# Strategie

### Aus MinesweeperWiki

Minesweeper strategy is the art of solving games. Techniques include learning patterns and where to click first, using guessing tactics and developing efficient clicking and mouse movement.

## Inhaltsverzeichnis |

## Patterns

A pattern is a common arrangement of numbers that has only one solution. If you memorise a pattern it will reduce the amount of time you waste thinking.

Before you start learning patterns, you should learn the basics. If a number is touching the same number of squares, then the squares are all mines. Here are some examples:

There are two basic patterns which combine to make all other patterns. The first is 1-1 and the second is 1-2. Whenever you see a 1-1 pattern starting from an edge (or where an opened square functions as an edge) the 3rd square over is empty. This makes sense because the first 1 touches two squares which must contain the mine, while the second 1 touches a third square, which must be empty. Whenever you see a 1-2 pattern the 3rd square over is always a mine. This makes sense because the first 1 touches two squares which must contain the mine, while the 2 touches a third square, which must contain the second mine. Here are some examples:

The two most famous patterns are 1-2-1 and 1-2-2-1. These are so common new players should memorise them immediately. If you look carefully they are just combinations of the 1-2 pattern.

At first it seems like there are many patterns. If you study them, they are actually 1-2-1 and 1-2-2-1 patterns or combinations. These in turn are variations of the basic 1-2 pattern. Each set of numbers reduces when you subtract known mines. Here are some final examples:

## Guessing

Sometimes in Minesweeper you have to guess. A typical case is a 50/50 situation where one mine is hidden in two squares. Guess quickly and move on. Thinking does not improve your chance of guessing correctly and wastes time. Waiting to see if you guessed right also wastes time. You will know soon enough if you are wrong! Do not delay taking these forced guesses - solving the rest of the board first is a waste of time unless you are going to guess correctly.

Many players get impatient and guess instead of solving. Do not guess unless it is necessary. The fastest way to solve 'Example A' is to click the unopened squares in a row. But if you click fast there is no time to react, so you will lose if the middle square is a mine. You have just guessed for no reason! A smart player will click the outer two squares first, which allows enough time to react to the initial click and decide if there is a mine.

Opening safe squares is as important as finding mines. If you can prove a square is safe, open it instead of guessing where the mine is. In 'Example B' there is a mine in the two yellow squares. Instead of guessing, open the safe 3rd square. This can allow you to open even more squares (marked blue) which may help you solve the original guess.

Often you can improve your chance of guessing right. There might be an arrangement of numbers with more than one solution, and the solutions require different amounts of mines. Instead of guessing, you can solve it by flagging the rest of the board and seeing how many mines are left. If you insist on guessing, think about the mine density of the level you are playing. For example, the solution with more mines is more likely on Expert than on Intermediate. Still keep in mind the density of each level is pretty low, so less dense solutions are more common overall.

Perhaps you have solved part of a board and need to guess in order to reach the rest of the board. You can improve your chance of surviving by clicking randomly! The average chance of hitting a mine is 0.206 on Expert and 0.156 on Intermediate and Beginner. These odds are much better than a 50/50 guess. Remember you are more likely to get openings by clicking on edges. Your bravery is often rewarded by finding that the original 'guess' becomes solveable when approached from a different direction.

Another important thing to remember is usefulness. If two solutions are equally likely, choose the one that will help most if it is correct. Sometimes one solution prevents another guess or creates an easier arrangement of mines.

When a guess has more empty squares than mines involved, it is always better to guess an empty square instead of guessing the mine. Flaggers often make the mistake of guessing the mine because they love to chord. Another mistake is to choose an empty square that creates a new guessing situation, such as turning a 33/66 problem into a 50/50 problem. Guess the empty square that solves the original problem.

Always choose the most likely solution. This can be very difficult to calculate! Sean Barrett has written Minesweeper Advanced Tactics as a guide. Local probability is easy to calculate but is usually wrong. For example, in the image below some squares are both 50/50 and 66/33 guesses! When all unsolved areas are considered, a simple 50/50 guess often has one square much more likely to contain the mine. A general rule of thumb is that if one square in a 50/50 situation touches a high number, it is more likely to be a mine than the other square.

The following example illustrates many of the above points. It looks like there are three unavoidable 50/50 guesses, and two unavoidable 66/33 guesses. One strategy is to guess quickly and hope for the best. This option will give the best score if you survive. A second strategy is to click a random square that does not touch any numbers. This usually has better odds of being safe and often helps solve the game. A third strategy is to determine the number of mines remaining by flagging the rest of the board. This reduces the number of solutions. In this example there are 79 possible solutions but only 2 contain 4 mines. A fourth strategy is to guess in the most useful place. Clicking square **I** has the potential to eliminate all the other guesses! For example, if it is a 4 or 7 the game can be solved no matter how many mines remain. A fifth strategy is to guess the most likely solution. A mine is more likely in **L** than **K** and more likely in **H** than **D**. A final strategy is to calculate the exact probability of each square taking the entire game into consideration. This is the hardest but most accurate method. Results for this example are available.

The guessing strategy you choose depends on whether you want to win more games or only fast games. Some strategies improve your chance of winning but are time consuming. Others are fast but risky.

## First Click

The first click in Minesweeper is always safe, but where is the best place to start? It depends whether you want quantity or quality.

Your best chance of finding an opening is in a corner, then on an edge, then in the middle. Emmanuel Brunelliere (France) calculated the theoretical odds as follows:

Beginner
| Intermediate
| Expert
| |

Corner
| 59.54% | 59.94% | 49.94% |

Edge
| 42.14% | 42.61% | 31.42% |

Middle
| 25.09% | 25.54% | 15.69% |

Tim Kostka then used his knowledge of Board Cycles to find the actual chance of finding openings on Windows Minesweeper. The first click is always safe because any mine is moved to the top left corner or nearest empty square to its right. This means the top left corner gives fewer openings than the other corners. It also means fewer openings result from the edge and middle squares touching the top left corner. Exact values for each square are on his website. Most of the variation is due to low outlier values near the top left corner.

Beginner
| Intermediate
| Expert
| |

Corner
| 50 - 60 % | 50 - 60 % | 40 - 50 % |

Edge
| 34 - 42 % | 36 - 43 % | 25 - 32 % |

Middle
| 19 - 24 % | 21 - 26 % | 12 - 16 % |

Your best chance of getting a *large* opening is in the middle, then on an edge, then in a corner. So far no one has calculated the theoretical advantage, but Tim collected actual results from Windows Minesweeper. The biggest openings occur in the very center of the board and decrease as you approach edges. The biggest openings from clicking on an edge are in the middle and decrease as you approach corners. This chart shows the variations in the average number of squares for each opening:

Beginner
| Intermediate
| Expert
| |

Corner
| 18 | 27 | 16 |

Edge
| 20 - 24 | 31 - 42 | 19 - 26 |

Middle
| 23 - 32 | 35 - 66 | 23 - 41 |

In summary, the best place to start depends on your preference for size or frequency. Large openings are more helpful but you will lose more games trying to find them. Small openings can be difficult but you will start more games. It is possible the benefits of either method cancel each other.

## Efficiency

The fewer clicks you take, the faster you will finish. Learn to be efficient.

The game ends when all safe squares are open, not when all mines are flagged. Beginners often waste time flagging every mine. The only good reason to flag is to clear more squares by chording. So before you place a flag, decide if it is useful.

Some players never flag because time spent placing flags could be better used to open more squares. This style is called No Flags, or NF. Flaggers argue that flags allow you to chord and clear multiple squares at the same time. It is generally agreed that NF is more efficient near high numbers (5,6,7,8) while Flagging is more efficient near low numbers (1,2,3,4). Near a high number like 7 a NF player needs only one click to open the safe square, but a Flagger needs seven flags and a chord. Near a low number like 1 a Flagger would place one flag and chord, but a NF player would need as many as seven clicks to open the safe squares. It is also generally agreed that NF is more efficient on low 3BV boards while Flagging is more efficient on high 3BV boards. For example, an Intermediate game with a 3BV of 40 has an average of one number touching each mine, while a 3BV of 120 has an average of three. A perfect NF player would need 40 and 120 clicks. A very inefficient and unlucky Flagger would need 80 clicks (40 flags, 40 chords) for both games. These examples are extreme cases but show the general reasoning. In reality NF players are not perfect and waste clicks at full speed, while Flaggers never need to flag all mines or chord on every number. If a player uses only NF or Flagging, there is probably no advantage to either method. The advantage comes when both techniques are combined and the player uses the most efficient solution for each situation. Often games can be solved in less clicks than its 3BV!

If you flag you can save time by using the 1.5 Click technique. The traditional way is moving the right button down and up to flag, and both buttons down and up to chord. The new method moves the right button down to flag, the left button down, and both up to chord. This eliminates one movement from every flag and chord combination. As long as the right button starts going down before the left button, the flag will get placed. The shorter the gap, the more time you save. You can nearly double your flagging speed with this method.

Here are some examples of efficient flagging:

Here are some examples of efficient NF:

It is usually easy to tell whether NF or Flagging is better in a given situation, but here is a complicated example:

Solving the whole board requires thinking ahead. If the yellow square is an opening, the most efficient start is to click on it. Then click the orange square. If it is a 3, 4 or 5 you are stuck. If it is a 2 click the green square. If the green square is a 3, 4, 5, 6 or 7 you are stuck, but if it is a 2 click on the red square for an opening. Apply the same strategy to the brown, purple and black squares. Total clicks vary from 3 to 7. |

Fritz Löhr has made an excellent demonstration of an Intermediate game being solved efficiently with explanations.

An important way to increase solving speed is to make fewer mouse movements. It takes time to move your mouse. New players often follow their eyes with the mouse instead of moving it intentionally toward a target. The next stage in reducing movement is learning to 'see' the solved board. This often allows you to solve at your current mouse location. If your mouse is near the 2 in 'Example A' you can flag the red square and chord instantly. This is obvious to a professional player because they have solved the adjacent squares in their head. A new player would have to move elsewhere and come back later. The red square in 'Example B' can be similarly solved. Less movement equals better scores.

The official statistic of efficiency is IOE, which compares the number of clicks taken to the 3BV of the board. Both Clone and Arbiter save IOE highscores as an incentive for improvement. Arbiter further breaks IOE into Correctness (clicks that changed the board) and Throughput (the potential IOE if all clicks had been correct). It also has a Path statistic that measures mouse movement in pixels.

The best way to improve efficiency is to play slowly. Find the most efficient solution and path to each problem before pressing any buttons. You will soon see improvements while playing at full speed.

## More Tips

- Do not use Questionmarks.
- Press 'F2' to start new games. Keep one finger on this button, it is faster than using the mouse.
- Avoid moving the mouse without a reason. New players often waste time moving the mouse everywhere their eyes look.
- Ignore the clock. Looking at the clock during a game wastes time, and will make you nervous if you are going fast.
- Many players listen to music while they play. This distracts them and lets them play on autopilot without nerves.
- Play in a warm room or heat your hands in hot water before you play. This increases blood flow and reaction time.
- Take short exercise breaks to increase blood flow and stimulate your brain.
- After a long playing session, it can help to change the version you are playing. This helps focus your eyes.
- If you accidentally click down on a mine, slide onto a different square before releasing the mouse button.
- Use the 1.5 Click.

## Links

- Strategy - Jim Loy was first to mention the 1-1 pattern (1996)
- Minesweeper Tips - Brian Chu was first to discuss the 1-2 pattern (1997)
- Minesweeper Page - Frank Wester wrote the first serious strategy guide (1997)
- Minesweeper Advanced Tactics - Calculating minesweeper probability, by Sean Barrett (1999)
- First Click - Emmanuel Brunelliere calculates the best theoretical place to start (2003)
- Minesweeper Tips - Various tips and strategy by Grégoire Duffez (2006)
- On the First Click - Tim Kostka determines the best actual place to start (2006)