2) because the normal factorial is !, i.e. of order 1, and this is of order 2, and of course u can extend it to 3, 4, etc

[@adrien5568]

Same.

[@VincentZalzal]

I was about to suggest the same (but with a superscript instead, same thing).

[@adrien5568]

A superscript could be confused with a power. Is 3!^2, 3!*3! or sf(3)? You would need a additional symbol, like writing the level into a square or something.

[@VincentZalzal]

@@adrien5568 Yeah, I was thinking of writing 3!^(2) for example, i.e. the superscript in round brackets, like you see sometimes for other things like derivatives in Lagrange notation, to avoid having to add new notation. But the subscript might be better. Anyway, the important point is that it should be a number representing the "order" of the factorial as Amir said, for generalization. This was the point I also wanted to make.

[@SkateTube]

Thanks for all the hard work ;)

[@endrankluvsda4loko172]

Thank you for all the great content, Brady! You rock almost as hard as the Mighty Black Stump! And Sophie was all kinds of awesome! I hope you have her back again sometime

[@tiagomarques9822]

In the former Portuguese currency before we changed to Euros, the Escudo, we used the $ sign after the numbers, not before: 5 escudos was 5$. But actually we always placed the centavos (cents) after the $, so 5 escudos would rather be represented by 5$00, and 2,5 escudos by 2$50.

[@geoffroi-le-Hook]

In Québec, they put the dollar sign after the number. The bus fare is 1,50 $ (un dollar cinquante)

[@TheAntibozo]

To denote superfactorial, follow it with the squirrel (technically chipmunk, but chipmunks are squirrels) emoji 🐿, e.g. 4🐿==288.

[@WAMTAT]

I love how Mathematicians often have multiple favorite numbers

[@Meta11axis]

You have to make a video explaining how it is possible to write accurately with such a grip of the marker. I was so puzzled I missed all the maths...

[@whiterottenrabbit]

Ditto! Like, WTF?

[@I.____.....__...__]

Yeah, it has always baffled me whenever I've seen one holding a pen or pencil like that, it looks so bizarre and odd that they can even have the dexterity and control to write like that. 🤨

[@rosiefay7283]

288 has another interesting property, connected to the Erdos-Straus conjecture. The conjecture is that for any integer n>1, there are positive integers x, y, z which represent n like this: 4/n = 1/x + 1/y + 1/z. Suppose a value is chosen for n. If there are positive integers a, b, d where 4ab=dn+a+b and d divides ab, then n is represented like this: x is the positive integer where ab=dx; y=an; z=bn. It is known that there are no such a, b, d if n is a square. Only three non-square n are known where there are no such a, b, d. The lowest of these is 288.

[@VG-or1nu]

I like her way/manner of explaining things.

[@1959Edsel]

17/12 is a pretty good approximation of the square root of two. 17 squared is 289. 12 squared, multiplied by 2 is 288.

[@ViliamF.]

Is this a version of Matt Parker from a parallel universe? This feels like a video Matt would have made, down to every detail, even the everchanging favorite number.

[@patton72010]

Every Numberphile's video comment section: X is so great!! Love watchin them. We need more videos with X.

[@ronniechilds2002]

I've been watching these Numberphile videos for years. Even though the math is almost always way over my head, I am still drawn to the videos and I think I finally realized why: The presenters are always cheerful, often to the point of humorous, in their demeanor. Whether they are young old, male, female, native- or non-native English speakers, their unfailingly bright and upbeat attitudes, just lift my spirits a little. Am I being nutty here?

[@Quantiad]

"We're running out of letters and symbols"... laughs in infinity.

[@BleuSquid]

Yay! Another reason to use the interrobang! I heckin love that bit of punctuation.

[@ronaldmullins8221]

I personally feel like the double factorial notation is better suited for super factorial. You're factorial-ing the factorials, so it would make sense to have two exclamation marks for it. Double factorial could be something different, maybe using a sub-script 2 if that's not already used. Could be expanded for triple or quadruple factorials if those are/can be things

[@poulanthrope]

The trouble with the Superman shield is it doesn't imply anything factorial, so how about the pentagon with a ! in the middle?

[@I.____.....__...__]

I would think the bigger problem is that it's not exactly present on most keyboards. 😒

[@jacobshirley3457]

​ @I.____.....__...__ Just start using emojiis. On Windows, press the WINDOWSKEY + PERIOD. 4😒 = 288

[@johncallan4788]

Thank you for introducing me to the 'Interrobang' ; ‽ I'm nearly 70 years old ; and I still learn something new every day.

[@NickEllis-nr6ot]

Her excitement is infectious 😀

[@Demo-critus]

From the definition of sf() it follows that sf(n) is equal to the product for k=1 to n of k^(n-k+1) [because n occurs in just one factorial, n-1 in 2 of them, n-2 in 3, etc. until 1 which occurs in all n factorials]. Isn't this a much simpler expression @numberphile ?

[@nottelling7438]

When someone ends an exclamation with a number, I enjoy reinterpreting that as a factorial, massively inflating the number they were expressing excitement about. The dollar sign notation could let me do that for people misusing dollar signs. I do like the interrobang, though.

[@jonathanallan5007]

288 is also the number of valid Sudoku arrangements for a 4 by 4 grid.

[@nymalous3428]

Sophie reminds me of Tom Scott. That last fact about 288 is nifty, I'll have to remember to show it to my students.

[@19seb85]

Great new collaborator. Looking forward to see more of her. She reminds me of Matt with her enthusiasm and new favorite number :)

[@terraqueo89]

Now this is some classic numberphile stuff! More numbers!

[@RalphDratman]

I like her teaching style. It is very lively!

[@TheNameOfJesus]

@1:05 - Instead of ending the even numbered sequence as 4! x 2!, you could end it at 2! x 0! since that would give you the same result. I've scrolled through the comments, and nobody has noticed that.

[@gabor6259]

For notation I suggest !n!. You can stack the exclamation marks on each side.

[@AbcDef-hi]

So much enthusiasm. I'd say this is a SUPER video.

[@BorlandC452]

My suggestion for super factorial notation is to put the exclamation mark sideways across the top, like x-bar.

[@justindesilets3526]

as a Canadian french speaker, I can confirm that we out the dollar sign after the number! so for me, it is 24$ and not $24.

[@DrewTrox]

3:29 I suggest moving on to Egyptian Hieroglyphs. For this equation I drew a kitty. This one uses a little bird.

[@TheDrinkingFood]

Since superfactorial seems like the next step up, much like tetration is to exponentiation, you could co-opt the up arrow notation in some way- for example, an up arrow with a dot below. I doubt theres a computer character for this tho.

[@cowsaysmoo51]

I think a combo of an ! and $ would be neat. Basically an exclamation point with a small "s" near the top.

[@numberphile]

Try the Halfsies challenge at brilliant.org/challenge/numberphile (and tell us your score!) - Brilliant sponsored this episode.

[@mcnica89]

FYI this link is giving me a 404 page not found right now

[@1vader]

The real link seems to be /halfsies instead of /numberphile

[@RollinLeonard]

broken link

[@bot24032]

@@1vader the numberphile link would work if it had challenge and not challenges (btw @numberphile i solved it)

[@Terigena]

Thanks @@1vader. I got a 94% with 4 perfect scores (including the "cover US with Australia" one!)

[@sk8pkl]

Very very interesting! It reminds me of the pythagorean tetractis. 4+3+2+1=10. 288 is part of the 432A "natural" scale. That factorial and n^n factorial relationship also reminds me of the perfect shapes wich have = values for perimeter and area... Oorr squaring the circle. Its like 4321 is a "perfect number" number in aloooottt of ways and i never heard of that one! Sweet! Thanks!

[@MxIzmir]

My notation for superfactorial is the number with a bowl of soup poured over it.

[@Weiszklee]

If all the standard symbols are used up, clearly we need to go into more exotic symbols. The most obvious oneds are, of cours,e emojis, and it has a perfect candidate for the super factorial: ❕

[@adamcionoob3912]

Cool video, also checked out the Halfsies challenge. Got 95% accuracy, 4 perfect cuts.

[@grafrotz5286]

To put the currency sign after the number is very natural. E.g. five Euro are written as 5,-€.

[@lanadragonfly]

Suggestion: Use an exclamation point with two dots instead of one. Sort of like an upside down ï. The nice thing about that is you can construct supersuperfactorials and n-number superfactorials and the symbol is just n number of dots under the exclamation point.

[@liliwheeler2204]

Maybe it's already been done, but somebody needs to make a playlist of all the "here's my current favorite number" videos 👀

[@jacksonstarky8288]

I like the interrobang or the Spanish punctuation suggested by @ceegers for the superfactorial notation.

[@MrBarberousse]

À dollar sign after the number feels wrong? You’ll be weirded out if you visit Canada.

[@flam1ngicecream]

My notation would be n!_2, where the 2 is a subscript. That paves the way for a super-superfactorial, written as n!_3: n!_3 = n!_2 × (n - 1)!_2 × (n - 2)!_2 × ... × 2!_2 × 1!_2. In general, you could even define a (super)^k-factorial as n!_k, where n!_k = n!_(k - 1) × (n - 1)!_(k - 1) × (n - 2)!_(k - 1) × ... × 2!_(k - 1) × 1!_(k - 1).

[@itsananimal4813]

I liked the video, but I loved that dollar sign! I had to pause it to make sure she hadn't printed it or used a stencil.

[@andypyne]

ŨF (the Latin letter U with tilde I'm using to mean "Soup", as in soup'r factorial)

[@ahseaton8353]

How about squaring the exclamation point, give or take some parens: 5{!}² or 5{!²) Then you could do a super-super factorial: 5(!³) or even higher terms: 5(!⁴) or 5(!^n)

[@stephenklump2421]

Useless side note: in Quebec they put the $ after the number, and often use , for the cents separator. So you'll see 5,99$ and eventually it will look normal.

[@jellezwaag]

Love her enthousiasm, instant part of the numberphile furniture 😅

[@AlaaGhazala]

I would go with !x, because it's really doing both operations.

[@dyhrdmet]

You broke my calculator with "super factorial" and you broke my keyboard with "interrobang"

[@dylanboekelman1471]

The notation should be !! with a strikeout or double strikeout so it’s effectively a # or H with points at the bottom.

[@Gna-rn7zx]

288 demonstrates the square theorem, too! SF(4(1))/(2(1))! = 288/2 = 144 = 12^2

[@zh84]

It would have been nice to get some examples of why the superfactorial is useful.

[@GenericAnimeBoy]

Safe to guess it comes up in combinatorics problems somehow.

[@Lexivor]

It probably isn't very useful.

[@JohnLeePettimoreIII]

SUGGESTION for super-factorial notation : ! + circumflex = ! (with a crown)

[@MarkStoddard]

"You never put a dollar sign after the number." Thank you so much. A not insignificant number of people actually do this, and I don't know why. Maybe just government school.

[@tristanc6967]

Alright, here's my bold suggestion for notion. We will now use the ! for super factorials, and use a subscript ! for the traditional

[@rockbloodystar]

A numberphile video about a number! Classic return to form.

[@pauljackson3491]

Even though all the normal symbols have been used up I still think the $ sign is better than the others because at least it is acsii. Are the others even in unicode? It's even more confusing if other mathematicians use different ways of defining superfactorial.