Estimating factorial size without a calculator
Eli Bendersky explores mathematical techniques for approximating the number of digits in large factorials.
What to know
- Bendersky demonstrates a practical method for estimating factorial sizes using Stirling's formula without computational tools.
- The approach uses the Gamma function and Laplace's method to approximate integrals, with results within a few digits of the actual answer.
- The mathematical technique is applicable to estimating sizes of extremely large numbers where exact computation is impractical.
Eli Bendersky Author and mathematician
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Post gains traction on Hacker News
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 · source -
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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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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background
Eli Bendersky publishes factorial estimation guide — Bendersky posted an article on his blog exploring mathematical methods to estimate the size of factorials without a calculator. The post demonstrates that 52! has 68 digits and explains the underlying mathematics using Stirling's formula and the Gamma function.
What people are saying 8 voices from 1 site · best of 12 · verbatim
- Sep 17
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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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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…
- Sep 16
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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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For another perspective, 52! is roughly the number of atoms in a galaxy. Galaxies are really quite large!
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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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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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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.