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Eighteen is the maximum number of pieces. I give two solutions. The
numbered diagram is so cut that the eighteenth piece has the largest
area--eight squares--that is possible under the conditions. The second
diagram was prepared under the added condition that no piece should
contain more than five squares.
No. 74 in _The Canterbury Puzzles_ shows how to cut the board into
twelve pieces, all different, each containing five squares, with one
square piece of four squares.