How many $2 \times 2 \times 2$ cubes must be added to a $8 \times 8 \times 8$ cube to form a $12 \times 12 \times 12$ cube?

[i]Proposed by Evan Chen[/i]
To solve the problem, we need to determine how many $2 \times 2 \times 2$ cubes must be added to an $8 \times 8 \times 8$ cube to form a $12 \times 12 \times 12$ cube.

1. **Calculate the volume of the $8 \times 8 \times 8$ cube:**
   \[
   V_{\text{small}} = 8^3 = 512
   \]

2. **Calculate the volume of the $12 \times 12 \times 12$ cube:**
   \[
   V_{\text{large}} = 12^3 = 1728
   \]

3. **Determine the additional volume needed:**
   \[
   V_{\text{additional}} = V_{\text{large}} - V_{\text{small}} = 1728 - 512 = 1216
   \]

4. **Calculate the volume of a single $2 \times 2 \times 2$ cube:**
   \[
   V_{\text{small\_cube}} = 2^3 = 8
   \]

5. **Determine the number of $2 \times 2 \times 2$ cubes needed to fill the additional volume:**
   \[
   \text{Number of small cubes} = \frac{V_{\text{additional}}}{V_{\text{small\_cube}}} = \frac{1216}{8} = 152
   \]

Thus, the number of $2 \times 2 \times 2$ cubes required is $\boxed{152}$.