> 1) White moves first.
> 2) The game is being played according to the usual chess rules, but
> if the white pawn reaches the Nth horizontal, the game immediately
> stops.
> 3) How could Black win? There are 5 actual possiblities
>    a) Beat the white pawn with the King
>    b) Block the movement of the white pawn with the King
>    c) Block the movement of the white pawn with the Pawn
>    d) Promote the black pawn to the Queen (or something else) and
> then beat the white pawn with this new piece
>    e) Promote the black pawn to the Queen (or something else) and
> then block the movement of the white pawn with this new piece
>
>

First, we consider that the site of white pawn is (x1,y1), (x2,y2)
for black pawn, (x3,y3) for Black King. We can consider the following
five lines:

>    a) Beat the white pawn with the King
We can count the least steps of Black King to get to y1+1 or y1-1,
and it could only go left-up or right-up. (I think it's the optimum)

>    b) Block the movement of the white pawn with the King
I think we can ignore it or put it into state a).

>    c) Block the movement of the white pawn with the Pawn
I think if and only if (x2=x1+1 or x2=x1-1) and (y2>=x2+2), then the
black pawn can block the movement of the white pawn and beat it...

>    d) Promote the black pawn to the Queen (or something else) and
> then beat the white pawn with this new piece
first i think because n>5, so we only can promote it to the Queen.
Then the best way to beat the white pawn is to move left to y1. If
and only if x1<=n-2, the Queen can beat it.

>    e) Promote the black pawn to the Queen (or something else) and
> then block the movement of the white pawn with this new piece
I think we can ignore it!!!

Only some ideas, I haven't programme yet. Because of the time...

> 1) White moves first.
> 2) The game is being played according to the usual chess rules, but
> if the white pawn reaches the Nth horizontal, the game immediately
> stops.
> 3) How could Black win? There are 5 actual possiblities
>    a) Beat the white pawn with the King
>    b) Block the movement of the white pawn with the King
>    c) Block the movement of the white pawn with the Pawn
>    d) Promote the black pawn to the Queen (or something else) and
> then beat the white pawn with this new piece
>    e) Promote the black pawn to the Queen (or something else) and
> then block the movement of the white pawn with this new piece
>
>
And what about beating with the pawn?