  We look at the unbiased Maker-Breaker Hamiltonicity game played on the edge
set of a complete graph $K_n$, where Maker's goal is to claim a Hamiltonian
cycle. First, we prove that, independent of who starts, Maker can win the game
for $n = 8$ and $n = 9$. Then we use an inductive argument to show that,
independent of who starts, Maker can win the game if and only if $n \geq 8$.
This, in particular, resolves in the affirmative the long-standing conjecture
of Papaioannou.
  We also study two standard positional games related to Hamiltonicity game.
For Hamiltonian Path game, we show that Maker can claim a Hamiltonian path if
and only if $n \geq 5$, independent of who starts. Next, we look at Fixed
Hamiltonian Path game, where the goal of Maker is to claim a Hamiltonian path
between two predetermined vertices. We prove that if Maker starts the game, he
wins if and only if $n \geq 7$, and if Breaker starts, Maker wins if and only
if $n \geq 8$. Using this result, we are able to improve the previously best
upper bound on the smallest number of edges a graph on $n$ vertices can have,
knowing that Maker can win the Maker-Breaker Hamiltonicity game played on its
edges.
  To resolve the outcomes of the mentioned games on small (finite) boards, we
devise algorithms for efficiently searching game trees and then obtain our
results with the help of a computer.
