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svobodamartin 45 minutes ago [-]
My favorite one is with the 52! seconds:
Start a timer that will count down the number of seconds from 52! to 0.
Then walk around the Earth’s equator with one step every billion years.
Then, after you make your way around the earth equator (by taking 1 step every billion of years), you take one drop of water out of the Pacific Ocean.
Then, you repeat the process of walking around the equator, and everytime you walk around, you keep draining one singular drop of water.
After the ocean is fully drained, you refill the ocean and put a piece of paper underneath you.
Now, you once again repeat this process of walking, draining, and placing papers.
After your stack of papers has reached the Sun, you repeat another 1000 times.
After all this, you have completed just about a third of the timer.
I feel like you could do something similar for these sorts of "fathom this large number" recipes.
davidnc 32 minutes ago [-]
For another perspective, 52! is roughly the number of atoms in a galaxy. Galaxies are really quite large!
andrewla 2 hours ago [-]
This brings to mind the analysis in Bender & Orszag; they approach this through difference equations (a bit of a lost art in formal mathematics; very 19th-century feel) rather than integration.
Instead of introducing the gamma function, they instead start from the observation that log(F_n) - log(F_n-1) = log(n), so treating this difference as analogous to integration, it says that F_n ~= nlogn + n as the leading asymptotic behavior. This is clear just by substitution and algebra; no calculus necessary (though it helps to "know the answer beforehand").
From there you can treat the error term in this as F_n = n^n * e^n * E_n and plug that into the same relationship (F_n = n * F_n-1) to derive what that error term looks like asymptotically, and end up in the same place that the integration on the OP leads to.
ninju 5 hours ago [-]
The author's casual mention of 52! at the opening of the article triggered an OLD webpage that I saw many years ago
Anyone know how to determine the age of this page (it's got be at least 20yrs old)
benatkin 9 minutes ago [-]
To me it feels like a smuggled exponentation. If I take half of 52 and raise it to itself and add a base unit to it like grams or meters, that's an incomprehensibly large amount.
stronglikedan 4 hours ago [-]
52 cards is the first thing I think of when I think factorials. It's such a great and relatable way to convey the subject to people, plus it usually ends up blowing their minds like it did mine when I first learned of it. Not from this page, but from a YT vid many moons ago.
Dwedit 4 hours ago [-]
It was made during the brief XHTML craze. (And it's also invalid XHTML)
TheRealPomax 5 hours ago [-]
The main.css file it imports dates itself to March 9 of 2005, and is housed in an "ancient history" section of the website that covers everything before October 26, 2010, so: "sometime between those two years" =P
DavidSJ 4 hours ago [-]
Its first appearance on the WayBack Machine is October 13, 2009, which narrows the range somewhat.
abetusk 4 hours ago [-]
lg(n!) grows roughly as (n lg n). Constants matter, of course, but to that's the rough estimate.
As an aside, if you take numbers from 0 to (n-1) in an array, there are n! configurations, so representing each configuration or differentiating each configuration take n lg n bits. So, in some sense, taking a mapping that's able to differentiate the input state to map to the ordered state takes at least O(n lg n) time, the standard runtime of a basic sorting algorithm.
Any additional assumptions (n larger than maximum element, distribution of elements) helps reduce this.
thechao 34 minutes ago [-]
My kids love taking about big numbers. TREE(3) is a family favorite. So, I was going over sequences with them, and I decided to go slow instead. My sequence was: 1 1 1 1 ... 1 ...
They accused me of using just "all 1s" (which is, naturally, cheating). Ai contraire!
The count of the number of digits in the decimal representation of the number of unique primes in the prime factorization of the natural numbers.
The best part is that even pretty young kids can compute this sequence; by the first "2" is at 2*3*5*7*11*13*17*19*23*29!
(Hopefully I got that right; the phone doesn't make it easy to type!)
3 hours ago [-]
movpasd 6 hours ago [-]
Stirling's approximation is also used a lot in statistical mechanics, because you often have to calculate logs of state space sizes, which means lots of combinatorics and thus lots of factorials. Plus it's continuous so you can do calculus.
Sharlin 5 hours ago [-]
A quick and dirty approximation of the number of digits in n! is n lg n, which approximates n! from above, via the inequality
1 * 2 * … * n ≤ n * … * n.
(This approximation should be familiar to many from an algorithmics class.)
For a tighter bound, use n lg n - n/2, or a better approximation of ln 10 in place of 1/2 if you wish. This comes from Stirling's approximation which notes that
ln n! = n ln n - n + O(ln n).
qsort 4 hours ago [-]
> (This approximation should be familiar to many from an algorithmics class.)
You need both sides though :)
What makes it interesting for estimating algorithmic complexity is that \log{n!} \in \Theta(n \log n). One side is obvious as you note, the other less so, but there's a famous trick to do both at once:
I don't know why that is surprising, but I'm teaching computer science at a university, and students seem surprised every time it comes up.
36 minutes ago [-]
pagade 6 hours ago [-]
Reminds me of: Professor asked us to find the biggest factorial using C programming language. And then using LISP. You can imagine our surprise.
pkaye 5 hours ago [-]
There is a algorithm call Prime Swing Factorial that can compute large factorials exactly in arbitrary precision math using prime factorization. Like 10000000! in under second depending of how optimized the math library it. Probably like 100x faster than the normal method.
anthk 2 hours ago [-]
With Lisp you can use iterative algos and get that under a second too.
SBCL can be ridiculously fast; and if you optimize the compilation for integers... the speed gets really close to your solution.
flcikfinder 4 hours ago [-]
[flagged]
brudgers 22 hours ago [-]
Factorial (n) for n > 24 is greater than 10^n.
saalweachter 53 minutes ago [-]
24! is approximately Avogadro's number (about a 3% difference).
anthk 2 hours ago [-]
I did factorials even under KLISP 23 with cons cells as fake integers:
Dog slow but the old n270 netbook (32 bit) handles big factorials >20 fine, and OFC it's instant under Common Lisp (SBCL) and Scheme (both S9 and Chicken).
Start a timer that will count down the number of seconds from 52! to 0. Then walk around the Earth’s equator with one step every billion years. Then, after you make your way around the earth equator (by taking 1 step every billion of years), you take one drop of water out of the Pacific Ocean. Then, you repeat the process of walking around the equator, and everytime you walk around, you keep draining one singular drop of water. After the ocean is fully drained, you refill the ocean and put a piece of paper underneath you. Now, you once again repeat this process of walking, draining, and placing papers. After your stack of papers has reached the Sun, you repeat another 1000 times.
After all this, you have completed just about a third of the timer.
https://sites.imsa.edu/hadron/2025/02/26/how-big-is-52/
I once made a little tool for getting more intuitive spatial scales for things in the universe at https://observablehq.com/@ikesau/scale-to-the-universe
I feel like you could do something similar for these sorts of "fathom this large number" recipes.
Instead of introducing the gamma function, they instead start from the observation that log(F_n) - log(F_n-1) = log(n), so treating this difference as analogous to integration, it says that F_n ~= nlogn + n as the leading asymptotic behavior. This is clear just by substitution and algebra; no calculus necessary (though it helps to "know the answer beforehand").
From there you can treat the error term in this as F_n = n^n * e^n * E_n and plug that into the same relationship (F_n = n * F_n-1) to derive what that error term looks like asymptotically, and end up in the same place that the integration on the OP leads to.
https://czep.net/weblog/52cards.html
Anyone know how to determine the age of this page (it's got be at least 20yrs old)
As an aside, if you take numbers from 0 to (n-1) in an array, there are n! configurations, so representing each configuration or differentiating each configuration take n lg n bits. So, in some sense, taking a mapping that's able to differentiate the input state to map to the ordered state takes at least O(n lg n) time, the standard runtime of a basic sorting algorithm.
Any additional assumptions (n larger than maximum element, distribution of elements) helps reduce this.
They accused me of using just "all 1s" (which is, naturally, cheating). Ai contraire!
The count of the number of digits in the decimal representation of the number of unique primes in the prime factorization of the natural numbers.
The best part is that even pretty young kids can compute this sequence; by the first "2" is at 2*3*5*7*11*13*17*19*23*29!
(Hopefully I got that right; the phone doesn't make it easy to type!)
For a tighter bound, use n lg n - n/2, or a better approximation of ln 10 in place of 1/2 if you wish. This comes from Stirling's approximation which notes that
You need both sides though :)
What makes it interesting for estimating algorithmic complexity is that \log{n!} \in \Theta(n \log n). One side is obvious as you note, the other less so, but there's a famous trick to do both at once:
\log{n!} = \log{\prod_{h=0}^{n} h} = \sum_{h=0}^{n} \log{h}
Therefore,
\int_0^n \log{x} dx \le \log{n!} \le \int_0^n \log{x+1} dx
with both integrals trivial by parts.
n! < exp(n log n)
> n! < exp(n log n)
Why would that be surprising? I can see that many wouldn’t know whether it’s true, but
and it’s not surprising that for n > 1https://t3x.org/klisp/22/index.html
Dog slow but the old n270 netbook (32 bit) handles big factorials >20 fine, and OFC it's instant under Common Lisp (SBCL) and Scheme (both S9 and Chicken).