# Find out the color of the the secretly square of a chessboard

Alice draws a standard chessboard in the plane. (This means that she draws it in the middle of a big piece of paper). She then secretly chooses a point inside some square of the board. Bob can now draw any polygon (without self intersections) in the plane and ask Alice whether her point is inside or outside his polygon. Bob can continue to ask about other polygons until he is sure of the color of the square that contains Alice’s point. What is the minimum number of polygons sufficient for Bob to find out whether Alice’s point is black or white?

If Bob always draw a polygon that contain half of the remaining possible squares, then it took $\log_2 64=6$ steps to find out the exact square that contains the secret point. However, to find out the color of such point, I believe there is a strategy that takes less than 6 steps. Since there are connected squares with the same color on the chessboard, I have no idea how to determine the color of the square that contains the secret point. Anyone can help me out? Thanks a lot.

• Bob's polygon may not intersect itself, but does it have to be convex? – David K Feb 27 '16 at 22:54