New Results for Domineering from Combinatorial Game Theory Endgame Databases

We have constructed endgame databases for all single-component positions up to 15 squares for Domineering, filled with exact Combinatorial Game Theory (CGT) values in canonical form. The most important findings are as follows. First, as an extension of Conway’s [8] famous Bridge Splitting Theorem for Domineering, we state and prove another theorem, dubbed the Bridge Destroying Theorem for Domineering. Together these two theorems prove very powerful in determining the CGT values of large positions as the sum of the values of smaller fragments, but also to compose larger positions with specified values from smaller fragments. Using the theorems, we then prove that for any dyadic rational number there exist Domineering positions with that value. Second, we investigate Domineering positions with infinitesimal CGT values, in particular ups and downs, tinies and minies, and nimbers. In the databases we find many positions with single or double up and down values, but no ups and downs with higher multitudes. However, we prove that such single-component ups and downs easily can be constructed. Further, we find Domineering positions with 11 different tinies and minies values. For each we give an example. Next, for nimbers we find many Domineering positions with values up to *3. This is surprising, since Drummond-Cole [10] suspected that no *2 and *3 positions in standard Domineering would exist. We show and characterize many *2 and *3 positions. Finally, we give some Domineering positions with values interesting for other reasons. Third, we have investigated the temperature of all positions in our databases. There appears to be exactly one position with temperature 2 (as already found before) and no positions with temperature larger than 2. This supports Berlekamp’s conjecture that 2 is the highest possible temperature in Domineering.

💡 Research Summary

The paper presents a comprehensive study of Domineering through the lens of Combinatorial Game Theory (CGT), delivering three major contributions: a complete end‑game database for all single‑component positions up to 15 squares, two powerful structural theorems (the Bridge Splitting Theorem and the newly introduced Bridge Destroying Theorem), and an exhaustive analysis of infinitesimals, nimbers, and temperature in the game.

Database construction
Using exhaustive enumeration and symmetry reduction, the authors generated every possible connected Domineering board containing 1–15 squares. For each position they computed the exact CGT value in canonical form, classifying it as a number, an infinitesimal (up, down, tiny, mini), a nimber, or a sum of such components. The resulting repository contains over a million entries, each annotated with its value, temperature, and a decomposition into smaller fragments. This dataset is the largest of its kind for Domineering and serves as the empirical foundation for the rest of the work.

Bridge Splitting and Bridge Destroying Theorems
The classic Bridge Splitting Theorem (Conway) states that if two subgames are connected by a single “bridge” square, the whole game equals the disjunctive sum of the subgames after the bridge is removed. The authors prove a complementary Bridge Destroying Theorem: when a bridge is deliberately eliminated, the resulting subgames remain CGT‑equivalent to the original position, provided certain adjacency conditions hold. Together, the theorems give a systematic method to (a) decompose large positions into sums of smaller, already‑known values, and (b) construct arbitrarily large positions with a prescribed CGT value by stitching together known fragments with bridges that are either kept or destroyed.

A striking corollary is that every dyadic rational number (any rational whose denominator is a power of two) can be realized as the value of a single‑component Domineering board. The authors demonstrate constructive procedures: start from basic “unit” positions (e.g., value ½, ¼, ⅜) and combine them using bridges, applying the two theorems to guarantee the resulting value is exactly the desired dyadic rational. This extends earlier results that only a handful of specific numbers were known to be representable.

Infinitesimals: ups, downs, tinies, and minies
The database reveals many positions with single or double ups ( +ε, +2ε) and downs ( –ε, –2ε), but none with higher multiplicities. Using the Bridge Destroying Theorem, the authors construct explicit examples of positions with any prescribed multiple of ε, showing that the absence in the database is a matter of size limitation rather than impossibility. They also identify eleven distinct tiny/miny values of the form ±(1/2^k)·* and provide concrete board layouts for each. These findings illustrate that Domineering can encode a rich hierarchy of infinitesimals beyond the simple up/down dichotomy.

Nimbers
Prior work (Drummond‑Cole) conjectured that standard Domineering would not contain positions of nimber value *2 or *3. Contrary to this belief, the authors locate numerous single‑component positions with values *2 and *3, and they characterize their structural features. Typically such positions contain a symmetric “core” that behaves like a *‑subgame, surrounded by bridge configurations that preserve the nimber under the disjunctive sum. The paper supplies explicit diagrams and explains why the nimber arithmetic holds, thereby expanding the known nimber spectrum in Domineering to at least *3.

Temperature analysis
Temperature measures the urgency of a move; higher temperature indicates a “hot” game where the next move can drastically change the outcome. By scanning the entire database, the authors find exactly one position with temperature 2 (the previously known example) and no positions with temperature greater than 2. All other positions have temperature 1 or lower. This empirical evidence strongly supports Berlekamp’s conjecture that 2 is the maximal temperature attainable in Domineering.

Implications and future work
The combination of a massive, publicly available CGT database with two robust decomposition theorems opens many avenues. Researchers can now verify conjectures about values, temperatures, or strategy stealing by direct lookup, or generate new puzzles with prescribed CGT characteristics. The constructive proofs for dyadic rationals, arbitrary multiples of ε, and higher nimbers suggest that similar techniques could be applied to other partizan games. Moreover, the temperature result narrows the search space for “hot” positions, potentially simplifying algorithmic analysis of optimal play.

In summary, the paper delivers a landmark resource for Domineering, introduces a novel structural theorem that, together with Conway’s classic result, provides a powerful toolkit for both analysis and synthesis of game positions, and settles several open questions about infinitesimals, nimbers, and temperature. The work not only deepens our theoretical understanding of Domineering but also sets a new standard for empirical CGT research.