OAI.SnakyConditional.snaky_winning_strategy_with_legal_states

The theorem states that there is a policy σ for a maker-breaker style game on the integer grid ℤ×ℤ, where a policy maps the number of remaining rounds, the maker's current finite set and the breaker's current finite set to a cell, such that σ always proposes a cell lying in neither set. Moreover, for every breaker reply sequence β from ℕ to cells that is legal before the final round, played for N=35 rounds from empty initial sets, several things hold. Here round k (counting from 0) has the maker add σ(35−k, M, B) to its set M and the breaker add β(k) to its set B, and legality means that for every k with k+1<35, β(k) is neither the maker's newly chosen cell, nor already in M, nor already in B. Under this assumption the maker's final set after 35 rounds contains a copy of the six-cell shape {(0,0),(1,0),(2,0),(3,0),(3,1),(4,1)} under one of the eight rotations and reflections of the square lattice, followed by a translation; the final maker set has exactly 35 cells; it is disjoint from the breaker's set after 34 rounds; and for every k<35 the maker and breaker sets after k rounds are disjoint and the breaker set has exactly k cells.