Question 29
The faces of a cube of size $1 \times 1 \times 1$ are painted in black, grey and white, two faces in each color, so that opposite faces have the same color. We have a squared $2018 \times 2018$ board divided into $2018^2$ cells of $1 \times 1$.
Ana and Beatriz play the following game. First, Ana puts the cube on a cell of the board so that the face of the cube that is in contact with the board coincides with that cell. Then, Beatriz and Ana, alternately, flip the cube, putting it on an adjacent cell, as shown in the picture:
The cube is allowed to come back to a cell only if the color of the face in contact with the cell is different from the colors of the faces that were in contact with the cell previously. Thus, the cube may visit a cell at most three times; one with a black face, another one with a grey face, and another one with a white face.
The player that in her turn is not able to make a valid move loses the game. Determine which player has a winning strategy.
Ana and Beatriz play the following game. First, Ana puts the cube on a cell of the board so that the face of the cube that is in contact with the board coincides with that cell. Then, Beatriz and Ana, alternately, flip the cube, putting it on an adjacent cell, as shown in the picture:
The cube is allowed to come back to a cell only if the color of the face in contact with the cell is different from the colors of the faces that were in contact with the cell previously. Thus, the cube may visit a cell at most three times; one with a black face, another one with a grey face, and another one with a white face.
The player that in her turn is not able to make a valid move loses the game. Determine which player has a winning strategy.