Question

A cube is divided into 343 identical cubelets. Each cut is made parallel to some surface of the cube. But before doing that the cube is colored with green on one set of adjacent faces, red on the second and blue on the third set. How many minimum cuts you have made?

• A. 15
• B. 18
• C. 12
• D. 7
  (e) 13
• E. 13

Hint:

To find minimum cuts for $n^3$ pieces, the formula is always $3(n-1)$. The coloring information in this specific question is a "distractor"—it doesn't change the number of cuts needed to reach 343 pieces.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The total number of small cubelets ($N$) produced from a larger cube is determined by the number of cuts made along the three axes ($x, y, z$). The relationship is $N = (x+1)(y+1)(z+1)$, where $x, y, z$ are the number of cuts in each direction.

Step 2: Key Formula or Approach:
For a perfect cube divided into $n^3$ cubelets, the number of cuts in each direction is $(n-1)$. Total cuts = $3 \times (n-1)$.

Step 3: Detailed Explanation:
1. We are given the total number of cubelets $N = 343$. 2. To find the number of cubelets along one edge ($n$), we take the cube root: \[ n = \sqrt[3]{343} = 7 \] 3. Since there are 7 cubelets along each edge, the number of cuts required in one direction is $7 - 1 = 6$. 4. A cube has three dimensions ($x, y, z$). To minimize cuts while maintaining identical cubelets, we apply the same number of cuts in all three directions. 5. Total cuts = $6 \text{ (horizontal)} + 6 \text{ (vertical)} + 6 \text{ (depth)} = 18$.

Step 4: Final Answer:
The minimum number of cuts made is 18.