Take a grid with dimensions $n\times n$ squares and randomly fill each square with $1$ of $n$ colors. What is the expected average group size of colors touching each other as $n$ approaches $\infty$? Groups of size $1$ count and they can touch either adjacently or diagonally.
For example, the following is a possible grid for $n=4,m=3$:
In this example, there would be $1$ group of size $4$, $2$ groups of size $3$, $1$ group of size $2$, and $4$ groups of size $1$. The average group size would be $\frac{16\text{ total squares}}{8\text{ total groups}}=2$.
I realize that the calculation for each iteration can be simplified to $\frac{n^2}{\text{total groups}}$ but I don't understand how to turn this problem into something that could be solved.
Here is the Desmos graph I created to test on my own: https://www.desmos.com/calculator/ar9kohfdv3
I would offer an approximation but I can't figure out a way to calculate the total number of groups. This means I would have to count them manually and I would not be able to get very much data.
Motivation: This question was inspired by A disc contains $n$ random points. Each point is connected to its nearest neighbor. What does the average cluster size approach as $n\to\infty$?
