this is a game that frogs play whenever
they are not being witnessed by humans
and the way that they play it you can
imagine that these are lily pads, and
there's five frogs. five frogs that want
to have a party on the same lily pad. One
frog jumps one space. Two frogs jump two
spaces. one frog jumps one space. got four
frogs on here so 4 frogs jump
four spaces these. These frogs are happy
because they are celebrating a party on
their own lily pad. now it's your job you
have to figure out: is this possible?
the day after whenever a new frog comes
to town. Is this possible? one extra rule:
you cannot jump frogs on to an empty
lily pad. so I could jump one frog there,
i can jump one frog here I could jump
one frog here and two frogs I could jump
here. 4 frogs must jump 4 spaces.
1,2,3,4, Oh, this is an empty lily pad so
you can't do that.
so this is an example of a failure. Can
we manage to do this with six frogs and
six lily pads? Can we manage to do this with
N frogs and N lily pads? so this is a
puzzle that doesn't have an obvious
answer before we solve six I want you to
imagine you've only got 2 lily pads
and 2 frogs. well that's easy to solve
so now we're going to use that i'm going
to split this into two and first of all
going to solve for the left and i'm going to
solve the right in exactly the same way.
this looks suspiciously similar to
having two frogs and two lily pads. here
we have three frogs and three frogs
separated by three. so this can be solved
in exactly the same way as 2 frogs can.
be solved in 2 lily pads. so now that
we've solved six, is it easy to solve 36?
of course it is because we can design 36
we can solve for this six in the same
way you can solve for this six, this six,
this six, this six. we've solved for all
the sixes. we now have six stacks of six
all separated by six. then that's just kind of
like a giant six. doesn't just give us a
solution for even numbers. it also gives
the solutions for numbers like 9. 3 x 3
because you could imagine with nine, with
nine we solve like this... okay?
now you can see we've really got a big
version of just the three. so we do the
same thing. we solved it. can we solve for
prime numbers? can we solve for prime numbers?
so the answer is yes and let me show you
this is another ...you should, you should
take time and try to solve for prime
numbers but let's set let's extend this
we'll go to 11 here. i'm going to show you
how to do it and you'll immediately see
that you could do all integers two three
four five six seven eight nine ten
eleven. Solved!
that will work for everything. To make
problem a little bit more interesting
now that we solve for every integer,
now what we're going to do is we're
going to say well one of these frogs is
actually the Queen frog. and the Queen
frog must always be on the top of any
pile, so you can think:
can you solve for that? Now for example
you could go to here but then you
couldn't move this on the top of the
Queen you could move to here. this then
becomes a challenge is this always
possible and visit isn't always possible
no matter where the queen is? If the Queen
starts over here, is this possible?
so you should take a moment and you
should figure out is this possible or
impossible? what I'm going to try to do
is I'm going to use this same technique
of back and forth and back and forth and
back and forth and and i'm going to end
up here. right, so let me do that first
so what do i need to do now? i need to
somehow end up on this place here and
that that is doable by the same
technique i go back and forth i'm going
to start here and I'll go back and forth.
and just by choosing correctly, i can
make sure that this stack is going to
end up on top so this is also solvable
for all integers. A little bit less obvious.
now let's remove the Queen and give
instead lazy told you where would you
like the lady told to be
in third position that ok so the lazy
told doesn't move so we have got to have
the party on this lily pad is this
possible
so take a second try to figure out it's
just always going to be possible or not
and the answer is yes this is always
going to be possible in the same with
the same system so here I'm going to
choose this one I'm going to move over
here and go on top here i'm going to
choose something in the middle and you
just have to figure out what I wanted to
jump this way and yes so I jumped this
way and we've solved it
this is solvable for all integers you
might want to experiment with one lazy
toad and one Queen that have to be on
top and everything else wrong and see if
that is possible
sometimes it's possible sometimes you
know now i'm going to share a problem
that i don't know the answer to
and this one we have the frogs and
unfortunately two of the lilypad are
waterlogged the end one second from the
end on either side so know frogs can go
on here i do not know if this is ever
solvable for any number of lily pads
I've never been able to solve a single
one of these so that's that's my open
problem here will show your failure so I
need to get this frog somehow because it
can't move here so i need to go over and
grab it so let me grab it by this
technique I'll go to that gives three
now I come over here
that's four 1234 I can come over and
rescue this one that's five one two
three four and I UCI i'm stuck here I've
got this one by itself not what can I do
well I can at least get this one but
then that's 12345 I can't move this back
how that goes under the waterlogged
lilypad I can't move this stack 123 and
I can't move this sec so I'm stuck here
so I've never been able to solve any of
these for any length but it's not
obvious at all of this can't be solved
for some length of lily pads just the 11
away from the end on each side being
waterlogged not low enough so what we do
we do we could split this into an ache
here and 12 we could do that and so now
our score is 12-4 is equal to 8