We introduce an impartial combinatorial game on Steiner triple systems called Nofil. Players move alternately, choosing points of the triple system. If a player is forced to fill a block on their turn, they lose. We explore the play of Nofil on all Steiner triple systems up to order 15 and a sampling for orders 19, 21, and 25. We determine the optimal strategies by computing the nim-values for each game and its subgames. The game Nofil can be thought of in terms of play on a corresponding hypergraph. As game play progresses, the hypergraph shrinks and will eventually be equivalent to playing the game Node Kayles on an isomorphic graph. Node Kayles is well studied and understood. Motivated by this, we study which Node Kayles positions can be reached, i.e. embedded into a Steiner triple system. We prove necessary conditions and sufficient conditions for the existence of such graph embeddings and conclude that the complexity of determining the outcome of the game Nofil on Steiner triple systems is PSPACE-complete.

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组合游戏nofil在Steiner Triple Systems上播放

我们在Steiner三元系统上引入了一种称为Nofil的公正组合游戏。玩家交替移动，选择三元组系统的点。如果玩家在回合中被迫填补一个障碍，他们将输。我们探讨了Nofil在所有Steiner三重系统上（最高15阶）以及19、21和25阶采样的玩法。我们通过计算每个游戏及其子游戏的nim值来确定最佳策略。可以根据对应的超图上的游戏来考虑Nofil游戏。随着游戏的进行，超图会缩小，最终将等同于在同构图上玩Node Kayles游戏。Node Kayles受到了充分的研究和理解。以此为动力，我们研究了可以达到哪些Node Kayles位置，即嵌入到Steiner三重系统中。