I apologize in advance if this question is not of sufficient level. Define a perfect magic hypercube of side length $k$ and dimension $n$ to be one in which the cells are filled with consecutive integers and the sum of numbers over cells in any geometric line is equal to the appropriate constant depending on $n$ and $k$.

From the density Hales-Jewett theorem it follows that for fixed $k$, there cannot exist perfect magic hypercubes of fixed side length $k$ and arbitrarily large $n$. My question is: what are simpler ways to prove the nonexistence of perfect magic hypercubes of fixed $k$ and arbitrarily large $n$? Thanks very much.