A large solid cube is assembled by gluing together identical unpainted small cubic blocks. All six faces of the large cube are then painted in red. If exactly 27of the small cubic blocks that make up the large cube have no red paint on them, how many small cubic blocks make up the large cube?
Only blocks that are on the inside of the cube will not be colored in red. Since the volume of cube is given by
where `a` is the side length of the cube, we have
Therefore, each side of the inner cube consists of 3 blocks. Since the inner cube must be completely surrounded by red blocks, we must add 2 blocks to the length of the side (if we were to add only 1 block, one side of the inner cube would not be surrounded by red blocks). Hence, the length of the side of the larger cube is
So the large cube is made of `5^3=125` small cubic blocks.
As we know that
the inner cube is constructed with 27,
so the inner cube has the length of cube root of (27) =3 units
As, there is no red paint on the inner block, it must be covered by a layer of blocks that has been painted. This outside layer is increased by the size of the block by 1 small cube in length, width, and height on both the sides of the inner block.
so the length of the larger cube is = 1+3+1 = 5 units
so the volume of the larger cube is = 5 x 5 x 5 = 125 units
and the volume of the small cube is 1 unit
so 125 small cubes are needed to make the large cube
Please check the figure given in the attachments for getting an idea
Let us think this is a cross section of the larger cube.
The white cube is the inner cube which is covered by a layer which is painted red in color.