Question

Directions: Each of the following questions is followed by two statements. Mark option (1) if the question can be answered using statement I alone. Mark option (2) if the question can be answered using statement II alone. Mark option (3) if both statements I and II together are needed to answer the question. Mark option (4) if the question cannot be answered even using both statements together.

A cube is painted on all sides and is cut into smaller cubes, all of the same size. How many of the smaller cubes do not have any side painted?

I. 8 of the smaller cubes are painted on three sides.
II. The number of smaller cubes is 64.

• A. The question can be answered using statement I alone
• B. The question can be answered using statement II alone
• C. The question can be answered only if both statements I and II are used together
• D. The question cannot be answered even using both statements together

Hint:

Recall that the corner cubes (3 faces painted) always number 8 for any cube size, so that fact alone fixes nothing. You need the total count of small cubes to pin down the cube's size.

Answer 1

The Correct Option is B

Step 1: Understand how a painted cube splits.
When a big cube is painted on all six faces and then cut into \(n^3\) identical smaller cubes, the small cubes fall into four groups based on position.
Corner cubes have 3 painted faces, edge cubes (not corners) have 2 painted faces, face cubes (not edges) have 1 painted face, and inside cubes have 0 painted faces.

Step 2: Check statement I alone.
Statement I says 8 of the smaller cubes are painted on three sides.
For any cube cut into \(n^3\) pieces with \(n \geq 2\), the number of corner pieces (3 faces painted) is always 8, no matter what \(n\) is.
So this fact is true whether \(n=2\), \(n=3\), \(n=4\), or any other size. It does not tell us the value of \(n\).
Without knowing \(n\), we cannot find how many small cubes have zero painted faces, since that count is \((n-2)^3\).
So statement I alone is not enough.

Step 3: Check statement II alone.
Statement II says the number of smaller cubes is 64.
Since \(n^3 = 64\), we get \(n = 4\).
Now we can directly find the number of unpainted (interior) cubes using \((n-2)^3\):
\[ (4-2)^3 = 2^3 = 8 \]
So statement II alone gives us the full answer, 8 unpainted cubes.

Step 4: Conclusion.
Statement I alone fails because it is true for every cube size and gives no new information. Statement II alone succeeds because it fixes \(n=4\) and lets us compute the unpainted count directly.
So the question can be answered using statement II alone, but not statement I alone. \[ \boxed{\text{Statement II alone is sufficient}} \]