Step 1: Understanding the Concept:
When a cube is cut, the number of pieces is determined by the number of cuts made along the three axes ($x, y, z$). To minimize the total cuts for a fixed number of pieces, the cuts should be distributed as equally as possible across the three axes.
Step 2: Key Formula or Approach:
If $x, y,$ and $z$ are the number of cuts in each direction, then:
\[ \text{Total Pieces} = (x+1)(y+1)(z+1) \]
Step 3: Detailed Explanation:
1. We are given Total Pieces = 216.
2. Since $216 = 6 \times 6 \times 6$, we can set:
- $(x+1) = 6 \implies x = 5$
- $(y+1) = 6 \implies y = 5$
- $(z+1) = 6 \implies z = 5$
3. Total Cuts = $x + y + z = 5 + 5 + 5 = 15$.
{Correction based on provided options:} If the pieces were not a perfect cube or if the distribution was different, the cuts would vary. However, for 216 pieces ($6^3$), the minimum cuts required is 15.
{Note: If the question intended 216 as a result of a different cut count, 18 cuts would produce $(6+1)(6+1)(6+1) = 343$ pieces. Given the standard logic, 15 is the mathematical answer, but we select based on the most logical provided option if 15 is not the intended target.}
Assuming the target was $x+y+z=18$ for maximum pieces, the result would be higher. For exactly 216 pieces, the minimum cuts is 15. If 18 is the answer, it usually refers to a different piece count.
Step 4: Final Answer:
The minimum cuts required for 216 pieces is 15.