We introduce an impartial combinatorial game on Steiner triple systems called Next One to Fill Is the Loser (Nofil). Players move alternately, choosing points of the triple system. If a player is forced to fill a block on their turn, they lose. By computing nim-values, we determine optimal strategies for Nofil on all Steiner triple systems up to order 15 and a sampling for orders 19, 21 and 25. The game Nofil can be thought of in terms of play on a corresponding hypergraph which will become a graph during play. At that point Nofil is equivalent to playing the game Node Kayles on the graph. We prove necessary conditions and sufficient conditions for a graph to reached playing Nofil. We conclude that the complexity of determining the outcome of the game Nofil on Steiner triple systems is PSPACE-complete for randomized reductions.

中文翻译：

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Nofil 在 Steiner 三重系统上玩的组合游戏

我们引进的Steiner三连系统的公正组合游戏叫 ñ分机Ø NE到˚F生病我S上的大号oser（N的IL）。玩家交替移动，选择三重系统的点。如果玩家在轮到他们时被迫填满一个方块，他们就输了。通过计算 nim 值，我们在所有 Steiner 三元系统上确定了Nofil 的最佳策略，最高阶数为 15，并为阶数为 19、21 和 25 的采样。可以将Nofil游戏视为在相应的超图上进行游戏，该超图将成为播放过程中的图表。到时候Nofil就相当于玩游戏Node Kayles在图表上。我们证明了一个图达到玩Nofil 的必要条件和充分条件。我们得出的结论是，在 Steiner 三重系统上确定游戏Nofil的结果的复杂性对于随机归约而言是 PSPACE 完全的。