Expert: Scott A Wilson - 3/9/2010

Question
125 small congruent cubes are combined to build a large cube. all the faces of the large cube are painted red. How many of the smaller cubes have no faces painted? How many of the cubes have exactely one face ainted? How many of the cubes hae eactely two faces painted? How many of the cubes have three faces painted?I do tknow even where to start with this question can u please explain

Answer
To have 125 smaller cubes, it is known that 125 = 5³, so the cube is 5x5x5.

With this, we know that there are 25 squares on the top and 25 squares on the bottom.
If these are removed (after the paint has dried, of course), that leaves a solid that
is still 5x5, but only 3 high.  Each face that is color is 3x5.  Note that the edges
are counted with both sides, so only count the right edge as being with the side.

This means that there are 4 faces to count, each 3 high, but 5-1 = 4 wide, since the
left corner is counted with the another side.  This gives 4(3x4) = 4*12 = 48 more cubes.

Thus, the total on the edge is 25 on top + 25 on bottom + 48 on the edge faces = 98.


That was the difficult methold of solving this problem.

Another way to look at this is to think of how many are on no face.
Since for each string of 5, on on either end is on a face,
so 3 cubes in the middle of each string are not.

That number would be 3x3x3 = 27 cubes on the inside.  The total, 125 cubes,
minus the inside, 27 cubes, is 125 - 27 = 98.