Re: Game with coins

Brian S wrote:
This is a really interesing variant. Consider the games {1,2,2} and
{2,1,2}. In ordinary nim these are the same game and the first player
has a win by removing the pile with one coin. But in this variant,
the middle pile cannot be touched, and these are two distinct games.
{1,2,2} is still a win for player 1, but with the one coin pile on the
inside, {2,1,2} is a win for player 2. No matter what player 1 does,
player 2 can force {1,1}, a win for him.

In general {x,y,x} with y>x is a win for player 2, {x,x,x} is a win
for player 1. {y,x,y} with y>x is trickier. If a player reduces
either outside pile to exactly x, he will lose, but otherwise looks
trickier to evaluate.

Take the game {3,2,3}, so far the only move which player 1 can make
and not lose is to reduce 3 to 1: {3,2,1}. Then player 2 wants to
avoid {1,2,1}, {2,2,1}, {0,2,1}, and {3,2,0}. But those are his
available moves, so he will lose.

I don't know how {4,2,4} might play out, the increase from 3 to 4
leaves some room between the sizes of the outside piles and the center
pile.

The game {4,2,4} is lost for the starting player, as is easy to see.

The game is basically a sequence of 2-pile Nim games, played one after
another (first with the outer two piles, then with the two piles 'just
inwards these', etc. (If the number of piles is odd, there remains only
one pile in the end, but this is equivalent to a game with two piles
containing 0 and n.)

Next you note that in a 2-pile Nim game, there is a strategy to take the
last coin yourself, and another one to force the other player to take
the last coin. So, in a sequence, you can manipulate who starts the next
2-pile-Nim-game, so to speak.

For example, in a game with piles p1, ..., p7, you first determine
whether the game with p2, ..., p6 is won by the starting player or by
the other, and when playing the p1,p7 Nim game, you manipulate the right
player into the starting position of the p2, ..., p6 game.

The winning conditions are easy to understand, but perhaps a bit tedious
to write down. But, as Gerry said, once you understand the 2-pile Nim
game, you know everything there is to know.

--
Cheers,
Herman Jurjus
.

Relevant Pages

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  • Re: Game with coins
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