A cube is painted Green on four adjoining side faces and Black on the other two faces, which are opposite each other (the top face and the bottom face). The cube is then cut by two evenly spaced cuts parallel to each of its three pairs of faces, which divides every edge into 3 equal parts and turns the big cube into a 3 x 3 x 3 arrangement of 27 smaller cubes of equal size.
How many smaller cubes are formed in all?
Show Hint
Two cuts parallel to each face split every edge into 3 equal parts, so the number of small cubes is (edge parts) cubed.
Step 1: Picture the cube as stacked layers.
Since every edge is cut into 3 equal parts, picture the big cube as 3 flat layers stacked one above the other, a bottom layer, a middle layer and a top layer, each layer being one unit thick.
Step 2: Count the small cubes in a single layer.
Look straight down at just one layer. Because each edge is divided into 3 parts, that layer looks like a flat 3 by 3 grid of small cubes when seen from above, which gives $3 \times 3 = 9$ small cubes in one layer.
Step 3: Multiply by the number of layers.
There are 3 such layers in total, and each one holds 9 small cubes, so the total count is
\[ 9 \times 3 = 27 \]
This layer by layer count lands on the same number as the direct $3^3$ calculation, which confirms the answer is correct.
Final Answer:
The cube is cut into 27 smaller cubes.
\[ \boxed{27} \]
