The rows and columns of a
table are numbered from
to
The cells of the table have been coloured with the following property being satisfied: for each
the
-th cell in the
-th row and the
-th cell in the
-th row have the same colour. (The indices of the cells in a row are considered modulo
.) Prove that the maximal possible number of colours is
.
Proposed by Hossein Dabirian, Sepehr Ghazi-nezami, Iran
![2^n \times 2^n](/media/m/e/4/4/e44ca411eab9f3d9482dd8cf6285f2bc.png)
![0](/media/m/7/b/8/7b8b0b058cf5852d38ded7a42d6292f5.png)
![2^{n}-1.](/media/m/3/d/1/3d13ba730a5fa38f762ba44f5e54be11.png)
![0 \leq i,j \leq 2^n - 1,](/media/m/2/4/6/246586dedf18bc99e1ed6864386bea2b.png)
![j](/media/m/7/9/e/79ebb10f98eb80d16b0c785d9d682a72.png)
![i](/media/m/3/2/d/32d270270062c6863fe475c6a99da9fc.png)
![(i+j)](/media/m/3/8/a/38a1f2f2ba6bfca3e4f672b8e53e2e79.png)
![j](/media/m/7/9/e/79ebb10f98eb80d16b0c785d9d682a72.png)
![2^n](/media/m/8/e/a/8ea40429bb1e68f68f9e7a97fd5351f7.png)
![2^n](/media/m/8/e/a/8ea40429bb1e68f68f9e7a97fd5351f7.png)
Proposed by Hossein Dabirian, Sepehr Ghazi-nezami, Iran