What is the maximum number of mines in a Minesweeper board?

[weevil]

I'm programming a minesweeper game, and I'm trying to come up with an equation to determine the maximum number of mines a board can have dependent on the amount of rows and columns given. I pulled some numbers from the Windows 7 version, but I'm not sure how to derive an equation from them.

I'll list a few and maybe someone could point me in the right direction? Sorry if this is in the wrong forum, I'm lost enough in mathematics that I don't even know what branch this falls under :P

Row | Col | Windows Max No. of Mines
9 x 9 = 67
10 x 9 = 76
11 x 9 = 84
12 x 9 = 93

9 x 10 = 76
10 x 10 = 85
11 x 10 = 95
12 x 10 = 104

9 x 11 = 84
10 x 11 = 94
11 x 11 = 105
12 x 11 = 115

9 x 12 = 93
10 x 12 = 104
11 x 12 = 115
12 x 12 = 126

9 x 13 = 101
10 x 13 = 113
11 x 13 = 125
12 x 13 = 138

9 x 14 = 109
10 x 14 = 123
11 x 14 = 136
12 x 14 = 149

I found a crappy approximation as Row * Col - .9 * (14 + Col - 9 + Row - 9)
where any floating point number in the solution is truncated due to being
stored in an integer data type. It's serving for now but I'd like to understand how to analyze something like this for curiosities sake.

 

[Borek]

Please explain what do you mean by "maximum number of mines" - what are constraints? Sounds to me like - if we assume each field can hold exactly one mine - maximum number of mines is just width*height of the board.

 

[weevil]

Please explain what do you mean by "maximum number of mines" - what are constraints? Sounds to me like - if we assume each field can hold exactly one mine - maximum number of mines is just width*height of the board.

Minesweeper needs empty cells on each new board. Windows gives a maximum number of mines you can have on any given board size, those are the numbers I gave above.

 

[Borek]

So basically you are trying to reverse engineer Minesweeper, and numbers you listed were checked experimentally?

 

[weevil]

So basically you are trying to reverse engineer Minesweeper, and numbers you listed were checked experimentally?

More or less :) The custom board size dialog gave those numbers as upper limits.