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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).

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What move should the first player start with?

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Up to symmetries of the board, there aren't very many possible moves for the first player:

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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.

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  • $\begingroup$ The two possibilities involving P and B are not uniquely solvable. rot13(Sbe juvgr va C, lbh pna erfcbaq jvgu juvgr va W, X be A. Sbe oynpx va C, lbh pnaabg jva.) This doesn't change the first move though. $\endgroup$ – Kruga Oct 7 at 8:43

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