the 'painted cube' exploration has plenty of potential for developing, testing and justifying general rules
the problem: a cube (made up of n by n by n 'cubelets') is dipped into paint so that only it's outside faces are covered
when broken apart, how many 'cubelets' will have one face, two faces, three faces and no faces painted?
and the task is good for looking at equivalent problems in different dimensions
for example, the 2-D equivalent would be painting around the edges of an n by n square and seeing how many of the 'squarelets' have 0 , 1 and 2 edges painted
4 - dimensions...