Question 19
A cube with edge length $3$ is divided into $27$ unit cubes. The numbers $1,2, \ldots, 27$ are distributed arbitrarily over the unit cubes, with one number in each cube. We form the $27$ possible row sums (there are nine such sums of three integers for each of the three directions parallel to the edges of the cube). At most how many of the $27$ row sums can be odd?