# Four has Four Letters

The word for the number four has four letters
It is the only number in the English language
that has the same number of letters as it's value.
And that's not really a mathematics fact
but we can make it a bit more 'mathsy' because
you can consider this a function, that takes a number, turns it to its word (in English)
and then counts how many letters that word has.
And for that function, four is a fixed point
It doesn't do anyway, it maps to itself.
So you can consider this like a sequence
So we map to five, that's got four letters
We map to four. There we are, and 4 maps to itself. We stay there.
If you repeat this process, you will always
end up trapped at four.
I'm quite jealous of other languages that have more interesting structures
in their networks for the letter-counting function.
So, Indonesian, of all languages, the Indonesian word
for four is 'empat'.
And you are like, "Well hang on, that's got five letters."
But the Indonesian word for five, "lima", has four letters.
So you get this nice little two-cycle.
And all Indonesian words feed into this two-cycle.
French is even better.
So, the word for three in French, 'trois', that has got five letters,
the word for five has got four letters,
the word for four has got six letters,
the word for six (which is spelt the same as English) has three letters
and we are back where we started.
All French words for numbers end up in this full cycle here.
That's absolutely fantastic.
It's only a shame that I can't pronounce these words
for you correctly in French.
You see, I can't speak French.
Traditionally, I let the funky music do the talking.
[Girl's Aloud - Can't Speak French reference]
Talking.
Now.
Back to boring English with its sinkhole at four.
But how do we know that that fixed point is the only real structure in the network?
How do we know that there is not a cycle somewhere else?
Well, for there to be a cycle somewhere,
you need to have words which contain more letters than the value of that number.
And that only occurs in English for numbers smaller than four.
All of one, two and three do have more letters than their value.
But above four, five onwards, all have fewer letters.
As you can see, four is a kind of threshold,
where above it, every single number has fewer letters than its value,
and so everything must drift down to this point.
The same thing happens in French, Indonesian and other languages,
I have a conjecture.
And my conjecture is for any linguistic number system,
there is some threshold k, such that for that numbers bigger than k,
they have m̶o̶r̶e̶ (fewer) letters than the number they represent.
And as a kind of, I guess, 'sub-conjecture' I suspect that
generally speaking, k is quiet small.
I'm ruling out trivial things like a tally.
Yeah, four could be DASH-DASH-DASH-DASH,
But, you are just multiplying up the value of the number by how many letters are in
whatever you are calling the line in your tally.
And that would give you a geometric sequence,
which, while wonderful and interesting,
is not what I'm after here.
Fun fact: The highest k value I've found for any linguistic number system is
18
And that is just reading it out in Binary.
Eighteen is one-zero-zero-one-zero, and that has eighteen letters.
And anything above eighteen, reading out binary,
you've got fewer letters than the value of that number.
And so, you get the fixed point at eighteen.
There's another fixed point at thirteen, because one-one-zero-one
has thirteen letters.
So all numbers in this system will hit one of those two points.
They actually hit eighteen a lot more than thirteen.
For smaller numbers it's not so bad.
if you do all numbers up to a 100,
sixteen of end up at thirteen and the remaining eighty-four end up at eighteen.
But as you get bigger, the vast majority hit eighteen
before they have any chance to get anywhere near thirteen.
Were you to write a program to check all the numbers up to a million,
(which I may have done)
you will find that 99.9904% of them hit eighteen,
and the remaining 0.0096% of them hit thirteen.
Please do double check I got that right.
If we don't have any loops in English, can we still have some fun?
Well, yes the first option is to cheat.
Nine has got four letters, but let's swap that out to be
Nine-r.
There we are, five letters now. And five could be,
ahh,
Five-arino.
That's nine letters.
There's...
the...
That's not going to work, is it?
Plan B, we have some near-misses.
So, fifty has five letters and it's ten times five,
and we are in base 10 that's kinda nice.
Likewise, seventy has seven letters which means
you get the kinda quirky things like,
Fifty-four is a five and a four if you look at the letters of the two words.
And, seventy-four — seven and four.
But, you know it doesn't...
It's not quite as satisfying, it's kinda a function where you split up all of the words,
count the letters and then use them in the columns of the base, I mean, c'mon.
When this all occurred to me yesterday, I started playing around with it and realised
just how boring the English language can be.
I realised there's only one real option for what to do here.
If we accept that everything is going to end up at four,
we can try to find the longest chain of numbers
that ends up at four.
And so, first of all, I checked all the chains for numbers below a hundred.
And there are several chains in there which are six numbers long.
And so, I pick the one with the smallest starting number
I've got it here.
So if you start at twenty-three,
twenty-three has eleven letters,
eleven has six letters,
six has three letters,
which has five letters
which gives us four.
I did find some seven-chains when I checked all the way up to a million.
And, there are quite a few of them.
And I now know the smallest seven-chain that ends up at four.
But, I'm not going to tell you what that is.
That's partly so you can find people who can try and find it yourself.
And partly, because it depends on how you say words above a hundred.
So you got an option.
So, for something like 496, that can be four hundred AND ninety-six, or just four hundred ninety-six.
I went with four hundred ninety six.
Other options are available.
That's my short introduction to the letter-counting function.
You've now got three optional pieces of homework.
You can try and find a longer chain.
I've found them up to seven.
I have no doubt there are bigger ones.
As long as you state your naming conventions, I don't mind what you use
as long as it is logical and semi-consistent.
Or, #2, you can try and find a language which has got a bigger loop than French.
To be fair, I haven't check very many.
French has a loop with four numbers in it.
See if you can locate a language with more than four in a cycle.
And, thirdly, your last bit of homework.
Can you find or come up with some kind of linguistic number system which has a k threshold of
greater than eighteen.
Bonus challenge, I guess, is to find three times square root 100
which does indeed equal thirty and has thirty letters.
But things like that, they are kinda medium-pleasing.
Obviously doing the maths is it's own reward.
But, for a literal less-figurative reward
I'm going to take my cheat notes here, of my six cycle
and I'm going to sign them down the bottom.
And I will pick whichever answer in the comments to this video I think is the most pleasing
and they can win this.
I can write "congratulations" and your name.
Then, I'll post that to you.
I've not got a huge amount of free time at the moment.
I'm meant to be writing a book.
I got a looming first-draft deadline coming up.
So, if you see someone else put any kind of finding in the comments,
I will hugely appreciate it if you can double check it four (for) me.
If you already know the language they're using,
or if you can work out what they are trying to do.
And you see if it works or doesn't.
And if you see someone's suggestion or someone's finding
and you think "That is really good",
do leave a comment. Say, "Hey, I think this is the person who should win the prize".
And that would help me immensely to whittle it down to a few good entries,
one of which will get this.
So, I don't know why you are watching this. [Music fades in]
Get to work.
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[THANKS FOR WATCHING]