Let's denote the true distance of the bubble from one side of the ice cube as $$x$$ and the refractive index of the ice cube as $$n$$. We will use the formula for apparent depth, which states that the ratio of the true depth to the apparent depth is equal to the refractive index:

$$n = \frac{\text{True depth}}{\text{Apparent depth}}$$

When viewing the bubble from one side, the true depth is $$x$$ and the apparent depth is $$12 \mathrm{~cm}$$. Using the formula:

$$n = \frac{x}{12}$$

When viewing the bubble from the opposite side, the true depth is $$24 - x$$ (since the side of the ice cube is $$24 \mathrm{~cm}$$) and the apparent depth is $$4 \mathrm{~cm}$$. Using the formula:

$$n = \frac{24 - x}{4}$$

Now we have a system of two equations with two variables:

1) $$n = \frac{x}{12}$$

2) $$n = \frac{24 - x}{4}$$

We can solve this system by setting the two expressions for $$n$$ equal to each other:

$$\frac{x}{12} = \frac{24 - x}{4}$$

To solve for $$x$$, first multiply both sides by $$12$$:

$$x = 3(24 - x)$$

$$x = 72 - 3x$$

Add $$3x$$ to both sides:

$$4x = 72$$

Divide by $$4$$:

$$x = 18$$

Now that we have the value of $$x$$, we can find the refractive index $$n$$ using either equation 1 or 2. Using equation 1:

$$n = \frac{18}{12} = \frac{3}{2}$$

Therefore, the refractive index of the ice cube is $$\frac{3}{2}$$.