Polymerization and Crystallization of Snowflake Molecules in Domineering

In this paper we present a combinatorial game-theoretic analysis of special Domineering positions. In particular we investigate complex positions that are aggregates of simpler fragments, linked via bridging squares. We build on two theorems that exploit the characteristic of an aggregate of two fragments having as game-theoretic value the sum of the values of the fragments. We extend these theorems to deal with the case of multiple-connected networks with an arbitrary number of fragments, possibly including cycles. As an application, we introduce several interesting Domineering positions, dubbed Snowflake molecules. A Snowflake molecule consists of a core subfragment with some "hooks" attached. We investigate four different cores, all with a different type of value: 1) the CrystalBall core, with number value 0; 2) the BigI core, with number value 1; 3) the DoubleT core, with nimber value *2; and 4) the GearWheel core, with switch value +/- 1*. We then show how from each such Snowflake molecule chains of Snowflakes can be built (a kind of polymerization) with known values, including flat networks of Snowflakes (crystallization).