This is a problem from HMMT 2015.
On an $8\times8$ chessboard, a rook starts at the lower left corner. Each minute, it moves to a square in the same row or same column with equal probability (however it cannot stay at the same square). What is the expected number of minutes until the rook reaches the upper right corner?
I saw a solution that stated the expected minutes it takes the rook from any square in the top row or any square in the right column (besides the upper right corner) to reach the upper right corner is equal. Additionally, they said that the expected minutes it takes for the rook to reach the upper right corner from any other square was equal. So basically if we number the grids on the coordinate grid, with $(1,1)$ being the lower left corner and $(8,8)$ being the upper right corner, the expected minutes it takes the rook from any of the squares $(8,1),(8,2)\cdots (8,7),(1,8),(2,8),\cdots(7,8)$ to reach $(8,8)$ is equal. Also, the expected number of minutes it takes the rook from any of the squares $(x,y), 1\leq x\leq 7, 1\leq y\leq 7$ to reach $(8,8)$ is equal as well.
They also mention that this is due to the fact that "swapping any two rows or columns doesn’t affect the movement of the rook" Can someone explain why this logic is true?