The combinatorial game N ofil played on Steiner triple systems

We introduce an impartial combinatorial game on Steiner triple systems called N ext O ne to F ill I s the L oser (N ofil ) . 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 N ofil on all Steiner triple systems up to order 15 and a sampling for orders 19, 21 and 25. The game N ofil can be thought of in terms of play on a corresponding hypergraph which will become a graph during play. At that point N ofil is equivalent to playing the game N ode K ayles on the graph. We prove necessary conditions and sufficient conditions for a graph to reached playing N ofil . We conclude that the complexity of determining the outcome of the game N ofil on Steiner triple systems is PSPACE‐complete for randomized reductions.