I get 16 pieces. Every time I cut the cake, I take all the resulting pieces and stack them on top of each other before making the next cut. This is much easier to do if the pieces are cut to equal size.
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Theoretically this can be a good solution - you had something similar to my previous cake-problem - but it will be pretty difficult to make the cut for the fourth time (even for the third, because the pile will be too tall...) :-)