Last Updated: May 7, 2015

# Minesweeper

## Introduction

This is a simple game based on the classic computer game minesweeper. It involves a grid of squares with one bomb on each row. The goal is the pick the column without the bomb at each step. The player can quit at any time or press his luck and keep going.

## Rules

1. First, the player must pick his game field. Following are the options. The first number is the number of columns, or choices at each step. The second is the number of row, or the number of decisions the player will need to make to reach the end. The default choice is 2x3.

• 2x3
• 3x6
• 4x9
• 5x12
• 6x15

2. The player must then make a wager.
3. With each row the player must choose one of the columns.
4. In each row there is a bomb hidden in one of the columns.
5. If the player chooses the column with the bomb, he loses the game and his wager. If the player chooses to play again he will do some from the same size field as chosen in step 1.
6. If the player does not choose the column with the bomb, then he will advance another row and his running balance will increase.
7. After advancing each new row, the player may try to advance again or quit and collect his winnings thus far.
8. An exception to the above rule is when the player reaches the end of the board, in which case he collects his winnings automatically.
9. The pay table is shown below.

Following is the pay table. It shows the win by the number of completed successful picks along the left row by the board size along the top column.

### Pay Table

Row 2x3 3x6 4x9 5x12 6x15
1 1.96 1.47 1.31 1.23 1.18
2 3.85 2.17 1.71 1.5 1.39
3 7.55 3.19 2.24 1.84 1.63
4 4.69 2.93 2.26 1.92
5 6.9 3.83 2.77 2.26
6 10.1 5.01 3.4 2.66
7 6.55 4.17 3.13
8 8.57 5.11 3.69
9 11.2 6.27 4.34
10 7.69 5.11
11 9.43 6.02
12 11.5 7.08
13 8.34
14 9.82
15 11.5

## Analysis

The following table shows the probability of attaining each number of possible successful picks, assuming the player always tries to reach the end.

Successful
Picks
2x3 3x6 4x9 5x12 6x15
0 0.500000 0.333333 0.250000 0.200000 0.166667
1 0.250000 0.222222 0.187500 0.160000 0.138889
2 0.125000 0.148148 0.140625 0.128000 0.115741
3 0.125000 0.098765 0.105469 0.102400 0.096451
4 0.065844 0.079102 0.081920 0.080376
5 0.043896 0.059326 0.065536 0.066980
6 0.087791 0.044495 0.052429 0.055816
7 0.033371 0.041943 0.046514
8 0.025028 0.033554 0.038761
9 0.075085 0.026844 0.032301
10 0.021475 0.026918
11 0.017180 0.022431
12 0.068719 0.018693
13 0.015577
14 0.012981
15 0.064905
Total 1.000000 1.000000 1.000000 1.000000 1.000000

The next table shows the expected return of moving forward at each decision point. The row column represents the row you are trying to successfully advance to. The "expected return" is defined as the ratio of expected money won to money bet at that point.

### Marginal Return

Successful
Picks
2x3 3x6 4x9 5x12 6x15
1 0.980000 0.980000 0.982500 0.984000 0.983333
2 0.982143 0.984127 0.979008 0.975610 0.981638
3 0.980519 0.980031 0.982456 0.981333 0.977218
4 0.980146 0.981027 0.982609 0.981595
5 0.980810 0.980375 0.980531 0.980903
6 0.975845 0.981070 0.981949 0.980826
7 0.980539 0.981176 0.980576
8 0.981298 0.980336 0.982428
9 0.980163 0.981605 0.980126
10 0.981180 0.981183
11 0.981014 0.981735
12 0.975610 0.980066
13 0.981638
14 0.981215
15 0.975900

The next table shows the overall expected ratio of all money bet to all money won by each stopping point goal. For example, the value for the 6x15 game with a stopping goal of row 10 is 0.981183. This means that if the player plays until he makes 10 correct picks, or dies trying, then the ratio of all the wins along the way to all the bets along the way is 0.981183.

### Overall Return

Successful
Picks
2x3 3x6 4x9 5x12 6x15
1 0.980000 0.980000 0.982500 0.984000 0.983333
2 0.981061 0.982043 0.980769 0.979839 0.981638
3 0.980884 0.981384 0.981320 0.980326 0.977218
4 0.981083 0.981249 0.980879 0.981595
5 0.981031 0.981081 0.980812 0.980903
6 0.980208 0.981079 0.980993 0.980826
7 0.981006 0.981018 0.980576
8 0.981040 0.980938 0.982428
9 0.980950 0.981006 0.980126
10 0.981022 0.981183
11 0.981022 0.981735
12 0.980617 0.980066
13 0.981638
14 0.981215
15 0.975900

## Strategy

If your goal is to maximize total money won to total money bet, the maximum return is 98.40% for the 5x12 board and stopping after just one successful pick.

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