立方体切割

关键思想是让每一刀产生尽可能多的块数。一个常见结论是：

若要把一个 $n \times n \times n$ 的立方体用贯穿整个木块的平面切割切成 $n^3$ 个小立方体，所需的最少切割次数为：

$$3(n-1)$$ $\text{推理：}$

• 沿 $x$ 轴方向切 $n-1$ 刀，把立方体分成沿 $x$ 方向的 $n$ 个薄层。
• 沿 $y$ 轴方向切 $n-1$ 刀，此时每个薄层又被切分成 $n \times n$ 个小块。
• 沿 $z$ 轴方向再切 $n-1$ 刀，就得到 $n^3$ 个单位立方体。

$\text{应用到本题：}$

这里 $n = 4$：

$$\text{最少切割次数} = 3(n-1) = 3(4-1) = 9$$ $\text{检查：}$

• 沿 $x$ 轴切 $3$ 刀 $\to 4$ 个薄层
• 沿 $y$ 轴切 $3$ 刀 $\to 16$ 个小块
• 沿 $z$ 轴切 $3$ 刀 $\to 64$ 个单位立方体

$\text{答案：}$ $$\boxed{9 \text{ cuts}}$$

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Original Explanation

The key idea is to maximize the number of pieces created with each cut. A well-known result is:

To cut a $n \times n \times n$ cube into $n^3$ smaller cubes using plane cuts through the entire block, the minimum number of cuts required is:

$$3(n-1)$$ $\text{Reasoning:}$

• Make $n-1$ cuts along the $x$-axis. This divides the cube into $n$ slabs along the $x$ direction.
• Make $n-1$ cuts along the $y$-axis. Each slab is now divided into $n \times n$ pieces.
• Make $n-1$ cuts along the $z$-axis. Each piece is now divided into $n^3$ unit cubes.

$\text{Application:}$

Here $n = 4$:

$$\text{Minimum cuts} = 3(n-1) = 3(4-1) = 9$$ $\text{Check:}$

• $3$ cuts along $x$ $\to 4$ slabs
• $3$ cuts along $y$ $\to 16$ pieces
• $3$ cuts along $z$ $\to 64$ unit cubes

$\text{Answer:}$ $$\boxed{9 \text{ cuts}}$$