I was unable to get the calculations to line up with the payouts I was actually getting, even when trying to factor in the house edge, so I'm pry missing something. That being said, here are the payouts I've collected from playing. Each number is the consecutive payouts for each correct square clicked.
1 Mine: 1.031, 1.076, 1.125, 1.179, 1.238, 1.303, 1.375, 1.456, 1.547, 1.65, 1.768, 1.904, 2.063, 2.25, 2.475, 2.75, 3.094, 3.536, 4.125, 4.95, 6.118, 8.25, 12.375, 24.75
3 Mines: 1.125, 1.286, 1.479, 1.712. 1.997, 2.35, 2.79, 3.349, 4.066, 5.004, 6.255, 7.962, 10.35, 13.8, 18.975, 27.107, 40.661, 65.057, 113.85, 227.7, 569.25, 2277
6 Mines: 1.303, 1.737, 2.35, 3.231, 4.523, 6.462, 9.445, 14.167, 21.894, 35.031, 58.385, 102.173, 189.75, 379.5, 834.9, (Last 4 guessed) 2087.25, 6261.75, 25047, 175329
24 Mines: 24.75
Let's take some exemplary looks at the calculations:
24 mines and one field is the same as 1 mine and 24 fields due to the symmetrical aspect of the payout structure (if you flip uncovered fields and number of mines, you always land at the same payout).
I've also color coded all other cases of this that are included in your list (3-1 + 1-3 as well as 1-6 + 6-1, and 3-6 + 6-3).
Since the first case is trivial, let us calculate the second one (1 mine, 24 uncovered fields).
(25/24)*(24/23)*...*(2/1) = 25, factor in the 1% house edge (same as *0.99) and you get 24.75
Now 6 mines and the last payout you could confirm (834.9), at 15 uncovered fields.
(25/19)*(24/18)*...*(11/5) = 843.33, factor in the 1% house edge -> 834.9
Calculations seem to fit the payouts you confirmed. Although I think you skipped one in your lineup for 1 mine, as that only lists 23 numbers.
If we want to save some time doing the calculations, we can always flip to the case with less steps involved.
Eg let's confirm the 2.75 number for one mine and 16 uncovered fields, flipping it to 16 mines and 1 uncovered field.
(25/9) = 2.77 (* 0.99) => 2.75
The basic logic of the calculations is quite simple.
You start at (# unknown fields / # non-mine fields) and you reduce both numbers by one with each non-mine field you uncover.
In other words, you always start at (25 / 25-#mines).
Continue that for as many steps as you want to calculate, or until you hit a base case (your number of non-mine fields hits one).
Then, and only then, apply the factor of 0.99 for the house edge (as it is a house edge per game, not per step) and you got your final payout.
Yeah, I found my error. I had a wrong number in my first calculation that I based the rest off of. And i added the number missing from the 1 mine list. Also updated the last 4 on 6 mines from Betwrong's calculations.
