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}

n=7 is impossible. Starting with 0 slices and 1 piece, each slice produces at least one additional piece, so 7 slices results in at least 8 pieces.

n = 6 is easy. The 6 lines cannot cross each other. Simplest are 6 horizontal or vertical parallel slices, each slicing off 1/7 of the cube. It is also possible with 6 slices sharing one common point on the square (any point will do). Or many other arrangements.

*Edited on ***February 25, 2011, 10:44 am**