centimeter cubes

Question

Suppose that 27 small 1 centimeter cubes are put together to make a solid 3x3x3 cube with a volume of 27 cubic centimeters and that the whole outside surface of this larger cube is painted. How many of the original small cubes will have: no face painted?

         one face painted?

         two faces painted?

         three faces painted?

Answer 1

Cicely, let's think about this...

 

I would recommend you draw this out.

 

If you have a 3 x 3 x 3 cube, there are three layers of 9 cubes, right? There are six 3 x 3 faces.

 

No matter how you lay the cube, the top layer and bottom layer will all have paint, because those layers face the outside. If we look at the middle layer, all the cubes that face the outside have paint. There is one cube in the very middle that doesn't face the outside, so it doesn't have any paint.

 

The six cubes that directly touch it are in the middle of each face and will have one side painted.

 

Each cube along an edge will have two faces painted, though the 8 corner pieces will have three sides painted. 

 

That leaves us with 12 cubes with 2 faces painted.

Answer 2

If you don't have a Rubik's Cube, picture a 3x3 one.

 

A cube has 6 surfaces.  For a 3x3 cube, each surface will have 9 little cube surfaces exposed.  That means that a total of 6*9=54 surfaces get painted.

 

Now, some cubes get more than one surface painted, some get exactly one surface painted, but:

    1 cube gets no surface painted             [the little cube in the center]

 

Let's work backwards:

 

How many get 3 surfaces painted? (the corner ones -- there are 8 of these).

 

How many get 2 surfaces painted? (the other edge little cubes --  there are 12 edges (PLZ consult your cube or picture), so that means 12 cubes have 2 edges painted.

 

Lastly, how many cubes get only 1 surface painted?  These are the center cubes of each big cube surface (and there a 6 surfaces, right?)

 

You may count in a different order.  Summary:

                                      cubes   surfaces

    no surface painted         1           0

    one surface painted       6            6

    two surfaces painted    12          24

    three surfaces painted    8          24

       total surfaces painted          54     (check!)