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

The game is played on a 5x5 grid.
The player starts by making a bet and choosing a number of mines to be hidden in the grid, from 1 to 24.
Spaces in the grid not occupided by a mine will have a diamond.
Then, the player will pick spaces in the grid, with the object of avoiding the mines.
As the player successfully find diamonds, his win will usually increase.
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.

Written by: Michael Shackleford

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