Define a

**slice** of a square to be a line segment with ends on

two different sides,

one corner and an opposing side, or

two opposite corners of the square.

Sequential slices may or may not cross previous ones, but a set of slices will subdivide the square into polygonal regions.

Find (or prove impossible) a way to slice a square into 7 pieces of equal area with n distinct slices for each n={3,4,5,6,7}

The hardest case, I think. By drawing three slices each of which crosses the other, we produce seven regions. By placing them artfully, it seems intuitively that it should be possible for all 7 regions to be equal. I just haven't done it yet.

Clearly, the first slice must divide the rectangle into a 4/7 and a 3/7 region. The 2nd slice must create a 1/7 and 3 2/7 regions. And the final slice must divide the 3 2/7 regions in half.

Is it really possible? Stay tuned for results.