A cube of side length $\frac{7}{3}$ is dropped into a random place on the 3Dlattice in the coordinate space (a coordinate plane with an extradimension), with the sides of the square parallel/perpendicular to theaxes. The expected number of lattice points (points with integercoordinates) in the interior of the cubes is $\frac{a}{b}$. In simplest terms,what's $a$+$b$? (JHMMC Grade 72020 R1/31)
The official solution uses symmetry and independence to calculate: :$$\left (\frac{1}{3} \cdot 3+\frac{2}{3} \cdot 2\right )^3=\left(\frac{7}{3}\right)^3=\frac{343}{27}$$
Is there a more elegant, (but still precise) way to do this that gets the answer based on a direct appeal to the volume of the space inside the box, and get the answer directly proportional to the volume (without going through the intermediate calculations).
Note: Considering JHMMMC grade 7, I would prefer solutions that within the reach of a (highly capable) 7th grader.
