In the game Minesweeper each numbered clue is surrounded by the corresponding number of mines in a Moore neighborhood, i.e. the 3x3 square lattice around the clue. Some years ago I asked myself how many different ways can mines surround a clue. It's clear that there's 512 possible ways, but treating those that are the same under rotation and reflection as one, the number drops down to only 51. I made a table with all of these and named them.
I am not the first one to do this, as you'd expect, but it wasn't until recently and under further inspection of the Game of Life algorithms section about non-totalistic rules (isotropic non-totalistic since they are indifferent to rotation and reflection, often abbreviated into INT) when I made the correlation: this naming convention could apply to Minesweeper as well. Developed by Alan Hensel, this notation is often referred to as Hensel notation. Read more about the topic here.
So, to get a better get the hang of these pattern names I decided to make an interactive grid, which would automatically identify the pattern and display which INT rule satisfies, in both Hensel notation and the more empiric name I call the patterns.
Putting the focus back on Minesweeper, it's only natural to ask how often these patterns appear in actual boards.
In order to estimate this, I ran some simulations. I am purposefully ignoring nuances like first click or 3BV and simply generating random boards of given size and number of mines, then identifying the configuration for each neighborhood. By default the algorithm does this to every cell, but we can limit the search to non-mines (clues or empty cells;) to cells not on the boundary, since those on the boundary can't have certain patterns (like 2n or 3q;) or both conditions.
Before showing you my results, I encourage you to try the interactive board yourself and see the results building up in real time. Alternatively you can draw on the board, and the results table will update for you. Just mind large size boards on mobile are a little hard to draw on.
If you want to actually play a non-totalistic Minesweeper, take a look at this variant by T6970, where there are no numbers, so c can either refer to 1c, 2c, 3c, and so on.
For each board size, I generated 10 millions boards. I didn't get any 8s in beginner 9x9, which is to be expected, as the theoretical probability for an 8 in this size is around 1 in 15 million, compared to the 1 in 2.8 million of beginner 8x8.
This is not an exhaustive search, the less common numbers like 7 and 8 may not robustly reflect the true probability of finding these on boards. Also we have the first click condition which I mentioned earlier. Feel free to take a look at the code on this page or its repository if you want to dig further in. Be mindful the code is probably far from being optimal.