1. What is the minimal number of 90-degree turns for a bishop to walk through all the black squares of a standard 8*8 chessboard, starting at h8 and terminating at a1?
2. What if in question 1 the word "black" is erased and the word "bishop" is replaced by "rook"?
The rook must travel in either all eight colums or all eight rows. If it does not then the square in the row and column not traveled in will never be reached.
Assume all eight rows are traveled in. It takes two 90 degree turns to change which row is traveled in and at least seven changes must be made to reach all eight rows. Therefore the minimum number of is 14. (The same argument applies for all eight columns.)
This minimum can be realized by a simple process: start at a1, march to a8 hitting all intervening squares, move up to b8, march to b1, etc.