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Sawing Chessboards
Challenger
With a jigsaw with an infinitely thin blade, we saw a chessboard in two. The cut is a straight line. How many of the 8 × 8 = 64 squares are maximally cut in two? Note: if two adjacent squares are separated because the saw goes exactly over the shared edge, we still consider both squares whole.
Mastermind
We now saw a 3-dimensional cubic chessboard consisting of 8 × 8 × 8 = 512 cubes with the same jigsaw in two, where the cutting plane is flat. How many of the 512 cubes are maximally cut in two?