So we're going to do a classic
Numberphile video, where we talk about a
number and a property of that number.
Today's number is 196. What's special about
196? Well we're going to do a process called
reverse and add. It's a very simple little game,
you can do it. So we're going to take a
number, what number shall we take? Give me
a two-digit number. - 23
23 alright. I'm going to reverse an add; so I'm going to
reverse the digits, so it's 32. And add
them together and if I'm correct it's 55,
okay? Now I'm going to stop there because
this is a palindrome number, so it's the
same forwards and back, and that's what I
want to get to. Let's see if I can do
another number, maybe it will take a
little bit longer to get to a palindrome.
What other number could we pick? Let's
pick a three-digit number- 254. We add 452,
so 706. So we haven't got a palindrome
number yet. So we might have to do that
again; so we take the result and apply it again.
...plus 607, add that together what do
we get? So 1313 - it's not palindrome
yet, might have to do another step. Question
is how many steps is this going to take?
So we add these together - hey! Only took
a couple of steps, 4444.
Will all numbers end up finally or
eventually with a palindrome?
It turns out all two-digit numbers do
end up in a palindrome, all three digit
numbers we're not so sure about. In fact
the number 196 is the smallest number
that we don't know if it ends in a
palindrome or not when you repeat this
process, and we've tried for a billion
iterations. We've done this over and over
and over more than a billion
iterations and still it hasn't ended
in a palindrome. Will it ever? A number
that doesn't end in a palindrome, it's
called a Lychrel number; 196 appears
to be a Lychrel number, it appears to be
the smallest one, but we're not sure, we
don't know, it's not proved - we've just
checked it a lot. So most numbers take
three, four steps? Not, you know, not
very long. 89, 89 is an interesting one, it
takes 24 steps to finally get to be a
palindrome number. I'm pausing in case
Brady says, go on then. In case Brady made
me do it, I did make a note of that
better, I haven't quite finished it yet
but Brady isn't going to make me do it!
- (Brady: I'll do it on screen for you)
Excellent! Fantastic! Because we would be here for a
long time, took me ages to do this.
(Brady: 196 might be one of these Lychrel numbers,)
(are there any definite Lychrel numbers?)
There are no definite Lychrel numbers.
We've got a list of candidates only. Now
I'm told that in lower basis, if we talk
about base 2 and base 3 and base 4, you
can show numbers that definitely go into
one of these loops or will never end up
in a palindrome and it turns out that base
10, not only at the base we use in day to
day life, but it's the smallest
interesting case for this. It's the
smallest base that we're not sure if we
can do it or not. In the smaller bases
apparently we have an answer.
Yeah, so 196 is the smallest candidate.
The next one is 295, the next one is 394,
and then it's 496 - so they're not irregular. It's
about every 100 or so. There's quite
a few of these, maybe some of them are Lychrel
numbers and some of them might just
finally become a palindrome at some
great distance away, we don't know. The
next one might be more obvious because
we started with 196, its palindrome
automatically is one of these candidates
as well; and so on and so on.
Yes we've checked 196 for billions of
iterations, the numbers get larger every
time you do it. The numbers are now
something like 600 million digits long.
Lychrel numbers- by the way that
name, Lychrel, let's write that down. That
came from one of the investigators in
this problem, it's actually a kind of
rough but not quite anagram of his
girlfriend's name which is Sheryl; oh
Lychrel strikes again. Why do we do this?
There's no application, we do
this for, you know, for fun, for the exercise,
for the mental exercise; for the practice
of it. Maybe it turns out that if there's
a proof the proof might use techniques
that are deeper then the problem looks
on its surface, and we can apply those
to serious problems. This problem
became kind of famous in the 1980s in
the community of hobbyist microcomputer
programmers. So they were using this to
test out their microcomputer programming
skills. And so they would make these
lists of these reverse and add
algorithms, they'd make these lists; and of
course all you need is when you want to
pass on the baton or pass on the torch
to the next person you just need the
last entry. All you do is just give them
the last entry and then they can carry
on the process. And that's what's
happened; since the 80s this process, this
baton, has been passed from person to
person - currently we are on these huge
numbers and still it hasn't been resolved yet.
We'd like to thank
audible.com for their support of this
Numberphile video. And with over 180,000
titles in their library
it is the best place to go for
audiobooks and other spoken materials.
Now I'm an audible customer myself, I
especially enjoy listening when I'm out
walking the dogs as I've spoken about
before, or embarking on a really long
drive across the country. Really makes
the whole drive a lot more enjoyable. Now
various Numberphile stars have books on
audible, I'm going to put some links to
them in the video description so you can
check them out. Or if you're after one
with a very number-y title you can read
the one I'm reading at the moment which
is 11.22.63, it's a really interesting
book by Stephen King about time travel
and also deals with the assassination of
John F Kennedy. It also weighs in at over
30 hours long unabridged so I'm
getting plenty of dog walks out of that
one. Now if you're interested in checking
out audible, and you really should, go to
audible.com/numberphile - then
they'll know you came from here and
that's good - and when you sign on you get
a free audiobook at the start, so you get
your first one for free you can see what
you think of the whole thing. It's a
great service, as I said I use it myself.
And importantly thanks to them for
supporting us here on Numberphile.