A standard 8 x 8 wooden chessboard has a straight line scratch in its surface, and is taken in for repair. The artisan who it is brought to decides to cover each affected square with a thin wooden veneer of the appropriate color.
Assuming that a different veneer is needed for each square of the board, what is the maximum number of such veneers that the artisan will require to do the job?
(1) If the scratch is assumed to have 0 width, then the answer is 15, because for maximal coverage of the line you must pass though alternating colors, and the path of maximum color alternation is just offset from the main diagnonal. (Of course, if the scratch is zero width, then you can't actually see it :)
(2) If the scratch has width, then the answer is 22 squares, because then the scratch must cover the maximum distance on the board, which is either of the long diagnonals.
Two solutions
