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Topic: A Sprouts Hypothesis re: Applegate, Jacobson, and Sleator
Replies: 0

 Danny Purvis Posts: 176 Registered: 12/6/04
A Sprouts Hypothesis re: Applegate, Jacobson, and Sleator
Posted: Dec 13, 2000 11:52 AM

I have a loose, perhaps partial explanation for the pattern noticed by
Applegate, Jacobson, and Sleator. They used a computer to analyze the
game exhaustively through n=11 and noticed that the first player wins
if and only if the remainder of n divided by 6 is 0, 1, or 2.

Let me restate this pattern recursively and loosely: there is a
tendency for the winner of n = x to be the loser of n = x + 3. Now,
in a game of n = x + 3 spots, either of the players is likely to be
able to separate 2 untouched spots from a very near equivalent of n =
x spots in a single move. (Since this single move will consume 1/3
of an untouched spot.) The winner of n = x spots will likely be the
winner of the very near equivalent of n = x spots but will then be the
loser of a game consisting of this very near equivalent of n = x spots
plus an extra move. Therefore the winner of n = x spots likely will
be forced to move to 2 untouched spots so separated. But any move to
a formation of 2 untouched spots, or to a formation of 2 untouched
spots + 1/3 untouched spot accessible to other spots not accessible to
the 2, changes the Sprague-Grundy number of that formation from 0 to 2
or greater, placing the initial mover to that formation at a distinct
disadvantage in the total game situation.