Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.



Re: the game of Sprouts
Posted:
Jan 19, 1998 5:49 PM


On 19 Jan 1998, Danny Purvis wrote:
> 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 oneinfinieth of the SpragueGrundy 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 "nimvalue" or "SpragueGrundy number". The definition is that a position has nimvalue n just if you can move from it to positions of all nimvalues from 0 to n1, but not to n. A game that necessarily lasts for a fixed number of moves has nimvalue 0 or 1 according as this number is even or odd (your "null"s and "inverter"s). However, there are other positions that have nimvalues 0 and 1, as well as those that have the larger nimvalues 2,3,4,... . You can find some tables of nimvalues in Sprouts in "Winning Ways"
John Conway



