Article 278670 of rec.puzzles
From: Dan Hoey <haoyuep@aol.com>
Newsgroups: rec.puzzles
Subject: Re: cotpi 17 - Pawns
Date: Tue, 16 Aug 2011 23:21:51 -0400
Organization: Aioe.org NNTP Server

cotpi wrote:
> Jeffrey Turner wrote:
>> cotpi wrote:
>>> There are 64 pawns placed on a chessboard. Each square contains a pawn.
>>> Two players are on the same side of the chessboard. Each player takes
>>> turns and removes a pawn along with all pawns above it and to its right.
>>> The player who picks the pawn at the bottom-left corner loses. Who can
>>> definitely win the game?

>> If the lower left corner is (0,0), and numbers go up to the right and
>> up, then if the first player picks the pawn at (1,1) they will win.

> If the first player picks the pawn at (1, 1) and all pawns at (x, y)
> such that x > 1 and y > 1, what forces the second player to pick the
> pawn at (0, 0)?

I think Jeffrey Turner is assuming this is the game of Chomp, where a
move at (1,1) removes all pawns at (x,y) where x >= 1 and y >= 1.  In
that case the only pawns left are at (x,0) and (0,y), and the first
player wins by mirroring the second player's strategy.

> The second player is free to choose a pawn at (7, 7) and the game
> would continue for a while. Can you show that the second player
> would be forced to pick the pawn at (0, 0) in the end?

No, the first player already removed the pawn at (7, 7), even in the
situation you described, since 7 > 1.  I think that in the game you
describe, where taking the pawn at (x0, y0) also removes pawns at (x, y)
where x>x0 and y>y0, the first player can continue by responding to the
second player's move (xn, yn) with a move at (yn, xn).  If that works,
the first player still wins.

Dan