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Encyclopedia > Minesweeper (computer game)
The game begins when the user clicks on a blank square.
A finished game

Minesweeper is a single-player computer game. The object of the game is to clear an abstract minefield without detonating a mine. The game has been rewritten for nearly every system platform in use today. The most well-known version comes bundled with later versions of Microsoft Windows. Image File history File links Minesweeper_genric. ... Image File history File links Minesweeper_genric. ... Image File history File links Minesweeper_diagram_end. ... Image File history File links Minesweeper_diagram_end. ... For information on interactive gaming in general, see video game. ... â€œMinefieldâ€ redirects here. ... â€œMinefieldâ€ redirects here. ... In computing, a platform describes some sort of framework, either in hardware or software, which allows software to run. ... â€œWindowsâ€ redirects here. ...

When the game is started, the player is presented by a grid of blank squares. The size of the grid is dependent on the skill level chosen by the player, with higher skill levels having larger grids. If the player clicks on a square without a mine, a digit is revealed in that square, the digit indicating the number of adjacent squares (typically, out of the possible 8) which contain mines. By using logic, players can in many instances use this information to deduce that certain other squares are mine-free (or mine-filled), and proceed to click on additional squares to clear them or mark them with flag graphics to indicate the presence of a mine.

The player can place a flag graphic on any square believed to contain a mine by right-clicking on the square. Right-clicking on a square that is flagged will change the flag graphic into a question mark to indicate that the square may or may not contain a mine. Right-clicking on a square marked with a question mark will set the square back to its original state. Squares marked with a flag cannot be cleared by left-clicking on them, though question marks can be cleared as easily as normal squares. The third question mark state is often deemed unnecessary and can be disabled so that right clicking on a flagged mine will set it back to its original state right away so mines flagged in error can be corrected with one right-click instead of two.

Clicking the left and right buttons at the same time on a number having as many adjacent flags as the value of the number reveals all the unmarked squares neighboring the number; however, one forfeits the game should the flags be placed in error. This method is a very useful tool when trying to beat a high score. Some of those implementations also allow the player to move the mouse with the right mouse-button held down after marking mines; the player can then left-click on multiple numbered squares while dragging with the right mouse-button, in order to clear large areas in a short time. As an alternative to clicking both buttons at the same time players can also middle-click or shift-click on fully-flagged numbers.

Some implementations of minesweeper "cheat" in favor of the player by never placing a mine on the first square clicked; some also change the board so the solution does not require guessing.

## History

### Gambling game

The earliest form of Minesweeper was a gambling game that was briefly popular during the 1950s. The player would buy a punchboard, which was made of paper and cardboard. The top layer was cardboard with a square lattice of holes, each of which corresponded to one of the buttons in the present-day Minesweeper game. The holes were blocked by a layer of paper underneath. The player could choose to punch the paper through any hole, and would then see what was printed on a third layer of cardboard underneath that hole. It revealed color-coded information equivalent to that which Minesweeper reveals from a button-click. If the player succeeded in punching a sufficient number of holes without punching a mined hole, he/she could return the board to the manufacturer for a prize.

These punchboards were sold in restaurants and bars, and probably made the manufacturer an enormous fortune before they became illegal. A familiar sound in a restaurant at that time was the wood-pecker sound of holes being punched at a rapid rate (usually with very little time for thought), which ended suddenly with a loud, vulgar shout.

### Computer game

The earliest known ancestor of Minesweeper as a computer game is Cube, found in the PDP-11 program library catalogue and credited only as "CONVERTED TO RSTS/E BY DAVID AHL, DIGITAL" [1] (referring to David H. Ahl). Cube was played in a 3x3x3 cube with 5 mines, where the player had to find their way from one corner (1,1,1) to the opposite corner (3,3,3). The player entered the co-ordinates of the next square they wished to explore. If the target was more than one square away or there was a mine there, the player lost. No information about the number of surrounding mines was given. The PDP-11 was a 16-bit minicomputer sold by Digital Equipment Corp. ... David H. Ahl is the founder of Creative Computing magazine. ...

The basic gameplay style became a popular but minor part of the puzzle game genre during the 1980s, with such titles as Mined-Out (Quicksilva, 1983), and Yomp (Virgin Interactive, 1983). Cube was further succeeded by Relentless Logic (or RLogic for short), by Conway, Hong, and Smith, which was available for MS-DOS as early as 1985. In RLogic, the player is a private in the United States Marine Corps, delivering an important message to the U.S. Command Center. RLogic is more similar to Minesweeper than to Cube in concept, but a number of differences exist: A puzzle is a problem or enigma presented as entertainment; that is written down, acted out, etc. ... Quicksilva magazine advert from 1984. ... Virgin Interactive was a successful and influential British video game publisher. ... The United States Marine Corps (USMC) is a branch of the United States military responsible for providing power projection from the sea,[1] utilizing the mobility of the U.S. Navy to rapidly deliver combined-arms task forces. ...

• In RLogic, the player must navigate through the minefield, from the top left corner to the bottom right corner (the Command Center).
• It is not necessary to clear all non-mine squares. Also, there is no mechanism for marking mines or counting the number of mines found.
• The number of steps taken is counted. Although no high score functionality is included, players could attempt to beat their personal best score for a given number of mines.
• Unlike Minesweeper, the size of the minefield is fixed. However, the player may still specify the number of mines.
• Because the player must navigate through the minefield, it is sometimes impossible to win — namely, when the mines block all possible paths.

## Game analysis

### Patterns and solving

There are many patterns of numbered squares that may arise during a game that can be recognized as allowing only one possible configuration of mines in their vicinity. In the interest of finishing quickly, it is often easiest to process the known patterns first, and continue on with the uncertain parts later. There are a few broad methods for solving problems in minesweeper games without guessing.

• $dagger$ denotes a square with a mine
• $boxdot$ is an unopened square
• Letters are unopened squares of special interest

#### Single-square analysis

Example case 2
$begin{array}{|c|c|c|c|} hline & 3 & dagger & boxdot hline & a & b & boxdot hline 1 & boxdot & boxdot & boxdot hline end{array}$
a and b must be mines, because the 3 is missing 2 mines, and the only squares that can provide with those mines are a and b.
Example case 1
$begin{array}{|c|c|c|c|} hline dagger & 3 & dagger & boxdot hline a & b & dagger & boxdot hline boxdot & boxdot & boxdot & boxdot hline end{array}$
a and b are safe to open, because the 3 is already satisfied with 3 mines.

There are two special cases that are of extra interest when solving a board. In Programmer's Minesweeper,[2] this is called "single point strategy".

• An opened square is satisfied if the number on the square is equal to discovered mines around it. It means that the rest of the unopened squares around must be safe.
• If a square needs more mines around it, and this number equals the number of adjacent unopened squares, then all these are mines.

#### Multiple square analysis

To solve more complex puzzles, one needs to consider more than one square at a time. One method that are commonly used in minesweeper AIs is to consider the board as a constraint satisfaction problem. AI redirects here. ... Constraint-satisfaction problems or CSPs are mathematical problems where one must find states or objects in a system that satisfy a number of constraints or criteria. ...

The variables, or unknowns, in minesweeper are the unopened squares, and the constraints are the adjacent squares that are opened. The algorithm consists of trying every combination of mines that satisfies all the numbers in the adjacent squares, and making a conclusion from there. For large puzzles, this is a time-consuming process for a computer, but expert minesweepers might be able to quickly see which squares need this procedure, and where one might expect it to succeed. The two rules above are such special cases.

Example
$begin{array}{|c|c|c|c|} hline & 1 & a & boxdot hline 1 & 2 & b & boxdot hline c & d & e & boxdot hline boxdot & boxdot & boxdot & boxdot hline end{array}$

Example: A corner square and the 3 adjacent squares have been opened, and the numbers given revealed. The letters here are unopened squares and they are our variables.

Blindly trying every combination gives the 4 valid configurations (out of 25), namely {a,b,c,d,e} = {1,0,1,0,0}, {0,1,1,0,0}, {1,0,0,1,0} and {0,1,0,1,0}, where 1 represents a mine.

The only common number in all these configurations is that the variable e is never a mine. The conclusion is that in all possible valid configurations, e is safe, and one can safely open that square. Analogously, if a square is marked as mine in every valid combination, then the square must be a mine.

One can also think of this as a system of equations, where the variables must be in {0,1}. In the above example, the constraints gives that a+b=1, c+d=1 and a+b+c+d+e=2. The third equation can be reduced to 1+1+e=2 and hence the square e must be safe. This strategy is more similar to the human approach, but is harder to implement as a computer program.

#### Final analysis

Used at the end of a game, this can be used to clear a square when all other squares on the board are either safe or can be shown to be mines. Often these final squares are on walls or in corners.

In some versions of the game the number of mines on the field is known. Near the end when almost all the tiles are lifted, knowing the number of mines remaining can give some insight to otherwise unresolvable patterns.

### Not always solvable without guessing

Minesweeper is not always solvable without guessing. For instance, in the following situation:

( represents a mine, and the numbers are the standard Minesweeper numbers. The position is at the bottom of the board.) Image File history File links Minesweeper_guess_1_generic. ... Image File history File links Minesweeper_flag_generic. ...

The player must guess which of the two squares marked with a ? is a mine.

The constraint satisfaction problem above might help a little to estimate the likelihood that a square is a mine; list all the valid combinations and count how many times each square is occupied by a mine. If the density of mines is known (or estimated during the game), the player can pick the square that is least likely to contain a mine.

Another apparent instance of required guessing is when an unclicked square is completely surrounded by either (1) mines, or (2) a combination of mines and the perimeter of the game window (the latter being much more common). In this case, since no numbers touch the unclicked square, a player has no information about the likelihood of the unclicked square being a mine. However, there is still a good strategy when facing this situation that will allow the player to avoid simple guessing: simply play the rest of the game and ignore this square. If the spot is in fact a mine, it will be automatically flagged when all other squares in the game window have been either clicked or flagged by the player. If the spot is not a mine, it will not be automatically flagged, and the player will be able to safely click it in the knowledge that it is not a mine. This only happens in some implementations of the game.

A few variants specifically focus on getting this aspect out of the game. At the simplest level, some programs give the solution away any time a guess is needed. Another one furthered the design and demands that the player decides if he or she has to guess or not. The resulting problem is always decidable (within an extended mathematical space). Yet another simply lets any guess the user makes (when a guess is necessary) automatically be the correct one.

### NP-completeness

In 2000, Kaye published a proof that it is NP-complete to determine whether a position in a Minesweeper game is consistent with some placement of mines. [3] Minesweeper is now mentioned in the Clay Mathematics Institute's unofficial description of the P versus NP problem. [4] In complexity theory, the NP-complete problems are the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. The reason is that if you could find a way to solve an NP-complete problem quickly, then you could use... The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Cambridge, Massachusetts. ... Diagram of complexity classes provided that P â‰  NP. The existence of problems outside both P and NP-complete in this case was established by Ladner. ...

### Mine probabilities must be balanced against rewards

If "playing Minesweeper perfectly" means finding a strategy that ensures the best probability of solving a random board, then there is more to playing perfectly than just choosing squares with lowest mines probabilities. Consider the following situation:

 $begin{array}{|c|c|c|c|c|c|} hline 2 & dagger & dagger & dagger & 3 & 1 hline 2 & dagger & a & dagger & 5 & dagger hline 2 & 4 & b & d & e & dagger hline 2 & dagger & c & dagger & 5 & dagger hline 2 & dagger & dagger & dagger & 3 & 1 hline end{array}$

( $dagger$ represents a mine, and the numbers are the standard Minesweeper numbers; a, b, c, d and e are the unknown positions. The other spaces/mines on the board are insignificant).

There is 2/3 probability of a mine on a, b, or c and 1/2 probability of mine on d or e; this can be derived by computing the six possibilities of mine placement on a...e. But playing d or e will give the player no useful information: if the player does not trigger a mine, he or she will see a 6 appear under e, or a 5 appear under d. Overall, playing d or e will let the player solve the area in only 1 of the 6 possible cases. If he or she plays a (or b or c) and he or she does not step on a mine, he or she will immediately know whether there is a mine on d or not; overall the player would solve the area in 2 of the 6 possible cases. So the moves a, b, or c, with the highest immediate danger, turn out to be the best in the long run.

This is a specific example of a more general principle that applies when prioritising squares: an unknown square should not be clicked on if more information may be gained by first clicking on an adjacent square; conversely, if there is no way to gain more information about a square, then a guess is inevitable and it should be clicked on to provide more information about the rest of the area.

## Measuring board difficulty

Beginner board with a 3BV of 25

The difficulty of a given minesweeper board is often measured using the 3BV measure (abbreviated from Bechtel's Board Benchmark Value). Image File history File links Minesweeper_diagram_3BV_25. ... Image File history File links Minesweeper_diagram_3BV_25. ...

#### Method

The 3BV of a board names the minimum number of left clicks required to open up all squares without a mine of a Minesweeper field.

• Each opening of a board counts as 1 3BV (white dots on the pictures).
• Each square without a mine but a number which is not a border (white lines) of an opening counts as 1 3BV (green dots on the pictures).

The sum of the 3BV is the 3BV of the whole board.

#### 3BV/s

3BV/s stands for 3BV per second.

• Formula: 3BV/s = 3BV ⁄ (time−1)

The subtraction of one from the time is required due to the fact that minesweeper begins with one second on the clock (as opposed to zero) and as such the time shown is always one second greater than the actual time taken. Thus, for example, if a Minesweeper board with a 3BV of 16 is finished with the clock displaying 9 seconds, the 3BV/s is 16⁄(9−1) = 2.

Because the time that is needed to finish a Minesweeper board depends highly on the difficulty of the board, it may not be the best way to compare records. 3BV/s on the other hand does consider the difficulty of the Minesweeper board as well as the time needed to finish it. Among the best Minesweeper players, 3BV/s records are not nearly as important as time records, but they give a picture of how fast someone can play with regard to mouse-handling.

If flags are marked, it is possible to require fewer clicks than the 3BV of the respective board. Using only left clicks is called non-flagging (nf) whereas marking mines with right-clicks is called flagging-style.

## Best times

The International Minesweeper Committee has compiled a "best ever" list[5] which includes videos of the fastest games submitted by players. In order to get on that list, records on beginner, intermediate and expert must add up to no more than 99.

• 37 on expert by Dion Tiu
• 10 on intermediate by Dion Tiu (and disputed scores of 9 and 10 by Jake Warner)
• 1 on beginner, tied with many players, some of whom have played games where one click reveals the entire board instantly.

## Variants

There are variations of Minesweeper available for download at various places on the Internet. These are generally differently shaped minefields in two and three dimensions, or various 2D layouts (such as triangular or hexagonal grids). For example, X11-based XBomb adds triangular and hexagonal grids, and Professional Minesweeper for Windows includes these and many others. In computing, the X Window System (commonly X11 or X) is a windowing system for bitmap displays. ...

• There is a game called "Nonosweeper", which applies Minesweeper-style graphics to a nonogram game. It shows a grid with groupings of numbers on the right side and bottom side. These numbers indicate clusters of mines. An example might be 2 1 2 3, denoting that there are clusters of 2, 1, 2, and 3 mines each separated by at least one empty space.
• MineSweeper3D is a 3D version of the classic Minesweeper. The rules are the same, but the game takes place on the surface of a three-dimensional model rather than on a flat grid.
• Hexmines was a variant on a hexagonal grid created by Macintosh developer Ingemar Ragnemalm. Apart from the different board geometry, it is largely identical to the original game.
• Michael Coan (COAN.NET) came up with a 2-4 player variant of the game in 2005. Super Sweeper / Frog Finder can currently be found on a few internet game sites with its popularity growing.

Example of a nonogram puzzle being solved. ... Ingemar Ragnemalm is a computer programmer, best known for writing the Sprite Animation Toolkit (SAT) and several games for the Apple Macintosh during the 1990s, including Bert. ...

## References

1. ^ http://pdp-11.trailing-edge.com/rsts11/rsts-11-013/
2. ^ Programmer's Minesweeper
3. ^ http://web.mat.bham.ac.uk/R.W.Kaye/minesw/ordmsw.htm
4. ^ http://www.claymath.org/millennium/P_vs_NP/
5. ^ http://www.planet-minesweeper.com/bestever.php

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