Eight identical cubes with of size $1 \times 1 \times 1$ each have the numbers $1$ through $6$ written on their faces with the number $1$ written on the face opposite number $2$, number $3$ written on the face opposite number $5$, and number $4$ written on the face opposite number $6$. The eight cubes are stacked into a single $2 \times 2 \times 2$ cube. Add all of the numbers appearing on the outer surface of the new cube. Let $M$ be the maximum possible value for this sum, and $N$ be the minimum possible value for this sum. Find $M - N$.
1. **Understanding the Problem:**
   - We have eight identical cubes, each with faces numbered from 1 to 6.
   - The numbers on opposite faces are: 1 opposite 2, 3 opposite 5, and 4 opposite 6.
   - These cubes are stacked to form a $2 \times 2 \times 2$ cube.
   - We need to find the sum of the numbers on the outer surface of this larger cube and determine the maximum (M) and minimum (N) possible values for this sum.

2. **Analyzing the Outer Surface:**
   - The $2 \times 2 \times 2$ cube has 8 small cubes.
   - Each small cube has 3 faces exposed on the outer surface.
   - Therefore, there are $8 \times 3 = 24$ faces visible on the outer surface.

3. **Maximum Sum Calculation:**
   - To maximize the sum, we need to place the highest numbers on the visible faces.
   - The highest numbers on each cube are 2, 4, and 6.
   - The sum of these numbers is $2 + 4 + 6 = 12$.
   - Since there are 8 cubes, the maximum sum is $12 \times 8 = 96$.

4. **Minimum Sum Calculation:**
   - To minimize the sum, we need to place the lowest numbers on the visible faces.
   - The lowest numbers on each cube are 1, 3, and 5.
   - The sum of these numbers is $1 + 3 + 5 = 9$.
   - Since there are 8 cubes, the minimum sum is $9 \times 8 = 72$.

5. **Finding the Difference:**
   - The difference between the maximum and minimum sums is $M - N = 96 - 72 = 24$.

The final answer is $\boxed{24}$.