"Jon Perry" <perry@globalnet.co.uk> wrote:

Neat. What do 1 and 2 imply?

For the purposes of a single heap game, 0 implies 2nd player win, anything else implies first player win. The different values 1 and 2 are useful if you wish to sum this game with other games (or copies of itself). I won't go into the details of what that means. A copy of Winning Ways I'm sure will explain that in great detail. I'll give this short technical description. A postion's value is zero if all the moves from that position are zero. The value is one if the current player can move to zero but not to one. Two if the current player can move to zero or one, but not to two.

If the increase range is generalised to l<=i<=r (l=1, r=2 is my original example - generally l>=0), how can you predict which player will win?

That looks easy to model. I can't say what the answer will be until I model it. I expect similar results, but I am concerned that the decision tree can get very complicated as r-l increases.

For the multiple heaps version, if a player can only remove from 1 heap at

a

time, which the additional play that if a player can exhaust a heap this signifies the end of their turn (i.e. they can pre-empt having to take k buttons), there are 2 options - the next player only need take i more buttons than the previous player took, or the pre-empting player can state how many they took and the next player must take more than this.

I don't know how to use the theory for the multiple heaps version other than to consider each case individually. I suspect that for more than 3 heaps it can get very complicated. Additionally, that rule that allows a player to state how many they "took" when emptying a heap can lead to a large number of choices for that player.