Re: Game with coins

Herman Jurjus wrote:
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...

No, this is not true, sorry.
For example, in the {4, 2, 5} game, if the first player removes all 4 coins from the first pile, then the second player is not forced to take coins away from the pile with 5 coins, but can now also take coins from the pile with 2 coins.

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Cheers,
Herman Jurjus
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