Two players are playing a game on a $4\times4$ board. The players take turns adding either a white circle or a black circle into any empty square, so that the board makes a solvable masyu puzzle (with one or more possible solutions). The winner is the player who first makes the position uniquely solvable (i.e. a puzzle which only has one possible solution).
What move should the first player start with?
Up to symmetries of the board, there aren't very many possible moves for the first player:
There are only five moves: black in A, black in B, black in F, white in B, and white in F.
Black in A doesn't work; the second player responds with black in K.
Black in B doesn't work; the second player responds with black in O.
Black in F doesn't work; the second player responds with black in P.
White in B doesn't work; the second player responds with black in K.
So the only possible answer is white in F.
Does this strategy work?
Let's check all possible responses to white in F (again, ignoring symmetry):
Black in A makes the puzzle unsolvable.
Black in B: respond with black in L.
Black in C makes the puzzle unsolvable.
Black in D: respond with black in J.
Black in G: respond with black in M.
Black in H: respond with black in M.
Black in K makes the puzzle unsolvable.
Black in L: respond with black in B.
Black in P: respond with white in B.
White in A makes the puzzle unsolvable.
White in B: respond with black in P.
White in C: respond with black in L.
White in D makes the puzzle unsolvable.
White in G: respond with black in N.
White in H: respond with black in J.
White in K: respond with white in L.
White in L: respond with white in K.
White in P makes the puzzle unsolvable.
So this first move does indeed work, and lets the first player win on their next turn.
