Imagine a rectangular solid that is made up of smaller unit cubes. The solid measures *x* by *y* by *z*. Now imagine building a shell of unit cubes around the entire solid.

First question: what is an equation for the number of unit cubes in the shell, in terms of *x*, *y*, and *z*?

Second question: for what *x*, *y*, and *z* is the number of cubes in the rectangular solid equal to the number of cubes in the shell?

The original cube has *xyz* unit cubes.

The new one has *x* + 2)(*y* + 2)(*z* + 2)

*x* + 2)(*y* + 2)(*z* + 2)

*xy* + 2*x* + 2*y* + 4)(*z* + 2)

*xyz* + 2*xy* + 2*xz* + 4*x* + 2*yz* + 4*y* + 4*z* + 8

So the shell alone has *xy* + *xz* + *yz*) + 4(*x* + *y* + *z*) + 8

Well, when the shell equals the original cube, then *xyz* = 2(*xy* + *xz* + *yz*) + 4(*x* + *y* + *z*) + 8.

Uh, not easily solved. Suggestions?