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Path achievement games

Starting with the empty graph on n vertices, two players alternately
add edges until the graph contains a p-path. The last player to
move wins. Assuming both players play optimally, the winner will depend
only on p and n. We analyse the game for p≤6 and arbitrary
n, determining the winner and providing a winning strategy.
The results illustrate how deceptive computing the results of small
cases can be.