July 11, 2009
Here is a sprouts-related game. Letís call it, just picking a name at random, g4g.
The elements of this game are entities and transformation laws.
The entities are designated as: v0, q1, s0, t1, d0, and d1. The transformation laws are as follows:
v0 => q1 (meaning that a v0 can be transformed to a q1)
v0 => s0
q1 => d1
q1 => t1
q1 => s0
t1 => s0
s0 => d0
s0 => d1
The numerical portions of the entity designations refer to a type of parity.
For a group of entities, these portions are to be nim summed to get total parity.
For instance, the total parity of the group v0, q1, and t1 is 0 while the total parity of the group q1, t1, s1 is 1.
This is two player game beginning with some number of entities all of type v0.
As in sprouts, the players alternate moves. A move consists of transforming zero entities (where this is legal) or transforming one entity.
A player to move is allowed to transform an entity if a transformation is available.
A player to move is required to transform an entity if a transformation is available and one of the following conditions apply:
(1) He is playing Left and parity is even; (2) He is playing Right and Parity is odd.
Play ends when no further transformations are possible. Left wins if the final position has parity of 0. Right wins if the final position has parity 1.
Here is a sample game:
v0, v0, v0 with Left to move
q1, v0, v0 with Right to move
q1, q1, v0 with Left to move
q1, q1, q1 with Right to move
d1, q1, q1 with Left to move
d1, t1, q1 with Right to move
d1, t1, d1 with Left to move
d1, t1, d1 with Right to move
d1, s0, d1 with Left to move
d1, d0, d1
Left won this sample game, but I believe that with perfect play Right should win every possible starting position.
In a future article I will try to show that g4g maps to a complex of sprouts positions.
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