The six faces of a wooden cube of side 6 cm are labelled A, B, C, D, E and F respectively. Three of these faces A, B, and C are each adjacent to the other two, and are painted red. The other three faces are not painted. Then, the wooden cube is neatly cut into 216 little cubes of equal size. How many of the little cubes have no sides painted?
- 125
- 135
- 91
- 108
- 100
The Correct Option is A
Solution and Explanation
Step 1: Break down the problem
The cube has side length = \(6 \, \text{cm}\). When cut into smaller cubes of side \(1 \, \text{cm}\), the total number of small cubes is: \[ 6^3 = 216 \]
Step 2: Painted vs unpainted sides
Faces A, B, C are painted. Importantly, these three faces are mutually adjacent (they meet at one corner). Faces D, E, F are unpainted.
Step 3: Condition for unpainted small cubes
A little cube has no side painted if it lies entirely within the "interior" of the big cube, i.e., away from the 3 painted faces.
Step 4: Effective interior cube
Since A, B, and C meet at one corner, any cube on those faces will be painted. To avoid them, we must exclude 1 layer from each of the painted faces. So the "completely safe" unpainted inner block has side: \[ (6 - 1)^3 = 5^3 = 125 \]
Step 5: Conclusion
Therefore, the number of little cubes with no painted sides is: \[ \boxed{125} \]
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