This is the infinite monkey theorem.

I've been thinking about how much monkey typing is needed, and decided to do some tests. I acquired a small gedanken monkey that can type randomly at 48wpm (4 keystrokes per second) without stopping for food or rest. I gave my monkey a small typewriter with 26 letter keys (capitals only) and a space bar five times as big as the others, and (forgiving soul that I am) excused it from any punctuation. I wanted to see if he would type the word MONKEY at some point over the next 5 years.

In fact, he did, and it was hidden away like this: "... SJKBV SDG FMMONKEYP SRGH DKAFJI ..." near the bottom of page 230 of volume 798 of a thousand volumes of monkeyprint. I have built a library to hold these volumes for the benefit of future generations.

To clarify my "it is likely": the probability of the word appearing in 5 years is about one half. So I was lucky: it's 50:50 whether the word MONKEY would be found at all.

I decide to raise the bar to MONKEY WANT BANANA. (If my monkey had typed this, I would have fed it. But it didn't, did it.)

Monkeys have been around for about 50 million years. What if I had acquired all the monkeys in the world from the very start of monkeys, and had them type continuously at 48wpm for 50 million years? If anyone can help me with how many monkeys there are and have been over the last 50 million years, please let me know. I'm going to suggest 10 billion, on average. They would have produced a hundred thousand trillion tons of monkeyprint by now. Might they have managed a MONKEY WANT BANANA? Yes: the probability that one of them would have done it is an impressive 89.5%.

This is promising. Now, what if the universe was filled with tiny monkeys right back to the Big Bang, typing as fast as possible until now?

So, how tiny? One per atom? That would be good. And how fast can they type? A billion keystrokes per second? Now we’re talking! We have a 98% chance of one of them managing a SHALL I COMPARE THEE TO A SUMMERS DAY THOU ART MORE LOVELY AND MORE TEMPERATE. 98% is good! But add one more word, and the probability of one of them knocking that out drops to one in fifty million, which is not so good.

I have to make my monkeys smaller! One per atom is not very many, as most of the universe is intergalactic space with less than one atom of matter per cubic metre. There should be more monkeys than this. I shrink the monkey until it occupies the smallest possible space there is. Quantum gravity physics tells us that there is a smallest possible distance, known as the 'Planck length'. So of course I want to use this.

If you were to try to examine anything on a smaller scale than the Planck scale, the sheer effort required to do the examining would be so great that you’d create a tiny black hole there, bigger than the size of the thing you were trying to see. (The black hole would vanish very quickly as soon as you stopped looking.) The same thing would happen if you tried to chop anything into smaller pieces than the Planck length – you’d make a black hole and end up with bigger things than you started with. In fact,

*nothing can happen*on a smaller scale than this – at least, not without radically altering the laws of physics so much that you would all but obliterate any meaning in the word 'smaller'. Experimentally, no-one has come anywhere near getting there, which is probably just as well.

As well as being the size limit on smallness, the Planck length is

**the**fundamental unit of distance – the only one that is not made up with reference to anything else, as are metres and miles and the like. This elegance and purity makes it very important in physics.

I can't see it catching on though – it's way too silly. You wouldn't really want to measure your inside leg in Planck lengths. One Planck length is a hundred billion billion times smaller than the distance across a proton, which is a tiny speck of a thing that sits at the very centre of a hydrogen atom (the smallest atom).

Back to the monkeys.

I’m going to divide the universe into Planck-sized regions, and put a monkey in each one. You will ask what the monkey is made of, when nothing can be smaller than the Planck scale, and I will say that it is not made of anything – it is a single, fundamental monkey particle. One in every Planck sized region of space. These regions are very small – there will be nearly as many monkeys inside the space occupied by a single atom as there are atoms in the universe. And there will be monkeys in the spaces not occupied by atoms too.

And they will type faster. How fast can a thing happen? Just as there is a shortest possible distance, there is a shortest possible time, and it’s called the Planck time. The Planck time is how long it would take you to cover one Planck length if you travelled at the speed of light.

My monkeys will type at a rate of one keystroke per Planck time.

They will type so fast because the energy required to confine a monkey to such a small region will make the monkey extraordinarily hot.

You will ask what the typewriter is made of, and I will say it is not separate from the monkey: typing is what a monkey particle does. (I don’t know what happens to the letters that the monkeys type. There is no room for them or anything else, as the cosmos is jam-packed with hot monkey particles. But I’m not going to let this stop me.)

So, from the Big Bang, with a monkey in every last tiniest unit of space possible, typing at the fastest speed there is, for the entire history of the growing Universe, and do we have a deal?

Yes! The first four lines of the sonnet “SHALL I COMPARE THEE TO A SUMMERS DAY THOU ART MORE LOVELY AND MORE TEMPERATE ROUGH WINDS DO SHAKE THE DARLING BUDS OF MAY AND SUMMERS LEASE HATH ALL TOO SHORT A DATE” will be knocked out somewhere in the cosmos several times a second!

This is good! In fact, every few dozen thousand years, it’ll come together with the next word – SOMETIME – to boot. Will we ever get the next

*two*words (SOMETIME TOO)? We might be lucky – there’s something like a one in three chance in the age of the universe.

So there we are. One in three. Ladies and gentlemen, I give you, from the Monkeys of the Cosmos, four lines and two words of a sonnet!

...FOESZH GIMCED GHN ASIO AKHPS WRSHALL I COMPARE THEE TO A SUMMERS DAY THOU ART MORE LOVELY AND MORE TEMPERATE ROUGH WINDS DO SHAKE THE DARLING BUDS OF MAY AND SUMMERS LEASE HATH ALL TOO SHORT A DATE SOMETIME TOOSFB L FPGPAAO XUN WVIKGXWS TX FSAOL PABK...

I don't know about you, but I think that's rather impressive.

If you want the Complete Works, as the theorem says, you'll have to wait.

[Back to blog]

## 34 comments:

Respect!

How is it possible that no-one has yet commented on this post?

I would like to see a book of little chapters like this.

I'd even help look for a publisher..

One last thought for the evening - what are the chances that it just so happens that, these comments and the word verification tests are being posted by a tiny little monkey? Just a little thought ecwnonz dnfsuyh embxwiwy very fast ifhcyaj and hot rjanykbl yykrqkh zynktqf

You can spank your Planck AND your monkey. Coincidence? I don't think so

(obviously, the subject of mental masturbation has absolutely no relevance here)

Hi Tom!

Glad you like. I'd do loads if it didn't take me so stupidly long. I'm not hot or fast enough for this game. But there will be more.

Hi jonathan!

Never thought of this as a porn site, but hey - whatever floats your boat. (I Googled "spank your Planck" and drew a blank. You got yourself a niche market! Go for it - I'll waive the commission, as it's you.)

I take your point though - it's not very useful, is it. And a bit silly.

Only taking the mick - it was very interesting. I've managed to work out the Planck length all for myself!

I hope you got the 2π right, Dr A, and didn't cheat.

2π, what 2π?

WHAT 2π!!!?????!!!???!!!

Now now.

Dimensional analysis doesn't tell you about the bar on the ℏ, is all. And if you did Compton = Schwarzchild, you'd be out by 2√π. Or something. Come to think of it, it's all a bit arbitrary, isn't it? How

didyou do it?I took a bit of paper and ripped it in half. Then I tore one of the halves in half. I carried on until I couldn't do it any more (the piece of paper was quite hard to hold down by this time).

Then I got my ruler out. I get

1.61624 x10^(-35) m

Cool. And did it turn into a little black hole?

He-he you funny!

Is your WV set by tiny monkeys?

Hallo Mr F!

Not sure what a WV is, I'm afraid.

Anyone inserting a set of random letters where one might expect a word is immediately suspect, I'm sure you'll understand...

...EWDSF AMAZING FUN THANKS ASPQWERJDFVG WER SDF PFJKEW C WERPAQW...

WV=Word Verification

TO BE OR NOT TO BE THAT IS THE GEZERNENKLAT.

Very impressive work.Would be nice of you, if you could post the probabilty calculations (the results of which you have used here).

Sure, Sumyodeep - or anyone else - drop me an email (address at top of blog) and I'll get on the case and send you some working.

...and all this time I thought it was 42....no, wait....

Hi Bob,

I'll keap this brief cause my typin ain't very gooad. I've just popped over to reemind you of Multiverses. Suffice to saye, I

I've got a mate who knocked out the Compleate Works of Shakespaere & Leo Tolstoy's War and Pease in one night!! Anyway, must dash, got a lion nibbling on my arse!!

Yours sincereley,

Cheetah (no not that one - a parallel one)!!

i think i would like to swim in a vast infinite ocean of quantum monkey/typers. i think it would feel squishy.

Hey there - I know this is an old post, but I enjoyed the heck out of it. I'm substitute teaching 8th grade math this week - my students will be in for a surprise when I hand this out as required reading! I'll have to check out the rest of your blog, I'm intrigued.

Awesome man. I love the way you wrote this article. I'm a writer myself, and I also write articles of this nature, but I have to say, this has earned you my respect, my friend.

In the seventeenth century, a monkey

didwrite the entire works of Shakespeare. It only took around ten million years, during which monkeys invented language, then writing. Later, we invented typewriters...Good point...

I love this post. I was arguing the theorem being quite literal with atoms, which of course confused the person I was arguing with. This left them speechless and angry.

If this is taken from an evolutionary standpoint, not purely a math one, then it is missing something vital. As Richard Dawkins points out in The Blind Watchmaker, the probability of randomly producing the works of WIlliam Shakespeare is phenomenally low. However, the odds go up dramatically when the concept of negative reenforcement is introduced. Assume instead that your monkeys were each individually poked in the back of the head after each incorrect keystroke, and the production of Mr. Shakespeare's works becomes much more believable.

This is an entertaining thought experiment on probability and the Planck length though.

Hello Bob,

I'm so impressed by this article. Is it possible if i can translate it into my own language?

Hi Doğukan

Yes, please do! And send me a link :)

Very interesting article, keep it up!

Really nice article. But may I know the mathematics of this thought experiment? And for determining when and where will the exact phrase appear, did you use a computer program to track the whole process?

Thank you

Hi Jivesh. Thanks.

It's four years ago now... I just put the numbers in a spreadsheet and used standard probability equations. I still have the spreadsheet - here is an entirely unhelpful view of it. :)

All the assumptions are given in the post, so it should be possible for someone with a basic grasp of probability to reproduce or check the numbers.

The only part that isn't explained here or anywhere is a calculation I did for the spacetime hypervolume of the cosmic history of the observable universe in Planck units, which turned out to be a 243-digit number beginning with a 7.

That's a handy piece of info to bring up in conversation over a pint. (Also handy if you ever meet a cosmologist and want to see them raise their eyebrows.)

I can't remember how I got that number. It involved some reasonably sensible cosmological model, some algebra and calculus on a piece of paper, and a pocket calculator at the end. I guess I should try to do it again sometime just to check, maybe try and figure out the second digit (out of 243).

If you want a very rough estimate (without involving general relativity!), you can always take the age of the universe and divide it by the Planck time and raise it to the fourth power.

(Now I bet you wish you'd never asked)

This is brilliant!

http://www.eloquentmath.com/

How would the probabilities change if the monkeys typed in binary (1 and 0) and we interpreted the binary output with ASCII or UTF-8?

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