2010 IMO Shortlist N6

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 $Prove that the maximal possible number of colors is$2^{n}$.

$2^{n} \times 2^{n}$ 表的行和列的编号从 0 到 $2^{n}-1$。表的单元格已按满足以下属性进行着色：对于每个 $0 \leq i, j \leq 2^{n}-1$，第 $i$ 行中的第 $j$ 个单元格和 $证明最大可能的颜色数为$2^{n}$。