A solid cube is painted yellow on all its faces. The cube is then cut into 60 smaller but equal pieces by making the minimum number of cuts. Which of the following statements is/are correct?

I. The minimum number of cuts is 9.
II. The number of smaller pieces which are not painted on any face is 6.

Select the correct answer using the code given below:

Correct Answer: Both I and II

Let us analyze the problem step-by-step to check the correctness of both statements.

Step 1: Analyzing Statement I (Minimum number of cuts)

When a solid cube is cut into smaller pieces by making cuts along its three perpendicular directions (axes), let the number of cuts along the three axes be $x$, $y$, and $z$.

The total number of smaller pieces obtained is given by the formula:

$N=(x+1)\×(y+1)\×(z+1)=60$

To minimize the total number of cuts $x+y+z$, the factors $(x+1)$, $(y+1)$, and $(z+1)$ should be as close to each other as possible.

Let us express 60 as a product of three integers:

$60=3\×4\×5$

Comparing $(x+1)\×(y+1)\×(z+1)=3\×4\×5$:

$x+1=3\⇒x=2$

$y+1=4\⇒y=3$

$z+1=5\⇒z=4$

Total minimum number of cuts = $x+y+z=2+3+4=9$.

Therefore, Statement I is correct.

Step 2: Analyzing Statement II (Number of unpainted smaller pieces)

The smaller pieces that are not painted on any face come from the inner core of the solid cube (excluding the outer layer along each dimension).

The formula for the number of unpainted pieces is:

$N_{\text{unpainted}}=(x-1)\×(y-1)\×(z-1)$

Substituting $x=2$, $y=3$, and $z=4$:

$N_{\text{unpainted}}=(2-1)\×(3-1)\×(4-1)$

$N_{\text{unpainted}}=1\×2\×3=6$

Therefore, Statement II is correct.

Since both Statement I and Statement II are correct, the correct option is Both I and II.