[Math] Possible number of bombs in minesweeper game


This question is about the minesweeper game, and I really don't know how to think about this.

Suppose that we are playing the game and we have already opened some number of cells. Then the closed cells which are adjacent to certain set of open cells form a group of cells, which I call clusters. Here are some examples of clusters (closed cells inside the black borders):


For each cluster $C$ we can consider the set $M(C)$ of possible number of mines it can contain. For example, on the image above, for the cluster adjacent to cells with numbers 2 and 3, this set is $\{3, 4, 5\}$.

Suppose that we have some cluster $C$ and $a,b$ are respectively the possible minimum and maximum number of mines it can contain, and moreover $b\geq a+2$. Is it true that for every integer $c\in(a,b)$ it can contain exactly $c$ mines?

Best Answer

By counterexample, consider the following cluster:

                                              enter image description here

Since it is impossible for only two of the three lowest flags to share a common mine, either a mine is placed in between the three flags, or one mine is placed for every flag. As such, this cluster can hold either $a = 13$ mines, or $b = 15 \geq a+2$ mines.