Just how big is Minesweeper? How many different games are possible?

Chapter 0: Definitions

We will define the following throughout the blog post.

• ${\displaystyle H}$: The height of the field.
• ${\displaystyle W}$: The width of the field.
• ${\displaystyle M}$: The number of mines in the field.

Chapter 1: Simple Case (A field with ${\displaystyle H=2}$, ${\displaystyle W=2}$, and varying ${\displaystyle M}$)

${\displaystyle M=0}$

With no mines in the field, there is only one possible Minesweeper game.

${\displaystyle M=1}$

With a mine in the field, there are four possible Minesweeper games, each for a mine in a different location.

${\displaystyle M=2}$

With two mines in the field, one might think there are twelve possible Minesweeper games (the first mine could go in four possible locations, the second mine could go in three possible locations), but since the order in which the mines are put does not matter (and affect gameplay), in reality, there are only six.

${\displaystyle M=3}$

Similar to the case of ${\displaystyle M=1}$, but this time we are choosing a place not to place a mine.

${\displaystyle M=4}$

Similar to the case of ${\displaystyle M=0}$.

Chapter 2: Conjecture

The conjecture is the following: If the field which has height ${\displaystyle H}$ and width ${\displaystyle W}$has ${\displaystyle M}$ mines, then the number of possible Minesweeper games is ${\displaystyle {\frac {\left(HW\right)!}{M!\left(-M+HW\right)!}}}$.

Chapter 3: Custom Field

The custom field is where the fun begins. We can customize to our heart's content, but there are some constraints:

• ${\displaystyle 9\leq H\leq 24}$
• ${\displaystyle 9\leq W\leq 30}$
• ${\displaystyle 10\leq M\leq \left(H-1\right)\left(W-1\right)}$

Ultimately, we must perform the following summation:

${\displaystyle \sum _{W=9}^{30}\sum _{H=9}^{24}\sum _{M=10}^{\left(H-1\right)\left(W-1\right)}{\frac {\left(HW\right)!}{M!\left(-M+HW\right)!}}}$

Now give a big drumroll please, for the number that I sought after is the following:

5515652596997352824536870236751423660351367657778256703333498595846274654961280124373345632294740641941024119283566140971758318650262798808356210976586493739847519720990392352891439114639467202636502202709734696970240

or approximately 5.516×10²¹⁶.

It is so big that Wolfram Alpha could not think of anything to compare to, so I will make it something to compare to. I dub thee number Grandmaster Minesweeplex.