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Today, we get to do ski jumps without
jumps, and ski jumps with jumps.

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Ski jumps without jumps, in some ways
isn't very interesting.

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Here's ski jumps without jumps.
We have two lines here.

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Left, this is left's piece, left moves
right, one space every turn.

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Right moves left one space every turn.
Now what happens?

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If left goes first, well actually, we can
do this with a board here.

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Here's left and here's right.
So, left moves, this is not very

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interesting because there's no choice for
anything.

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Right moves, left moves, right moves, left
moves, right moves, left moves, right

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moves.
Left doesn't have a new move.

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You can't jump off, off, off, off the
side.

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Right has now moved, so left though, was
the first person that had no moves.

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Let's try that again.
Let's do that real quick.

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Beep, beep, beep, beep, beep, beep, beep,
beep.

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Left turn, left has no moves.
So, if left goes first, left loses.

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Now, in exactly the same way, if right
goes first, right, right is the dime down

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here.
Right goes first, left moves right.

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Left, right, left, right.
Left, right has no moves here.

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Right loses.
So, whoever moves first loses.

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This says that this game is zero.
So, ski jumps without jumps is not very

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interesting.
So, let's look at ski jumps with jumps.

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Alright.
What's the, the deal here?

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Here's the deal.
Left starts over here, right starts over

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here.
Now the additional rule and the way the

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board is set up, only left can do this.
But you could, you could move the board

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like this and only right could do it.
So, so what happens if, if the pieces ever

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like this with left on top of right, and
left instead of moving right can jump over

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Left down to here.
Now, that doesn't remove the, the, you

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jump over your opponent, your opponent
doesn't disappear, it just stays there,

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but this allows you an additional option
that the penny here can jump down to here

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or move to the right whenever they're
lined up exactly like this.

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Now let's play it.
Again for many of the moves, there aren't

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very many choices.
So, let's look at left first.

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Left moves there, right moves here.
Left moves here, right moves here.

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Now what's the best left move?
The best left move is to jump at this

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point.
Right moves left.

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Left moves right.
Right moves left.

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Left moves right.
Now right has no move, right loses.

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So, if left goes first, left wins.
Let's do the other one.

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If right goes first, there is, there's
never any chance to jump.

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And now it's right's turn.
Right has no move, so right loses.

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So left wins going first, and left wins
going second.

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So in the ski jumps with jumps, left
always wins, that says is positive.

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Now with a little bit of work you can
show, and try this yourself.

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That, that if we play ski jumps with jumps
and we allow right an extra move, here's,

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and this is minus 1.
It allows, right, one extra move.

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Then, whoever moves first loses.
If left moves first, left loses.

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If right moves first, right loses.
Basically, what happens is the same thing

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as before.
But you play in here.

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And, and, when, when right gets stuck at
the end, right has an additional move over

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here in this game.
So that, that's how that works out.

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So, so what we have for ski jumps with
jumps, is there's this gain, this

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somewhere in between 0, and 1.
And the interesting problem, which you

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might want to think about, is where is it
in there?

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Can we say it's, it's closer to 0, is it
closer to 1?

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What's going on?
We'll eventually answer that.