Question 28
The rows and columns of a $2^{n} \times 2^{n}$ table are numbered from 0 to $2^{n}-1$. The cells of the table have been colored with the following property being satisfied: for each $0 \leq i, j \leq 2^{n}-1$, the $j$ th cell in the $i$ th row and the $(i+j)$ th cell in the $j$ th row have the same color. (The indices of the cells in a row are considered modulo $2^{n}$.)
Prove that the maximal possible number of colors is $2^{n}$.
Prove that the maximal possible number of colors is $2^{n}$.