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?

Draw the diagonal you spoke of. It will pass through the 8 squares of the same color (let's say Black). Notice how it passes through the points where the two diagonally neighboring squares meet.

Now see what happens if you move this line sideways (without changing its angle of inclination). The line now passes through the corners of the neighboring (White) squares as well for a total of 15. So 15 is definitely possible.

Now if i could find a proof that 16 is not, the problem'ssolution would be complete.

Posted by levik on 2002-05-05 07:58:25