painted cube

the 'painted cube' exploration has plenty of potential for developing, testing and justifying general rules

the general problem: a cube (made up of n by n by n 'cubelets') is dipped into orange paint so that only it's outside faces are painted

when broken apart, how many 'cubelets' will have one face, two faces, three faces and no faces painted orange?

and the task is good for looking at equivalent task in different dimensions: 1D, 2D, 3D, 4D...

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

students can be invited to say what will happen in
4 - dimensions...