How many ways to color an NxN grid?

[adamFah]

Say you have a white NxN grid, but you have the choice to make any cell you want black. How many ways are there to color the grid, without the ith column and ith row being the same for any i?

For example

W W B
B B B
B W B

would be a valid coloring, but

W W B
W B B
B W B

would not be because the first row and column are equal.

 

[Romsek]

I suggest you find the number of ways there are to make the ith row and ith column identical and subtract this from the total number of ways you can color the grid without restrictions.

 

[pka]

I suggest you find the number of ways there are to make the ith row and ith column identical and subtract this from the total number of ways you can color the grid without restrictions.

Good suggestion! If I had written a computer in the last twenty years, I would try that route.
Noting that a valid colouring of a \(N\times N\) matrix means that the matrix and its transpose must not share a row.
It seems to my old memory that several programing languages are ideal for checking that sort of situation.

 

[Steven G]

Adam, this is a math help site where we help students solve their problems. We prefer not to do the problems for you. So please follow the guidelines that you read and post back telling us where you are stuck and what work you have done.