> zovich - the nim building block; null - a zovich which will yield an > even number of moves; inverter - a zovich which will yield an odd > number of moves; switch - a zovich which the mover can convert to her > choice of null or inverter; trap - a zovich which will become a switch > after one move; biosphere - a hermetically sealed Sprouts life zone; > countable - a Sprouts dot with 0 or 2 lines attached. > > A Sprouts zovich with an even (odd) number of countables which will > yield an odd (even) number of biospheres is an inverter. An inverter > plus a trap wins. > > These concepts are very useful because of a ubiquitous trap, the > biosphere consisting of two unattached dots.
You've discovered one-infinieth of the Sprague-Grundy theory! This applies when games decomose as sums of smaller ones (for Sprouts these are your "biospheres", and assigns to each summand a number called it's "nim-value" or "Sprague-Grundy number". The definition is that a position has nim-value n just if you can move from it to positions of all nim-values from 0 to n-1, but not to n. A game that necessarily lasts for a fixed number of moves has nim-value 0 or 1 according as this number is even or odd (your "null"s and "inverter"s).
However, there are other positions that have nim-values 0 and 1, as well as those that have the larger nim-values 2,3,4,... . You can find some tables of nim-values in Sprouts in "Winning Ways"