## Cube Layer Problem

*Visualise that you have an unlimited number of small cubes (all the size*1×1×1

*) in different colours. Then, imagine, you are building bigger cubes from these small unit cubes by wrapping layers (like in an onion or Russian nesting dolls) such that each layer has a different colour.*

*Next, imagine the layers of small unit cubes and try to answer the following questions:*

*Level 1.**Imagine a cube, C*3×3×3

_{1}, of the size*(each layer has a different colour).*

*a)*

*How many layers does cube C*

_{1 }have?*b)*

*Draw a picture, how the small unit cubes of the outer layer touch the faces of the previous layer inner cube. How many unit cubes of the outer layer have a face touching the inner layer face?*

*c)*

*How many unit cubes of the outer layer touch the inner cube along the edges? Draw a picture.*

*d)*

*How many unit cubes of the outer layer touch the inner cube at the vertices? Draw a picture.*

*e)*

*How many small unit cubes are there in total?*

*Level 2.**Imagine a cube, C*

_{2}, that has one more layer than the previous one (each layer has a different colour).*a)*

*How many small unit cubes did you add to the previous cube C*In Level 2, the questions b), c), d) and e) are the same as in Level 1.

_{1}?