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}
One way to do n = 4:
a) Draw one slice parallel to an edge, dividing the square into a 3/7 rectangle and 4/7 rectangle
b) Draw a second slice through the center of the 4/7 rectangle, angled so that it divides the 3/7 rectangle into a 2/7 trapezoid and a 1/7 trapezoid. There is exactly one line that will do this, and it can be found by rotating the slice around the center point until the 3/7 rectangle is properly divided. This divides the 4/7 rectangle into two new 2/7 trapezoids.
c) Draw a third slice through the same center of the original 4/7 rectangle, angled so that it divides the two new 2/7 trapezoids into a 4 1/7 quadrilaterals. Again, there is exactly one line that will do this, and it can be found by rotating the slice around the center point until the 2/7 trapezoids are divided.
d) Draw the 4th slice in any one of many ways, entirely within the remaining 2/7 trapezoid, dividing it into a 1/7 triangle and a 4-sided or 5-sided polygon (your choice).
Edited on February 25, 2011, 10:45 am
n = 4 (spoiler)
