Question

A cube of side 125 cm is painted red on all the faces and then cut into smaller cubes of side 25 cm each. Find the number of smaller cubes having at least two faces painted.

• A. 48
• B. 36
• C. 44
• D. 52

Hint:

For any cube of side \( L \) cut into smaller cubes of side \( s \) with \( n = L/s \):
- Cubes with 3 faces painted = 8
- Cubes with 2 faces painted = \( 12(n - 2) \)
- Cubes with 1 face painted = \( 6(n - 2)^2 \)
- Cubes with 0 faces painted = \( (n - 2)^3 \)
Remembering these standard formulas helps in solving cube partitioning questions rapidly during examinations.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
In this problem, a larger cube is painted on all outer faces and then partitioned into smaller, identical cubes.
We are asked to find the number of these smaller cubes that have at least two of their faces painted.
The phrase "at least two faces painted" refers to cubes that have either exactly two faces painted or exactly three faces painted.
No smaller cube can have more than three faces painted, so this covers all possibilities of two or more painted faces.

Step 2: Key Formula or Approach:
Let the side length of the larger cube be \( L \) and the side length of each smaller cube be \( s \).
The number of divisions along each edge of the larger cube is given by the ratio \( n \):
\[ n = \frac{L}{s} \]
The standard formulas for the number of painted faces on the smaller cubes are as follows:
1. Cubes with exactly 3 faces painted (located at the corners): \( N_3 = 8 \) (always constant for any \( n \ge 2 \)).
2. Cubes with exactly 2 faces painted (located on the edges): \( N_2 = 12(n - 2) \).
The total number of cubes with at least two faces painted is:
\[ N_{\text{at least 2}} = N_2 + N_3 = 12(n - 2) + 8 \]

Step 3: Detailed Explanation:
1. We begin by identifying the side of the larger cube, which is \( L = 125\text{ cm} \).
2. Next, we find the side of each smaller cube, which is \( s = 25\text{ cm} \).
3. We compute the number of parts \( n \) along each edge of the larger cube:
\[ n = \frac{125}{25} = 5 \]
4. We determine the number of smaller cubes that have exactly three faces painted. These cubes are situated at the 8 corner vertices of the larger cube. Thus, \( N_3 = 8 \).
5. We calculate the number of smaller cubes that have exactly two faces painted. These cubes lie along the 12 edges of the larger cube, excluding the corner cubes.
Using the formula \( 12(n - 2) \), we substitute \( n = 5 \):
\[ N_2 = 12 \times (5 - 2) = 12 \times 3 = 36 \]
6. To find the number of smaller cubes with at least two faces painted, we sum the two quantities:
\[ N_{\text{at least 2}} = N_2 + N_3 = 36 + 8 = 44 \]
7. This gives us a total of 44 smaller cubes that satisfy the given condition.

Step 4: Final Answer:
Therefore, the number of smaller cubes having at least two faces painted is 44, which corresponds to option (C).