What is the command to yield the expansion of Z^N in terms of x and y.

Z complex = x+iy,

N integer >1

e.g. Z^2 = x^2 - y^2 + 2.i.x.y

A mathematical friend wrote to me recently with the following problem:

a + b + c = 1

a² + b² + c² = 2

a³ + b³ + c³ = 3

Find the values of  a^4 + b^4 + c^4 and a^5 + b^5 + c^5.

Note the question doesn’t ask for the values of a, b and c, but can you find them?

A sphere, radius one unit with centre at the origin of the x, y and z axes is cut along the planes x=0, y=0 and z=0 yielding 8 total pieces, none of which are cubes.

For a sphere radius 2, cut along the planes x=2,y=2,z=2, x=-2,y=-2,z=-2, you get  64 pieces of which 8 are cubes.

Derive an expression for the number of  pieces you will get with a sphere radius integer R, cut along the planes x,y,z=1, x,y,z=-1, x,y,z=2,…     x,y,z=R-1, x,y,z=1-R  

How many of those pieces will be cubes?

The question arose from slicing onions!