A large cube is formed by using many small 1"x1"x1" cubes. Once the large cube is formed, all the outside faces are painted. There are now exactly 27 small cubes that do NOT have any paint on any face. How many 1"x1"x1" cubes were used to form the large cube?
3^3 = 27 = # of cubes not painted. There needs to be one cube outside of every one of these cubes on each side of all the non-painted cubes= 5^3 = ?
125
To find the answer, we need to determine the dimensions of the large cube.
Let's assume that the edge length of the large cube is n. This means that each face of the large cube has n x n small cubes, and there are 6 faces in total.
If we remove one layer from each face of the large cube, we will get a smaller cube with edge length (n-2). This smaller cube will be made up of (n-2) x (n-2) small cubes on each face.
Thus, the total number of small cubes in the large cube is given by:
Total number of small cubes = Number of small cubes on the outside faces + Number of small cubes in the smaller cube formed by removing one layer from each face
Number of small cubes on the outside faces = 6 x (n x n)
Number of small cubes in the smaller cube = (n-2) x (n-2) x 6
Together, these two quantities should equal 27, since there are exactly 27 small cubes with no paint on any face.
So, we can set up the equation:
6 x (n x n) + (n-2) x (n-2) x 6 = 27
Simplifying this equation will give us the value of n, which will tell us the number of small cubes used to form the large cube.