Step 1: Understanding the Concept:
When a large painted cube is cut into smaller, identical cubes, the smaller cubes located along the edges have 2 faces painted, while those at the corners have 3 faces painted. The phrase "at least two faces painted" means we need to find the total number of cubes that have exactly 2 faces painted plus those that have exactly 3 faces painted.
Step 2: Key Formula or Approach:
First, determine the number of layers or segments ($n$) along one edge:
\[ n = \frac{\text{Side of larger cube}}{\text{Side of smaller cube}} \]
The formulas for counting the painted cubes are:
- Cubes with exactly 3 faces painted (always at the 8 corners):
\[ N_3 = 8 \]
- Cubes with exactly 2 faces painted (located on the 12 edges, excluding corners):
\[ N_2 = 12(n - 2) \]
- Cubes with at least 2 faces painted:
\[ N_{\text{at least 2}} = N_2 + N_3 \]
Step 3: Detailed Explanation:
Calculate the value of $n$:
\[ n = \frac{125}{25} = 5 \]
Now, calculate the number of cubes with exactly 2 faces painted ($N_2$):
\[ N_2 = 12(5 - 2) = 12 \times 3 = 36 \]
Next, identify the number of cubes with exactly 3 faces painted ($N_3$):
\[ N_3 = 8 \]
Sum these two values to find the cubes with at least 2 faces painted:
\[ N_{\text{at least 2}} = 36 + 8 = 44 \]
Step 4: Final Answer:
Therefore, the total number of smaller cubes having at least two faces painted is 44.