You have four 4s: 4 ,4 ,4 and 4,
and using all four 4s
you have to create, using the mathematical operators,
as many numbers as you can.
So, the way that we start this normally, if we're
doing "level 1" four 4s, will be,
all that you're allowed to use is, addition, subtraction,
multiplication and division.
Let's start at 0, and we gonna then go to 1, then go to 2.
So, with 0, how do you get four 4s to equal 0?
It is kind of easy.
You can go 4 minus 4 plus 4 minus 4. That's 0.
Brilliant! We've got 0.
Okay, let's put the four 4s again to equal 1.
Well, we could do 4 divided by 4, now that's 1,
plus 4 minus 4, or we could have done
4 divided by 4, divided by 4 divided by 4 which is 1 divided by 1, which is 1.
Again that's 1. There are lots of different ways,
and they get a little bit more complicated.
And we can go and do 2. And we can
carry on upwards.
How far? Ten? A hundred? A thousand?
I'm going to show you today,
Brady, your mind will be blown!
I will prove the four 4s all the way up to
infinity.
[Brady]: You can make every number?
You can make EVERY number up to infinity.
It's a fantastic little proof and we're
going to get there. We need to actually be,
we need to clarify the operators first.
We can get up to 10 just with these:
plus, minus, multiplication and division.
To get up to 20, we need to introduce
some other things: what we need to
introduce is concatenation,
which means that just that we've got 4 4 4 4, we can
bring them together, so we can actually
bring them together and consider that, say forty-four.
We introduce the decimal point so one of these could be zero point four.
And we introduce the factorial, which means it's 4 times 3 times 2 times 1.
And also, we need the square root,
so square root of 4 is obviously 2.
You might think, well, that's spoiling this other
little beautiful simple puzzle, but
actually factorial, concatenation,
decimal, and square root are really basic.
They're kind of in our basic armory.
so I think that's fine. Well, how far can we do
with all of these? We can definitely get
up all the way to 50. Higher than 50?
We can get up pretty high, we can get up to 100,
but in fact we need to start introducing other ones, and
I'll show you-- one of the ones that doesn't work
with all these, is, say, 99.
I'm gonna show you how to do 99.
So 99, there are different ways to do this, but this
is the one that I quite like. It all depends
which is the new symbol you're going to do.
4 over 4, as a percentage is 100
minus 1 is 99.
So that's quite a clever way
and the good fun thing about the four 4s
well, just say you're bored on a
plane or a train, you can just start at
the beginning and then just go up and
if you've got a long haul flight
you might be able to get to 500 or something.
Some purists might disagree
and say this is really ugly and prefer
something else. Some people introduce
things like "the triangle number of" or
there's lots of combinatorial symbols that you can do,
you know, "4 choose 4" or
"4 factorial choose 4."
But already, with the operations that I've mentioned here,
where we have pretty much everything we need.
The only other one that we need to get
up to infinity is log, and log is a
pretty simple thing, mathematically speaking.
Brady: "Do you mean log ten, the natural log
or all different types of log?"
We're going to get there because actually the
log that we need is log base 4.
Brady: "Of course."
Of course, it had to be 4.
And we're also going to use log base half.
This is your brief refresher on what a log is, okay?
So, if a equals b to the c, then
the log base b of a equals c.
So that's basically the same thing.
Let's put some numbers in, because we're
gonna be doing log 4, so 4 is equal to
4 to the 1, okay? Because 4 to the 1 is 4.
So the log base 4 of 4 is 1.
16 is 4 squared,
so the log base 4 of 16 equals 2.
64 equals 4 cubed,
so log base 4 of 64 equals 3.
Can you see there's a pattern here?
Basically, log base 4 of 4 to the n equals n.
n doesn't have to be a whole number
so the square root of n is the
same thing as n to the half.
Let's translate that here.
We've got log the base 4 of the square root of 4
which is actually log base 4 of 4 to the half equals half.
And let's do the square root of the square root of n which is
the same thing as n to the half
times half, which is the same thing as n to the quarter.
We get log base four of square root of the square root of 4
equals log base 4 equals a quarter. I'm just gonna carry on because it's all over.
So the log of log of base 4 of the square root of the square of the square root of 4
equals log the base 4 of 4 to the 1/8 equals 1/8.
So, we are half the way there to proving the four 4s to infinity.
Because what we've actually done, we've got two 4s.
We know that this expression here,
we know log 4 of the square root of 4 is half.
Log 4 of the square root of the square root of 4 is a quarter.
So we know that just using two 4s we can
get the half the halfing sequence
which is half quarter eighth. So we've got two 4s left,
and let's see how we can get there.
Another bit of paper, and we're gonna do all on this next bit of paper.
We had log 4 of 4 to the n is n.
It's actually the same.
Whatever we have here that would be the same.
Log half half to the n equals n
Right?
And so let us now try to find the
log base half of these things here.
What I'm gonna do, the answer is always
log base half of half. That's 1.
Log base half of a quarter,
okay we've got two 4s to do that, is the same thing
as log base half of half squared, which equals 2.
Log base half of an eighth
is log base half of half cubed, which equals 3.
So, we know that two 4s goes into here,
and in here and here. We just need to
find a way of expressing log base half
with two 4s. So we need to find a way of expressing half with two 4s,
that half is obviously 2 in 4
but that same thing is root 4 over 4.
So here, we can say that
log to the base root 4 over 4 of log 4 root 4 is 1.
Because that is half, and
that is log of half, so that's 1
Brady: "So you used all four 4s then."
I've got all four 4s.
Earlier I showed you how you can use the four 4s to make 1
in a very very simple way.
Use 4 divided by 4, plus 4 or minus 4 is 1
But you can also do it here.
Log to the root 4 over 4 of log to the base 4 of root 4 is 1,
but log to the base 4 over 4 of log base 4
of the square root of the square root of 4 equals 2.
So this will expand out, and
Brady: "So however many square root signs you put there"
is equal to the number.
Brady: "so you could put, if you put a billion square root signs there, you'd get a billiion."
Yeah. If you have a bit of paper big enough to write,
log root 4 over 4 of log 4
the square root the square root the square root
so much that you have a billion square root signs
of square root 4,
including everything there are a billion square root signs,
that would equal a billion.
The four 4s is a puzzle that was first written
down in the late 19th century and you
see very clearly in 1890s, there's a boom of this puzzle.
Then in like the 1930s, there's another boom.
Paul Dirac has a go at it. And Paul Dirac,
who famously loved mathematical puzzles,
basically beats the puzzle, completely
defeats the puzzle. Before it was considered how high
can you go, what are the operators that
you're going to need, and he just says, well,
with just logs and a square root
I can bust this.
It's broken. I can solve it all the way to infinity.
Eccentricity of one, well actually, it's no longer an ellipse, then it's a parabola,
and the shortest you can get is an eccentricity of zero. Eccentricity means away from the circle.