Post gains traction on Hacker News
1 Sep 15 9:25 AM · 8d ago · 1 article · 2 posts · 3 sources · development 1 of 1
The article reached Hacker News frontpage with 70-83 points and 30-39 comments, indicating community interest in the mathematical content.
“The other day, I found myself wondering how big 52! (52 factorial) is, and that led me to ponder how these could be estimated without a calculator or a computer.”
Eli Bendersky, Author · hn ↗Eli Bendersky Author and mathematician
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Reported in the same hours no headline names this development itself — these 1 claim were published in its stretch
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1 outlet How big are factorials?
first by HN Frontpage, 8d ago
What people said 11 voices · best of 12 · verbatim
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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…
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H
How Big Are Factorials? L: https:// eli.thegreenplace.net/2026/how -big-are-factorials/ C: https:// news.ycombinator.com/item?id=4 9712185 posted on 2026.09.15 at 09:25:38 (c=1, p=3)
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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…
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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…
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I came across an interesting feature of factorials while making the puzzle books at https://www.kakurokokoro.comThe widest two rows are nine digits across, but while the first can be any of arrangements of the digits 1-9, the second cannot repeat any digits in the same columns, and so it limits allowable permutations to the number of derangements…
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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…
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This reminded me of tetration and Knuth's up-arrow notation. It's basically repeated exponentiation.Searching now, I just learned of tetrofactorial, which is a factorial using tetration operations. There's also pentation which is repeated tetration.And there's a whole wiki for it here: googology.fandom.comIt's fun because the numbers are so big…
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Holy hell, this is great.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-universeI feel like you could do something similar for these sorts of "fathom this large number" recipes.
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The author's casual mention of 52! at the opening of the article triggered an OLD webpage that I saw many years agohttps://czep.net/weblog/52cards.htmlAnyone know how to determine the age of this page (it's got be at least 20yrs old)
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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.
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For another perspective, 52! is roughly the number of atoms in a galaxy. Galaxies are really quite large!
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