Today, we get to do ski jumps without
jumps, and ski jumps with jumps.
Ski jumps without jumps, in some ways
isn't very interesting.
Here's ski jumps without jumps.
We have two lines here.
Left, this is left's piece, left moves
right, one space every turn.
Right moves left one space every turn.
Now what happens?
If left goes first, well actually, we can
do this with a board here.
Here's left and here's right.
So, left moves, this is not very
interesting because there's no choice for
anything.
Right moves, left moves, right moves, left
moves, right moves, left moves, right
moves.
Left doesn't have a new move.
You can't jump off, off, off, off the
side.
Right has now moved, so left though, was
the first person that had no moves.
Let's try that again.
Let's do that real quick.
Beep, beep, beep, beep, beep, beep, beep,
beep.
Left turn, left has no moves.
So, if left goes first, left loses.
Now, in exactly the same way, if right
goes first, right, right is the dime down
here.
Right goes first, left moves right.
Left, right, left, right.
Left, right has no moves here.
Right loses.
So, whoever moves first loses.
This says that this game is zero.
So, ski jumps without jumps is not very
interesting.
So, let's look at ski jumps with jumps.
Alright.
What's the, the deal here?
Here's the deal.
Left starts over here, right starts over
here.
Now the additional rule and the way the
board is set up, only left can do this.
But you could, you could move the board
like this and only right could do it.
So, so what happens if, if the pieces ever
like this with left on top of right, and
left instead of moving right can jump over
Left down to here.
Now, that doesn't remove the, the, you
jump over your opponent, your opponent
doesn't disappear, it just stays there,
but this allows you an additional option
that the penny here can jump down to here
or move to the right whenever they're
lined up exactly like this.
Now let's play it.
Again for many of the moves, there aren't
very many choices.
So, let's look at left first.
Left moves there, right moves here.
Left moves here, right moves here.
Now what's the best left move?
The best left move is to jump at this
point.
Right moves left.
Left moves right.
Right moves left.
Left moves right.
Now right has no move, right loses.
So, if left goes first, left wins.
Let's do the other one.
If right goes first, there is, there's
never any chance to jump.
And now it's right's turn.
Right has no move, so right loses.
So left wins going first, and left wins
going second.
So in the ski jumps with jumps, left
always wins, that says is positive.
Now with a little bit of work you can
show, and try this yourself.
That, that if we play ski jumps with jumps
and we allow right an extra move, here's,
and this is minus 1.
It allows, right, one extra move.
Then, whoever moves first loses.
If left moves first, left loses.
If right moves first, right loses.
Basically, what happens is the same thing
as before.
But you play in here.
And, and, when, when right gets stuck at
the end, right has an additional move over
here in this game.
So that, that's how that works out.
So, so what we have for ski jumps with
jumps, is there's this gain, this
somewhere in between 0, and 1.
And the interesting problem, which you
might want to think about, is where is it
in there?
Can we say it's, it's closer to 0, is it
closer to 1?
What's going on?
We'll eventually answer that.