This movie shows the width of the 'missing white rectangle' or 'gap'.
What can you say about this width when the area of the P-shape is a minimum?

It is fairly clear that the minimum area of the P-shape (or the maximum area of the 'gap') occurs when the 'gap' is a square, the same size as the square 'leg' of the P-shape.
We prove this finding later, but first we can use it to explain why the minimum area of, for example, the M-shape occurs when w = 1/6: NEXT PAGE