word problems and turning them into algebraic equation to slove

Question

This will be hard to visualize, but I'll try to explain.

 

Suppose that n³ sugar cubes are stacked to form an n x n x n cube. ( cube is 6 cubes wide, 6 cubes length and 6 cubes in height ) The large cube is now painted yellow. depending on where the sugar cube is positioned in the large cube, it may have one or several of it's 6 faces painted yellow.( For example, if n=1, all 6 faces are painted.)  For larger n, there are some sugar cubes with unpainted faces.

 

Give expressions in the variable n, n>3, for the number of sugar cubes in the n x n x n cube with 0,1,2 or 3 painted faces, respectively.  The expressions obtained , when added together, should give n³ for all n larger then equal to 3.

Why?

Use algebraic simplification to check your identity in the variable n.

Answer 1

f(0) = (n-2)^3

f(1) = 6(n-2)^2

f(2) = 12(n-2)

f(3) = 8

 

f(0)+f(1)+f(2)+f(3) = (n-2)^3+6(n-2)^2+12(n-2)+8 = n^3