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MATT PARKER: I'm going to show you two of my all-time
favorite diagrams.
And it's a way of linking numbers as they go
from one to the next.
And a lot of people will have heard of-- or I'm sure you'll
have a video on this somewhere--
about happy numbers.
DR RIA SYMONDS: 0 squared equals 1.
This makes my number 7 a happy number.
MATT PARKER: And I'm not going to do happy numbers.
I'm going to do a thing on the number 145, if that's OK.
And I'll get to that in my second diagram.
My first diagram is the idea of happy numbers.
So for example, let's do the happy number--
let's do 13.
You can square both those digits and add them together.
So if you square 1, you get 1.
If you square 3, you get 9.
And so that will give you 10.
OK?
And if you repeat the process--
you then square 1, you get 1.
You square 0, you get 0.
That will give you 1.
And so 13, through a chain, ends up at 1.
And there are other numbers higher up.
So let's say you had 23.
And you square 2, you get 4.
You square 3, you get 9.
You get 13.
23 will lead into 13, as will 32.
Doesn't matter what order they're in.
In fact, over here, we could have gone in that way via 31.
And what I'm going to do now is very quickly draw for you
all the two-digit numbers, because there are some
three-digit numbers.
We could have come into 10.
130 would work just as well.
And that will have come in from 97, because if you square
9 and square 7, add them together, you get 130.
Which of course means that 79 will do the same thing.
And we could have got there from 94, or we could have come
in from 49.
And if we come in from 49, we could have got there from 70
or from 7, would give us 49, because it's just the 7.
And on the other side over here, we could have
got to 32 via 44.
And we actually could have got to 1 from 100.
And we could have got into 100 from--
86 would have got us to 100, as would have 68.
The great thing about 68 that's different to 86 is you
can get there from 28.
Or you can come in from 82.
And you can get into 82 from either 19 or 91.
And those are all the two-digit numbers that end up
in this happy tree.
And I call that process of squaring both the digits-- or
all the digits-- and then adding them "happification."
So you could call this the happification tree, and these
are all the extreme points.
So actually, that there is the top of that branch.
That's the top of that branch.
That's the top of that branch, and the rest
of them feed down.
And obviously, a lot of the forks you just get the
redundant--
vestigial, I guess-- number which leads in at the same
point, because it doesn't matter on the
order of the digits.
OK.
BRADY HARAN: Happification tree.
MATT PARKER: Happification tree.
There you are.
Now, the diagram I really want to show you-- the reason I'm
talking about 145--
is not the happification tree.
We'll do this with 83.
That's probably a good start.
83 is not a happy number.
And the reason 83 is not a happy number-- we'll put it
over here-- is that if you keep happificating--
bear with me--
83, you do not end up at 1.
But where else can you end up, right?
What happens?
Square 8 and square 3.
And you add them together.
You end up with 73, which, as you will have realized by now,
could also happen if you had started at 38.
So put that in for completeness sake.
73--
if you now square 7 and square 3, that will give you 58.
If you square 5, square 8, that will give you 89.
If you square 8, square 9, you'll get the 145 that I was
mentioning before.
If you square 1, 4, and 5, and add all those together, you
get 42, a rather popular number.
If you square those, you'll get to 20.
If you go from 20, you'll get to 4.
4 goes straight to 16.
16 will give you 37, and then 37 will give you 58.
And at this point, we're back in the loop.
So now, two things happen.
When you start with a two-digit number and start
squaring and adding the digits, you either end up down
here at 1, so you can consider this like a tree.
So I call this the happification tree.
And it all grows out of 1.
Or you end up in this coil, right?
And all the other numbers feed into this coil.
So these are the happy numbers.
And I call these the--
melancoiling numbers?
That'll work.
So you've got the happy numbers and the
melancoiling numbers.
And because I am the thorough, fastidious kind of guy, I am
going to fill in very quickly-- if you don't mind
bearing with me for a second, Brady--
I'm going to fill in all the other melancoiling numbers.
37 has quite a tree that comes off it.
You can get to 37 from either 106--
that is a 6--
which feeds in from 95 and 59.
Or alternatively, you can also get there via 61, which is
kind of cool.
61--
you can get to there from 65, which gives us 74 and 47,
which comes off 75 and 57.
Whereas 61 also gives us 56, which leads into--
OK, 45-- nearly there.
63, that's 36.
And down here, we end up with 6, and 60, and
there's another end.
This is the map, which consists of two parts of how
all the two-digit numbers link together.
This is the structure for all two-digit numbers when you
happificate them, so where you square all the digits and add
them together.
And you'll notice it requires a few three-digit numbers to
form the structure.
So there's a three-digit number.
There's the 145, which the video is all about.
And there's a few other ones.
There's 113 hiding down here.
Where else does one sneak in?
There's 130 sneaks in over there.
And they're vital for the structure.
All other three-digit or higher numbers filter into
this diagram.
This is the complete structure for all the two-digit numbers.
BRADY HARAN: Let's draw the three-digit one as well, then.
MATT PARKER: OK.
I'll get some-- no!
So you can if you want, but it doesn't add any
further detail to this.
And what I love is all numbers of any size will filter down
into one of two situations.
They either filter down to 1.
Or they will filter down into the melancoil.
And this is it.
And of course, from here, you can go absolutely wild.
And I'm going to refrain from this level of wildness.
But you could look at what happens when you cube all your
digits, right?
So if you cube all your digits,
you get more scenarios.
You get some numbers that go-- there are different sizes and
types of coils.
There are different ways to end up at 1.
Because you always get 1, because things like 1,000 will
drop you there.
There are points--
which there's no points here.
It's either 1, or it's the spiral.
You get points in three because-- as again, I'm sure
there are other videos about narcissistic numbers.
DR RIA SYMONDS: It's the same number.
It's in love with itself.
MATT PARKER: When you hit, let's say, 371, then 1 cubed
plus 3 cubed plus 7 cubed gives you 371.
And so you end up stuck.
And things feed into that point.
And you can do it for powers of 4 or 5, or there's a type
of infinite number of options there.
And you can do it in different bases.
Mathematicians get a bit emotional when we become
base-10-centric.
And there's no reason why you can't generalize
this to other bases.
But there you are.
If you limit yourself just to squaring in base 10, these are
the two diagrams you get.
And so I got bored.
And I made a quick spreadsheet.
And I went through.
And I classified all of this and drew the diagram myself.
So this was me, again, bored in the pub with some mates.
And we went through and classified all the numbers.
So there you are.
I came up with this myself, but other people very well
could have done this.
I don't want to claim ownership.
This somewhat pathetic name of melancoiling numbers I did
come up with.
But again, I've seen them described as not happy numbers
or unhappy numbers.
But then again, why are they called happy numbers in the
first place?
What a ridiculous name.
So this particular interpretation of the happy
numbers is of my invention.
And as always, I gave it a suitably pathetic name.