Hello and welcome to Seoul for the solution for my chain number problem
or number chain problem... Whatever!
So, in the previous video
the problem video
there was
it was a big hint
The hint was: the solution is unique
I asked you to find the right one
the one
that has
the longest chain.
and this is what we're going to use to find THE solution
So, if you remember
I gave you as an example 25
Imagine that 25
is THE solution.
25 makes
if you remember
a chain of 3
2 × 5 = 10; 1 × 0 = 0 That makes a 3-step chain.
So, imagine this is the solution
if this is the solution,
there is a another solutions, 52 because 25 and 52,
if you multiply them
you get the same result
and I told you the answer is unique.
So obviously
25
25 or 52,
are not the solution
so which numbers among here
which numbers among
between 1 and 99
Well, you have all those
one-digit because when
you reverse them it's still the same number so if one of them is the solution, it's unique.
obviously it's not the solution because we know what it is a 1-step chain
because we don't multiply and we don't do anything with those so,
those are not the solution
but there are others like 10
because the equivalent of 10, 01, doesn't exist.
the one that exists is 1 and it's a stop
So this one has no equivalent when you reverse it.
so if this one is the solution it's unique.
unfortunately all those numbers that end by 0
make a very very short chain because 0 × 1 = 0 and
over, it's always a 2-step chain
and we know
because we tried already with 25 that 25 is already longer
it's a three-step chain
so
so so so
which numbers can work?
There are few of them remaining actually
They are 11
22
33
44
55
66
77
88
and 99. Those can work
Only those
Because when you reverse them of course i'ts the same numbers so if
one of them is the solution
then it's unique
and that's exactly what we're looking for
So, we just have to compute those
1 × 1 = 1,
it's over
4, it's over
9 it's over
this is a little bit more interesting
but still it's a 3-step chain
and it's over.
This one
it's a 4-step
we recognize our 25
which is our
3-step chain
And this is the 55. 4 steps... Getting good, but
this is also
like this
like this
and there's another 4-step so
since the solution is unique there's great chances that those are not
the solution and the solution is bigger. Bigger? Let's try bigger actually
Well, bigger is a little bit
not so good
bigger is not better, not always better.
This one, maybe?
4-step chain, another one.
One, two, three 4-step chains, for a unique solution it's a little bit of a problem
But if we try this one,
of course I kept it for the end,
everybody understands why,
it makes 49
36
18
and finally... 8
And you get a 1, 2, 3, 4,
and 5-step chain
which is the longest
among the numbers from 1 to
99
so 77
is the answer
because the answer
we said it, is unique
so we could find it
easily.
I hope you enjoyed this problem
and see you next time for the next one.
Bye, bye!
