Biased Weak Polyform Achievement Games

In a biased weak $(a,b)$ polyform achievement game, the maker and the breaker alternately mark $a,b$ previously unmarked cells on an infinite board, respectively. The maker’s goal is to mark a set of cells congruent to a polyform. The breaker tries to prevent the maker from achieving this goal. A winning maker strategy for the $(a,b)$ game can be built from winning strategies for games involving fewer marks for the maker and the breaker. A new type of breaker strategy called the priority strategy is introduced. The winners are determined for all $(a,b)$ pairs for polyiamonds and polyominoes up to size four.

💡 Research Summary

The paper studies biased weak achievement games on infinite regular tilings, where two players alternately mark previously unmarked cells: the Maker marks a cells per turn, the Breaker marks b cells. The Maker’s objective is to occupy a set of cells congruent to a given polyform (polyomino, polyiamond, polyhex, or polycube), while the Breaker tries to prevent this. The authors first formalize the notion of a “proof sequence,” a chain of situations (C, N) that describes how the Maker can progress toward the goal despite the Breaker’s moves. They recall the classic pairing (or paving) strategy for the Breaker, which works well for (1, b) games but does not extend naturally to games with a ≥ 2.

A central contribution is the introduction of the “priority strategy” for the Breaker. In this approach each board cell is assigned a priority level; on each turn the Breaker occupies the highest‑priority unoccupied cells among those that could help the Maker. The paper illustrates the strategy with concrete (2, 2) and (2, 5) examples, showing how the Breaker can systematically block the Maker’s critical cells. Moreover, a more sophisticated “history‑dependent priority strategy” is defined, allowing the Breaker to adapt priorities based on the sequence of previous moves. The authors provide an algorithm that verifies the correctness of such strategies, enabling computer‑assisted proofs.

On the Maker side, the authors prove a decomposition theorem (Theorem 3.1). Any (a, b) game can be expressed as a collection of simpler subgames (a₁ ⊕ a, b₁), …, (a_s ⊕ a, b_s) played on disjoint finite subboards. By tracking a progress vector p and a supply vector q, they construct a stage‑diagram that quantifies how many turns are needed in each stage and how many subboards must remain “alive.” This framework yields Corollary 3.3: if a polyform is a (1 ⊕ a, ⌊b·a⌋)‑winner, then it is also an (a, b)‑winner. The decomposition technique thus reduces many biased games to a finite set of base cases.

The authors then define the “threshold sequence” for a polyform: the list of (a, b) pairs where the Maker’s status changes from loser to winner as a increases. Using the new Breaker strategies and the decomposition theorem, they compute the full threshold sequences for every polyomino and polyiamond of size up to four cells. For instance, the triangular polyiamond T₃,₁ is a (2, 5)‑winner, while various L‑shaped polyominoes become winners at (3, 5) but remain losers at (3, 4). The results are presented in tables and diagrams, and the paper discusses the computational effort required to verify each case.

Finally, the paper lists several open problems, such as extending the analysis to larger polyforms, to three‑dimensional polycubes, and to other bias regimes where a and b grow together. It also suggests investigating optimality of the priority strategy and possible refinements of the stage‑diagram method. Overall, the work advances the theory of biased achievement games by providing a systematic Maker decomposition, a novel Breaker priority framework, and a complete classification for small polyforms.