  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.
