Newsgroups: sprouts-theory
From: Dan Hoey <Hoey@AIC.NRL.Navy.Mil>
Date: Wed, 31 Dec 2008 15:21:21 -0500
Subject: A lousy periodicity theorem

In _Sprouts Game on Compact Surfaces_, Julien Lemoine and Simon
Viennot proved that beyond a certain number (depending on the region),
increasing the genus of an orientable region cannot affect a sprouts
game, nor can increasing the genus of a non-orientable region by two
affect the game.

This reminds me of an observation I made that is provable in the same
sort of way (unless I've overlooked something).  A "louse" is a boundary
consisting of a single degree-2 point that does not appear anywhere
else; "2." in Glop notation.  We can change a sprouts game by adding
a louse to a region.  The theorem is that beyond a certain number
(depending on the region), adding two lice to a region does not affect
the game.

I'm pretty sure that the result can be strengthened to show that (in the
presence of enough lice) adding one louse to a region will change the
Sprague-Grundy value of the position by the nimber *1.  This should
hold in both normal and misère play.

Dan Hoey