A lousy periodicity theorem

[Dan Hoey]

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

[Dan Hoey]

I wrote:
> 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 regret having to withdraw this theorem. Given new observations in
the
"Fwd: Sprouts equivalences" thread, I see problems in my proposed
method
of proof that render this claim quite dubious.

Dan