A cube is painted red on two adjacent faces and on one opposite sides face, yellow on two opposite faces and green on the remaining face. It is then cut into 64 equal cubes. How many cubes have one red coloured face only?
Show Hint
For any cube of side $n$, the number of small cubes with exactly one face painted is $6 \times (n-2)^2$. If only $k$ faces are of a specific color, then $k \times (n-2)^2$ small cubes will have only that color painted.
Step 1: Understanding the Concept:
A cube of 64 small cubes implies a $4 \times 4 \times 4$ structure. The distribution of paint on the faces determines the coloring of the small internal and external cubes. Step 2: Detailed Explanation:
1. Total cubes = 64, so the side of the large cube is $n = \sqrt[3]{64} = 4$.
2. Based on the description (Red on two adjacent faces and one opposite to one of them, Yellow on two opposite, etc.), there are effectively 4 faces painted Red (Front, Back, Left, Right) if we consider the phrasing "two adjacent and one opposite sides face" as covering the vertical faces.
3. For an $n=4$ cube, the center cubes of any face are $(n-2) \times (n-2) = (4-2) \times (4-2) = 2 \times 2 = 4$ cubes.
4. These center cubes on a face have only one face painted, and it is the color of that large face.
5. If there are 4 faces painted red, then the total number of cubes with only one red face (and no other colors) is:
Number of red faces $\times$ Center cubes per face = $4 \times 4 = 16$. Step 3: Final Answer:
There are 16 cubes that have one red colored face only.
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