**How would you cut a plane through a cube, such that
its intersection with the cube would be a regular hexagon? **

**Solution:**

**Let the cube
be centered at the origin of a cartesian coordinate system, and the length of each side
be 2 units. Cut the plane so that it passes thru the following
points on the edges of the cube:**

**(1,
0, -1), (0, 1, -1),(1,-1,0),(0,-1,1),(-1,1,0),(-1,0,1)**

**The
points above represents the vertices of the regular hexagon.
These points are at the midpoints of their corresponding edges. The center
of the regular hexagon coincides with the center of the square at the
origin. The sides of the hexagon are on the faces of the
cube. **