Students will work in groups to find a solution to this problem:

How many 1x1x1 cubes are painted on three faces, two faces, one face, or no faces when the following occurs?

Different sized cubes are constructed from unit cubes
The surface areas are painted
The larger cubes are taken apart

Place students in groups and give each group a large quantity of unit cubes, graph paper, and colored pencils or markers.

Hold a cube up and tell students it is the cube’s first birthday. Write down words that describe the cube, like number of corners, faces and edges). Have them build what a cube may look like on its second birthday.

Hold up a unit cube. Tell students this is a cube on its first birthday. Ask students to describe the cube (eight corners, six faces, twelve edges). Have them build the third, fourth and fifth birthday of the cube, then ask this question for them to solve:

If this cube was ten years old, dipped into paint, dried and taken apart into the unit cubes, how many cubes have three faces, two faces, one face or no faces painted? Have them draw the cube on each of its birthdays up to ten and look for patterns.

Have them also record the exponents for the number of cubes painted on three faces, two, one and no faces. Then, they need to display their findings in graph form, once again looking for patterns. They should notice the following:

The 3 painted faces are always the corners – 8 on a cube, cubes with two painted faces occur on the edges between the corner and increase by 12 each time. Cubes with one face painted occur as squares on the 6 faces of the original, first cube. The cubes with no faces painted are the cube within the cube.

You can also download a PDF version of this game to file away for later here on my site or on TeachersPayTeachers ($2).