# Mines

## Introduction

Mines is a simple game of chance involving clicking on fields in a 5x5 grid, hoping to avoid hidden mines. The more you safely pick and the more mines to avoid, the more you win. As far as I know, the game can be played exclusively at BetFury.

## Rules

1. The game is played on a 5x5 grid.
2. The player starts by making a bet and choosing a number of mines to be hidden in the grid, from 1 to 24.
3. Spaces in the grid not occupided by a mine will have a diamond.
4. Then, the player will pick spaces in the grid, with the object of avoiding the mines.
5. As the player successfully find diamonds, his win will usually increase.
6. After each turn, the player may choose to cash out or pick another space, as long as there are more diamonds to be found.

The following table shows the win for by number of diamonds found for 1 to 6 total mines. Wins are on a "for one" basis.

### One to Six Mines

Diamonds
Found
One
Mine
Two
Mines
Three
Mines
Four
Mines
Five
Mines
Six
Mines
1 1.03 1.03 1.07 1.13 1.18 1.25
2 1.04 1.12 1.23 1.35 1.5 1.66
3 1.05 1.23 1.41 1.64 1.91 2.25
4 1.05 1.35 1.64 2 2.48 3.1
5 1.06 1.5 1.91 2.48 3.25 4.34
6 1.07 1.66 2.25 3.1 4.34 6.2
7 1.07 1.86 2.67 3.92 5.89 9.06
8 1.08 2.09 3.21 5.04 8.15 13.59
9 1.09 2.37 3.9 6.6 11.55 21
10 1.1 2.71 4.8 8.8 16.8 33.61
11 1.12 3.13 6 12 25.21 56.02
12 1.14 3.65 7.63 16.8 39.21 98.04
13 1.16 4.31 9.93 24.27 63.72 182
14 1.19 5.18 13.24 36.41 109.25 364.16
15 1.23 6.33 18.2 57.22 200.29 801.16
16 1.28 7.91 26.01 95.37 400.58 2000
17 1.34 10.17 39.01 171.67 901.31 6001
18 1.44 13.57 62.42 343.55 2400 24004
19 1.59 19 109.25 801.16 8410 168000
20 1.82 28.5 218.5 2400 50470
21 2.24 47.5 546.25 12020
22 3.06 95 2190
23 5.12 285
24 23.74

The following table shows the win for by number of diamonds found for 7 to 12 total mines. Wins are on a "for one" basis.

### 7 to 12 Mines

Diamonds
Found
7
Mines
8
Mines
9
Mines
10
Mines
11
Mines
12
Mines
1 1.31 1.39 1.48 1.58 1.69 1.82
2 1.86 2.09 2.37 2.71 3.13 3.65
3 2.67 3.21 3.9 4.8 6 7.63
4 3.92 5.04 6.6 8.8 12 16.8
5 5.89 8.15 11.5 16.8 25.21 39.21
6 9.06 13.59 21 33.61 56.02 98.04
7 14.34 23.48 39.91 70.96 133.06 266.12
8 23.48 42.26 79.83 159.67 342.15 798.36
9 39.91 79.83 169.65 387.77 969.44 2710
10 70.96 159.67 387.77 1030 3100 10860
11 133.06 342.15 969.44 3100 11630 54290
12 266.12 798.36 2710 10860 54290 380000
13 576.59 2080 8820 47050 353000 4940000
14 1380 6230 35290 282000 4230000
15 3810 22830 194000 3110000
16 12690 114000 1940000
17 57080 1030000
18 457000

The following table shows the win for by number of diamonds found for 13 to 18 total mines. Wins are on a "for one" basis.

### 13 to 18 Mines

Diamonds
Found
13
Mines
14
Mines
15
Mines
16
Mines
17
Mines
18
Mines
1 1.97 2.15 2.51 2.63 2.96 3.39
2 4.31 5.18 6.33 7.91 10.17 13.57
3 9.93 13.24 18.2 26.01 39.01 62.42
4 24.27 36.41 57.22 95.37 171.67 343.35
5 63.72 109.25 200.29 458 901.31 2400
6 182.08 364.16 801.16 2000 6010 24030
7 576.59 1380 3810 12690 57080 457000
8 2080 6230 22830 114000 1030000
9 8820 35290 193000 1940000
10 47050 282000 3100000
11 353000 4230000
12 4940000

The following table shows the win for by number of diamonds found for 19 to 24 total mines. Wins are on a "for one" basis.

### 19 to 24 Mines

Diamonds
Found
19
Mines
20
Mines
21
Mines
22
Mines
23
Mines
24
Mines
1 3.95 4.75 5.93 7.91 11.87 23.75
2 19 28.5 47.5 95 285
3 109.25 218.5 546.25 2190
4 801.16 2400 12020
5 8410 50470
6 168000

## Analysis

The following table shows the probability of winning and expected return for all combinations of mines and picks.

### Mines Analysis

Mines Picks Pays Prob. Win Return
1 1 1.03 0.960000 0.988800
1 2 1.04 0.920000 0.956800
1 3 1.05 0.880000 0.924000
1 4 1.05 0.840000 0.882000
1 5 1.06 0.800000 0.848000
1 6 1.07 0.760000 0.813200
1 7 1.07 0.720000 0.770400
1 8 1.08 0.680000 0.734400
1 9 1.09 0.640000 0.697600
1 10 1.1 0.600000 0.660000
1 11 1.12 0.560000 0.627200
1 12 1.14 0.520000 0.592800
1 13 1.16 0.480000 0.556800
1 14 1.19 0.440000 0.523600
1 15 1.23 0.400000 0.492000
1 16 1.28 0.360000 0.460800
1 17 1.34 0.320000 0.428800
1 18 1.44 0.280000 0.403200
1 19 1.59 0.240000 0.381600
1 20 1.82 0.200000 0.364000
1 21 2.24 0.160000 0.358400
1 22 3.06 0.120000 0.367200
1 23 5.12 0.080000 0.409600
1 24 23.74 0.040000 0.949600
2 1 1.03 0.920000 0.947600
2 2 1.12 0.843333 0.944533
2 3 1.23 0.770000 0.947100
2 4 1.35 0.700000 0.945000
2 5 1.5 0.633333 0.950000
2 6 1.66 0.570000 0.946200
2 7 1.86 0.510000 0.948600
2 8 2.09 0.453333 0.947467
2 9 2.37 0.400000 0.948000
2 10 2.71 0.350000 0.948500
2 11 3.13 0.303333 0.949433
2 12 3.65 0.260000 0.949000
2 13 4.31 0.220000 0.948200
2 14 5.18 0.183333 0.949667
2 15 6.33 0.150000 0.949500
2 16 7.91 0.120000 0.949200
2 17 10.17 0.093333 0.949200
2 18 13.57 0.070000 0.949900
2 19 19 0.050000 0.950000
2 20 28.5 0.033333 0.950000
2 21 47.5 0.020000 0.950000
2 22 95 0.010000 0.950000
2 23 285 0.003333 0.950000
3 1 1.07 0.880000 0.941600
3 2 1.23 0.770000 0.947100
3 3 1.41 0.669565 0.944087
3 4 1.64 0.578261 0.948348
3 5 1.91 0.495652 0.946696
3 6 2.25 0.421304 0.947935
3 7 2.67 0.354783 0.947270
3 8 3.21 0.295652 0.949043
3 9 3.9 0.243478 0.949565
3 10 4.8 0.197826 0.949565
3 11 6 0.158261 0.949565
3 12 7.63 0.124348 0.948774
3 13 9.93 0.095652 0.949826
3 14 13.24 0.071739 0.949826
3 15 18.2 0.052174 0.949565
3 16 26.01 0.036522 0.949930
3 17 39.01 0.024348 0.949809
3 18 62.42 0.015217 0.949870
3 19 109.25 0.008696 0.950000
3 20 218.5 0.004348 0.950000
3 21 546.25 0.001739 0.950000
3 22 2190 0.000435 0.952174
4 1 1.13 0.840000 0.949200
4 2 1.35 0.700000 0.945000
4 3 1.64 0.578261 0.948348
4 4 2 0.473123 0.946245
4 5 2.48 0.383004 0.949850
4 6 3.1 0.306403 0.949850
4 7 3.92 0.241897 0.948237
4 8 5.04 0.188142 0.948237
4 9 6.6 0.143874 0.949565
4 10 8.8 0.107905 0.949565
4 11 12 0.079130 0.949565
4 12 16.8 0.056522 0.949565
4 13 24.27 0.039130 0.949696
4 14 36.41 0.026087 0.949826
4 15 57.22 0.016601 0.949897
4 16 95.37 0.009960 0.949930
4 17 171.67 0.005534 0.949953
4 18 343.55 0.002767 0.950534
4 19 801.16 0.001186 0.949992
4 20 2400 0.000395 0.948617
4 21 12020 0.000079 0.950198
5 1 1.18 0.800000 0.944000
5 2 1.5 0.633333 0.950000
5 3 1.91 0.495652 0.946696
5 4 2.48 0.383004 0.949850
5 5 3.25 0.291813 0.948391
5 6 4.34 0.218859 0.949850
5 7 5.89 0.161265 0.949850
5 8 8.15 0.116469 0.949223
5 9 11.55 0.082213 0.949565
5 10 16.8 0.056522 0.949565
5 11 25.21 0.037681 0.949942
5 12 39.21 0.024224 0.949807
5 13 63.72 0.014907 0.949863
5 14 109.25 0.008696 0.950000
5 15 200.29 0.004743 0.949992
5 16 400.58 0.002372 0.949992
5 17 901.31 0.001054 0.949997
5 18 2400 0.000395 0.948617
5 19 8410 0.000113 0.949746
5 20 50470 0.000019 0.949934
6 1 1.25 0.760000 0.950000
6 2 1.66 0.570000 0.946200
6 3 2.25 0.421304 0.947935
6 4 3.1 0.306403 0.949850
6 5 4.34 0.218859 0.949850
6 6 6.2 0.153202 0.949850
6 7 9.06 0.104822 0.949689
6 8 13.59 0.069881 0.949689
6 9 21 0.045217 0.949565
6 10 33.61 0.028261 0.949848
6 11 56.02 0.016957 0.949904
6 12 98.04 0.009689 0.949953
6 13 182 0.005217 0.949565
6 14 364.16 0.002609 0.949983
6 15 801.16 0.001186 0.949992
6 16 2000 0.000474 0.948617
6 17 6001 0.000158 0.948775
6 18 24004 0.000040 0.948775
6 19 168000 0.000006 0.948617
7 1 1.31 0.720000 0.943200
7 2 1.86 0.510000 0.948600
7 3 2.67 0.354783 0.947270
7 4 3.92 0.241897 0.948237
7 5 5.89 0.161265 0.949850
7 6 9.06 0.104822 0.949689
7 7 14.34 0.066203 0.949358
7 8 23.48 0.040458 0.949946
7 9 39.91 0.023799 0.949803
7 10 70.96 0.013387 0.949922
7 11 133.06 0.007140 0.949994
7 12 266.12 0.003570 0.949994
7 13 576.59 0.001648 0.949988
7 14 1380 0.000686 0.947368
7 15 3810 0.000250 0.951113
7 16 12690 0.000075 0.950364
7 17 57080 0.000017 0.949948
7 18 457000 0.000002 0.950697
8 1 1.39 0.680000 0.945200
8 2 2.09 0.453333 0.947467
8 3 3.21 0.295652 0.949043
8 4 5.04 0.188142 0.948237
8 5 8.15 0.116469 0.949223
8 6 13.59 0.069881 0.949689
8 7 23.48 0.040458 0.949946
8 8 42.26 0.022476 0.949856
8 9 79.83 0.011899 0.949922
8 10 159.67 0.005950 0.949982
8 11 342.15 0.002777 0.949982
8 12 798.36 0.001190 0.949994
8 13 2080 0.000458 0.951945
8 14 6230 0.000153 0.950420
8 15 22830 0.000042 0.949865
8 16 114000 0.000008 0.948617
8 17 1030000 0.000001 0.952315
9 1 1.48 0.640000 0.947200
9 2 2.37 0.400000 0.948000
9 3 3.9 0.243478 0.949565
9 4 6.6 0.143874 0.949565
9 5 11.5 0.082213 0.945455
9 6 21 0.045217 0.949565
9 7 39.91 0.023799 0.949803
9 8 79.83 0.011899 0.949922
9 9 169.65 0.005600 0.949985
9 10 387.77 0.002450 0.949982
9 11 969.44 0.000980 0.949996
9 12 2710 0.000350 0.948445
9 13 8820 0.000108 0.949791
9 14 35290 0.000027 0.950061
9 15 194000 0.000005 0.949596
9 16 1940000 0.000000 0.949596
10 1 1.58 0.600000 0.948000
10 2 2.71 0.350000 0.948500
10 3 4.8 0.197826 0.949565
10 4 8.8 0.107905 0.949565
10 5 16.8 0.056522 0.949565
10 6 33.61 0.028261 0.949848
10 7 70.96 0.013387 0.949922
10 8 159.67 0.005950 0.949982
10 9 387.77 0.002450 0.949982
10 10 1030 0.000919 0.946258
10 11 3100 0.000306 0.949320
10 12 10860 0.000087 0.950195
10 13 47050 0.000020 0.949993
10 14 282000 0.000003 0.948984
10 15 3110000 0.000000 0.951431
11 1 1.69 0.560000 0.946400
11 2 3.13 0.303333 0.949433
11 3 6 0.158261 0.949565
11 4 12 0.079130 0.949565
11 5 25.21 0.037681 0.949942
11 6 56.02 0.016957 0.949904
11 7 133.06 0.007140 0.949994
11 8 342.15 0.002777 0.949982
11 9 969.44 0.000980 0.949996
11 10 3100 0.000306 0.949320
11 11 11630 0.000082 0.949729
11 12 54290 0.000017 0.950020
11 13 353000 0.000003 0.950330
11 14 4230000 0.000000 0.948984
12 1 1.82 0.520000 0.946400
12 2 3.65 0.260000 0.949000
12 3 7.63 0.124348 0.948774
12 4 16.8 0.056522 0.949565
12 5 39.21 0.024224 0.949807
12 6 98.04 0.009689 0.949953
12 7 266.12 0.003570 0.949994
12 8 798.36 0.001190 0.949994
12 9 2710 0.000350 0.948445
12 10 10860 0.000087 0.950195
12 11 54290 0.000017 0.950020
12 12 380000 0.000002 0.949945
12 13 4940000 0.000000 0.949945
13 1 1.97 0.480000 0.945600
13 2 4.31 0.220000 0.948200
13 3 9.93 0.095652 0.949826
13 4 24.27 0.039130 0.949696
13 5 63.72 0.014907 0.949863
13 6 182.08 0.005217 0.949983
13 7 576.59 0.001648 0.949988
13 8 2080 0.000458 0.951945
13 9 8820 0.000108 0.949791
13 10 47050 0.000020 0.949993
13 11 353000 0.000003 0.950330
13 12 4940000 0.000000 0.949945
14 1 2.15 0.440000 0.946000
14 2 5.18 0.183333 0.949667
14 3 13.24 0.071739 0.949826
14 4 36.41 0.026087 0.949826
14 5 109.25 0.008696 0.950000
14 6 364.16 0.002609 0.949983
14 7 1380 0.000686 0.947368
14 8 6230 0.000153 0.950420
14 9 35290 0.000027 0.950061
14 10 282000 0.000003 0.948984
14 11 4230000 0.000000 0.948984
15 1 2.51 0.400000 1.004000
15 2 6.33 0.150000 0.949500
15 3 18.2 0.052174 0.949565
15 4 57.22 0.016601 0.949897
15 5 200.29 0.004743 0.949992
15 6 801.16 0.001186 0.949992
15 7 3810 0.000250 0.951113
15 8 22830 0.000042 0.949865
15 9 193000 0.000005 0.944701
15 10 3100000 0.000000 0.948372
16 1 2.63 0.360000 0.946800
16 2 7.91 0.120000 0.949200
16 3 26.01 0.036522 0.949930
16 4 95.37 0.009960 0.949930
16 5 458 0.002372 1.086166
16 6 2000 0.000474 0.948617
16 7 12690 0.000075 0.950364
16 8 114000 0.000008 0.948617
16 9 1940000 0.000000 0.949596
17 1 2.96 0.320000 0.947200
17 2 10.17 0.093333 0.949200
17 3 39.01 0.024348 0.949809
17 4 171.67 0.005534 0.949953
17 5 901.31 0.001054 0.949997
17 6 6010 0.000158 0.950198
17 7 57080 0.000017 0.949948
17 8 1030000 0.000001 0.952315
18 1 3.39 0.280000 0.949200
18 2 13.57 0.070000 0.949900
18 3 62.42 0.015217 0.949870
18 4 343.35 0.002767 0.949980
18 5 2400 0.000395 0.948617
18 6 24030 0.000040 0.949802
18 7 457000 0.000002 0.950697
19 1 3.95 0.240000 0.948000
19 2 19 0.050000 0.950000
19 3 109.25 0.008696 0.950000
19 4 801.16 0.001186 0.949992
19 5 8410 0.000113 0.949746
19 6 168000 0.000006 0.948617
20 1 4.75 0.200000 0.950000
20 2 28.5 0.033333 0.950000
20 3 218.5 0.004348 0.950000
20 4 2400 0.000395 0.948617
20 5 50470 0.000019 0.949934
21 1 5.93 0.160000 0.948800
21 2 47.5 0.020000 0.950000
21 3 546.25 0.001739 0.950000
21 4 12020 0.000079 0.950198
22 1 7.91 0.120000 0.949200
22 2 95 0.010000 0.950000
22 3 2190 0.000435 0.952174
23 1 11.87 0.080000 0.949600
23 2 285 0.003333 0.950000
24 1 23.75 0.040000 0.950000

The table above shows that in cases of 2 to 24 mines, the expected return is always close to 95%. In general, the expected return for the first pick is close to 95% and for all subsequent picks is close to 100%.

An exception is the case of one mine. There, the expected return is 96% for one pick, close to 95% for 24 picks, and much less, as low as 41% for 2 to 23 picks.

My advice is this game is to either choose at least two mines and play as long as like, or one mine and either stop at one diamond or go all the way to all 24.