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MINESWEEPER FAQ

version 1.11

by Michael Spencer (michael.spencer@balliol.ox.ac.uk)

last updated 2 October 2003

Contents:

1. Overview of game

2. How to become a moderately good player

3. How to become an amazingly good player

4. Dealing with closed cells

5. Minesweeper Challenge

6. Speed tips

7. Other stuff

- - -

Version History

1.11: new best times for Expert and Intermediate modes added;

Minesweeper Challenge given its own section

1.10: three new examples added to Section 3; more information added

about the colours; more wording improvements; MINESWEEPER CHALLENGE

added

1.04: three new examples added to Sections 3 and 4; a new best

time for Expert mode has been added

1.03: new best times for all three modes have been added; a further

example has been added in Section 4; the wording has been slightly

improved in some of the tips and examples

1.02: new best time for Expert mode added

1.01: corrections have been made to one of the examples in Section 4,

which as it stood was an impossible position

1.00: first release

- - -

Overview of game

The object of the game is to find a set number of mines that have

been concealed in a rectangular grid. To do this, you must click on

the grid squares to open them up. If you open a square that contains

a mine, it will blow up and you lose. However, if you open a square

that does not contain a mine, you will be told how many mines are in

the squares adjacent to the square you opened. Using this information

it is usually possible to pinpoint the mines' locations more

accurately than you would be able to do by guesswork alone.

The game provides three difficulty levels (Beginner, Intermediate,

Expert), plus the option of creating your own. On each of the main

levels, the game will record your best time, and so the ultimate

object is to get this as low as possible.

You can mark a square as mined by right-clicking it. This will make

it easier for you to work out which squares are mined and which are

free, as you do not have to hold the mines' locations in your head.

Right-clicking a square twice will mark it with a question-mark,

which you can use to indicate uncertainty about a square. Right-

clicking a square three times returns it to its unmarked state.

The following example demonstrates the game's method:

Key: ? unopened square

- blank square

X square that contains a mine

???? Step 1. You have no information and must choose a starting

???? square. Useful fact: the FIRST square you click on

???? is always "free", so you can get started. We'll try

???? the square in the top-left corner.

---- Step 2. The top-left square was blank, i.e. not mined and

-111 not adjacent to any mines. Hence all surrounding

12?? squares were opened automatically for us. Now, the

???? square to the right of the "2" must contain a mine.

---- Step 3. Notice that the "1" inside the nook is only adjacent

-111 to one unopened square, so we have marked this as

12X? mined. The "1" above the marked square now touches a

???? known mine, so the other squares it touches are free.

---- Step 4. We have revealed a "1" next to the known mine, so,

-111 by repeating the process of Step 3, two further

12X1 squares on the bottom row are known to be free.

????

---- Step 5. Now, the "2" on the bottom row is adjacent to one

-111 known mine, and there is only one square its other

12X1 mine could be on. Once this square is marked, we can

??21 deduce that the bottom-left corner square is free.

- - -

How to become a moderately good player

First, look for places where you know there must be mines.

Here are the most common formations:

1. --1??

-11??

11X??

?????

To deduce where a mine is from a "1" square, there must be only one

unopened square touching it, so the only possible formation is the

"nook". In the diagram above, the square X must contain a mine.

2. ----1 ----1

-1221 11211

12XX? ?X3X?

????? ?????

The most common formation with only two unopened squares adjacent to

a "2" occurs when two mines are together just next to a nook. Another

possiblity is a straight edge, if the square orthogonally adjacent to

the "2" has for some reason already been opened. Both "X"s are mines.

3. ---1? --11?

2322? -13X?

XXX?? 12XX?

????? ?????

The most common formation with a "3" occurs when three mines are

touching along a straight edge, as in the first diagram. Also

important is the case where a "3" sits inside a corner, and so is

only adjacent to three squares, as in the second diagram.

4. ---1?

--12?

124X?

?XXX?

The most common formation with a "4" (which is, however, not very

common) requires a nook just one square away from a straight edge,

and one mine in the nook and three next to it along the edge.

5. --1?? ----1

--2X? 11211

125X? ?X5X?

?XXX? ?XXX?

With a "5", two important formations are the outside corner, and the

straight edge with a "5" jutting out from it.

Once you have marked all known mines in this manner, the next step is

to look for numbered squares whose quota of mines is already filled.

For example, in the left-hand "5" diagram above, once you have marked

all five mines, the "2" just to the left of the "5" is adjacent to

two known mines, and so the bottom-left corner square is free.

By opening all squares you find that are known to be free, and then

marking any new mines that you can definitely locate using your new

information, and repeating the process, you should be able to

regularly complete the Beginner and Intermediate levels. Sometimes,

of course, you will just be unlucky and not be able to find out

enough information. Just don't get upset if this happens!

- - -

How to become an amazingly good player

PART ONE

With an enormous amount of luck, the Expert level as well can be

completed with just the techniques described above. However, just one

addition to these will make a huge difference, and you should be able

to complete Expert with at least a reasonable success rate - although

you will still often have bad luck or reach an impossible position.

--1??

111??

ABC??

?????

Imagine you get this formation in the top-left corner of a level. The

squares labelled "A", "B", and "C" are all unopened; I have labelled

them merely for convenience.

The "1" immediately above square A is adjacent to exactly one mine;

in other words, exactly one of the squares A and B contains a mine.

But the "1" immediately above square B is also adjacent to this mine,

whether it turns out to be in A or B! Therefore, without knowing

whether it is square A or B that is mined, we can deduce that square

C is free!

Of course, if we had the identical formation with a "2" instead of a

"1" above B, we would know instead that square C contained a mine.

Before reading on, read through this carefully until you understand

completely the logical steps taken in the last four paragraphs.

The core idea is that we can make a deduction of the following type:

"exactly one of these two specific squares contains a mine". There

are, in fact, enormously many more formations than the above in which

this type of deduction can be made - and can prove useful.

PART TWO

The following technique is a more elaborate version of that described

above.

----1

11221

ABCD?

?????

Again, the labels "A", "B", "C", and "D" are merely for convenience.

The "2" immediately above square C tells us that exactly two of

squares B, C, D contain mines. But it cannot be the case that both

B and C contain mines: if they did, the "1" above B would be adjacent

to two mines, which is impossible. Therefore the two mines are in

square D and one of squares B and C. This in turn tells us that

square A is free - because the one mine adjacent to the "1" is in

either square B or square C.

Alternatively, the "1" immediately above square B tells us that

exactly one of squares A, B, C contains a mine, so the other two

must be free. But squares B and C cannot both be free: if they were,

there would be no room to place two mines adjacent to the "2". As

before, we deduce that square A is free and square D is mined.

PART THREE

Once these techiques have been grasped, there is no limit to the

complexity of the situations in which they may apply.

-1???

12???

??A??

?????

This adds merely one extra level of complexity. If this formation

occurs in a corner of the grid, each "1" is adjacent to one mine;

this accounts for both mines adjacent to the "2", so A is free.

---1?

1113?

?ABCD

?????

Here, the top "1" is adjacent to one mine, and there is one mine in

squares A, B, and C, hence at most one in B and C. Therefore square

D is mined, from which we can deduce that square A is free. In

general, something useful can be deduced when a "2" occurs next to a

"1" along an edge, or a "3" next to two "1"s around a corner.

--1?? --1?? --1??

--1?? --1A? --1A?

112?? 1121? 1121?

????? ?B1C? ?B2C?

In the first diagram, we can use the methods already described to

show that the squares immediately to the right of and below the "2"

are free. Suppose that we uncover these to reveal "1"s, as in the

second diagram. Now, only three uncovered squares (A, B, and C) are

left touching the "2", so two of these are mines. But A and C cannot

both be mines because of the "1" between them; neither can B and C,

for the same reason. Therefore A and B are the mines.

(It is to be noted that the reasoning exactly parallels that in the

case where a "1" and a "2" are adjacent along an edge. Also, if we

slightly modify the example, as in the third diagram, we can still

deduce by the same reasoning that the square B contains a mine and

that the three squares to the right of the rightmost "1" are free.)

X2-2X X2-2X

X414X X424X

X???X X???X

????? ?????

There are two special cases to be considered when just three squares

in a row are adjacent to a row of numbered squares. In the first

diagram, one of the squares next to each "4" is a mine, and the "1"

therefore means that the squares immediately below each "4" are free,

so that the mine is in the square below the "1". Conversely, in the

second diagram there are two mines, below the "4"s, while the square

below the "1" is free.

--1?? --1A?

112?? 112B?

????? ?C1D?

????? ?????

Again, in the first diagram (assuming this formation is in the corner

of the grid) we know that the square under the "2" is free. Suppose

that opening this reveals a "1". Now, exactly one of A and B contains

a mine; so exactly one of C and D contains a mine; therefore square

B and the three squares under the "1" are all free (which means that

it is A that contains the mine).

This is an example of a two-stage deduction, which is not at all

unusual; much longer chains of deduction do occur, but more rarely.

I will not give examples here, because it should be clear from the

examples already given how the methods of deduction work.

PART FOUR

Having learned these techniques, it helps to commit to memory the

following common formations in which they occur.

----1

11111

??A??

?????

(Any situtation in which two "1"s next to each other come out from

one of the edge walls.) The third square in the row below the "1"s

must be free.

----1

12111

??X??

?????

(As above, with the second square changed to a "2".) The third square

in the row below the "1"s and "2" must contain a mine.

----1

11111

??1??

?????

(A common follow-up to the first example above, if clicking on

square A revealed a "1".) The three squares below the "1" are free.

----1

11111

??1??

?122?

?????

(And a common follow-up to the follow-up.) The squares to the left

of, under, and diagonally under the lowest "1" are all free.

----1

11111

??1??

1122?

--1X?

--1??

The square marked X contains a mine. (Make sure that you can follow

this example and the previous two - the logic is the same for all of

them. Which three squares in the above diagram are certainly free?)

-----

11222

A??X?

?????

(Any situation in which a "1" and a "2" are adjacent along a straight

edge.) The square diagonally below the "1" (square A) must be free,

and the square diagonally below the "2" must contain a mine.

-----

11221

??XX?

?????

(Any situation with the combination "1221" occurring along a straight

edge.) By the reasoning above applied twice, the squares directly

under both "2"s must contain mines.

-----

11211

?X?X?

?????

(Any situation with the combination "121" occurring along a straight

edge.) Again by the reasoning above applied twice, the squares

diagonally below the "2" on both sides must contain mines.

???X?

?14X?

???X?

?????

(Any situation in which a "1" is adjacent to a "4".) To fit four

mines around the "4", only one of them being adjacent to the "1", all

three squares on the side opposite the "1" must be mined. Likewise,

all three squares touching the "1" on the side opposite the "4" are

free.

112??

1X2??

233A?

??B??

The "2"s and "3"s with a known mine (the X) already adjacent to them

function like "1"s and "2"s - because they have one and two of the

remaining squares next to them mined, respectively. So, using what we

already know about two "1"s or a "1" followed by a "2" along an edge,

we deduce that square A is free and that B contains a mine.

- - -

Dealing with closed cells

?11??

?32??

XX2??

221??

A closed cell is an area, like the two unopened squares in the upper-

left corner in the diagram above, about which no further information

can ever be gained; and yet the information you have does not allow

you to find the remaining mines. In this case, it is obvious that

exactly one of the two squares contains a mine, but there is no way

to find out which other than by guessing.

Closed cells are, of course, extremely annoying. Once you've got

started on a grid, you can usually almost complete it by pure logic;

but if you get a two-square closed cell then you MUST guess where the

mine is. My advice is: as soon as the cell comes up, guess. It will

only be more annoying if you complete the rest of the grid and then

return to the cell only to make the wrong guess.

Four-square closed cells are another matter entirely.

X22X1

2??21

2??21

X22X1

Imagine that this formation occurs in the middle of a large grid,

and that we do know that the four "X"s are all mines. The closed

cell must contain two mines, one in each row and one in each column,

and we cannot tell where they are without guessing. Like the two-

square closed cell, it is a precisely fifty-fifty chance.

??11?

??22?

12X1?

1111?

However, four-square closed cells much more commonly occur in the

corner of the grid. Here, we know that the bottom row and the right-

hand column of the cell each contain exactly one mine, but we know

nothing about the top row or the left-hand column. Therefore, it is

correct to solve the rest of the grid before dealing with the cell;

this allows us to be armed with the knowledge of the total number

of mines the cell contains. If it contains ONE mine, this must be in

the bottom-right corner; if it contains THREE, these must be in all

the squares except that corner; only if the cell contains exactly TWO

mines are we necessitated to guess.

??11?

??22?

23X1?

1111?

Don't fall into the common trap of seeing something that looks like

a closed cell and assuming it is one. Here, we do have enough

information to solve the cell: the lower two squares are mined, the

top-right square is free, and the number in this square will tell us

whether the remaining square is mined or free.

11111

X22X1

3??21

???1-

Some closed cells are much more complex. The one above occurred in a

real game; this formation was in the bottom-left corner of a grid,

and I knew that it contained exactly THREE mines.

Probability is the key to this one. If the square just to the left of

the "1" is a mine, then so is the square to the right of the "3" and

exactly one of the two squares under the "3". Conversely, if the

square to the left of the "1" is free, then the square to the left

of the "2" and both squares under the "3" are mined. The former of

these is more probable, simply because there are more arrangements

of the mines that fit it; therefore I clicked in the square to the

left of the "2", knowing that, whether this square turned out to be

a "3" or a "4", I would be able to find all the mines.

Sometimes, as in the above example, going for the guess that has the

greatest probability of being right is the correct method; more

often, though, you need to go for the guess that will provide the

most new information if you are right.

122?2

2X4?X

3X??X

2X33X

This one is also a position from a real game. There is exactly one

mine among the top two unopened squares, and exactly one among the

bottom two. If you click on one of the top two and are right, then

whichever one you clicked, it will provide no new information about

which of the bottom two is mined, so you will have a 25% chance of

solving the cell completely. However, if you open one the bottom two

squares and are right, then you will certainly be able to solve the

top half of the cell. This is therefore the right tactic, giving you

a 50% chance overall.

- - -

Minesweeper Challenge

This interesting variant of the game is played by choosing a Custom

grid, 16x30 (the same size as Expert) but with 100 mines instead of

99. Each time you complete it, add one more mine! See how far you can

get...

(Contributed by INSANE)

Playing this Challenge is a REALLY good way to improve at Minesweeper.

Because the positions and deductions you will meet tend to be more

complex than those in the Expert game, you will get more used to

making these deductions under pressure. I highly recommend a crash

diet of Minesweeper Challenge only, preferably on a machine with a

slow mouse, for a couple of months, and then returning to the normal

Minesweeper modes on a good machine, as an excellent way to improve

your best times.

The furthest I have got is 127 mines.

- - -

Speed tips

1. There's an option to turn off the "question-marks" feature, so

that right-clicking on a marked square unmarks it immediately.

Hooray. Use this option. Question-marks are useless anyway, and you

will lose less time if you mark the wrong square by accident.

(Thanks to the FAQs by AlaskaFox and KJobst for drawing this

option to my attention.)

2. If you have already marked a mine next to a "1", there are two

ways to open all the unopened squares next to the "1": either click

on the "1" with both mouse buttons simultaneously, or shift-click it

with the left button only. Obviously, being able to open many squares

at once will increase your speed. My advice is to use the shift-click

method, so that you do not get confused and click a square with the

wrong button - which could easily have fatal consequences!

Of course, this also applies when you have marked exactly two mines

next to a "2" and similarly with the other numbers (except "8"s).

3. Think ahead. Here's an example of what I mean:

--1A?

111B?

?????

?????

You get this formation when you click the top-left corner of a grid.

Well, exactly one of A and B contains a mine, so the three squares

underneath the corner "1" are all free. But, if you think ahead,

instead of opening them by clicking on them, you will mark a mine

directly under the leftmost "1" (think about it) and use the shift-

click method to open these squares much more quickly.

4. Don't waste time marking mines and opening single squares where

this will not lead to any new information; instead, try to open up

another area of the grid. If you have multiple areas to work on,

switching between them will give you time to think each time you turn

up some new information.

5. Learn to recognise the patterns of mines around a square, so that

if you have a "4" (for example) in the middle of a cluster of marked

mines and other numbers, with one unopened square adjacent to the

"4", you will be able to tell at a glance whether that square is free

or contains a mine.

6. Start with a corner square. Corners are adjacent to fewer squares,

so you are more likely to find a blank square, which will open up an

area of the grid quickly. If you start in the centre, you are likely

to waste valuable time clicking around until you can get started.

7. Don't always mark mines. You complete the grid not by marking all

mines, but by opening all unmined squares - which I still think is

rather odd. If you can, visualise where the mines are and work out

from that which squares are free without marking the mines. This tip

is especially important for getting good times on the Beginner and

Intermediate levels.

8. You can stop the timer by holding down both mouse buttons over it

and pressing Escape, then releasing both buttons. Using this, you can

get a time of 1 second on all three levels! Except that it won't be

very satisfying, and it will wipe out your genuine best times. Oh,

and the "best times" box will record your time as "1 seconds".

- - -

Other stuff

The colours. Just in case you wanted to know, the colours of the

eight possible numbers are: 1 blue; 2 green; 3 red; 4 dark

blue/purple; 5 reddy brown; 6 light greeny blue. Some versions of the

game have 7 dark brown/black and 8 grey; others have 7 magenta and

8 black.

My best times so far, again just in case you really want to know:

Beginner - 3 seconds

Intermediate - 28 seconds

Expert - 94 seconds

The sizes of the different grids:

Beginner - 9x9 (81 squares) with 10 mines. This is also the smallest

grid you are allowed to choose under Custom, and the fewest mines.

Intermediate - 16x16 (256 squares) with 40 mines.

Expert - 16x30 (480 squares) with 99 mines.

Custom - up to a maximum of 24x30 (720 squares) with 667 mines.

There is a glitch in choosing a Custom grid size. If you click on

Custom to choose a grid size, then change your mind and click

"Cancel", you will keep the grid size, but the program will consider

it as a custom grid size and will not record your best times. To

avoid this problem, re-choose the grid size from the menu.

Interesting fact: the clock moves from 0 seconds to 1 at the instant

you open the first square, which means that, even if your first click

opens all the squares (as it might on a maximum-sized custom grid

with only ten mines), your cannot record a time of zero seconds. It

also follows that a final time of (for example) 60 seconds actually

means "more than 59 seconds, up to and including 60 seconds exactly".

And that's all I know about Minesweeper. If you wish to e-mail me

about the subject, or to suggest improvements to this FAQ, you are

welcome to do so.

version 1.11

by Michael Spencer (michael.spencer@balliol.ox.ac.uk)

last updated 2 October 2003

Contents:

1. Overview of game

2. How to become a moderately good player

3. How to become an amazingly good player

4. Dealing with closed cells

5. Minesweeper Challenge

6. Speed tips

7. Other stuff

- - -

Version History

1.11: new best times for Expert and Intermediate modes added;

Minesweeper Challenge given its own section

1.10: three new examples added to Section 3; more information added

about the colours; more wording improvements; MINESWEEPER CHALLENGE

added

1.04: three new examples added to Sections 3 and 4; a new best

time for Expert mode has been added

1.03: new best times for all three modes have been added; a further

example has been added in Section 4; the wording has been slightly

improved in some of the tips and examples

1.02: new best time for Expert mode added

1.01: corrections have been made to one of the examples in Section 4,

which as it stood was an impossible position

1.00: first release

- - -

Overview of game

The object of the game is to find a set number of mines that have

been concealed in a rectangular grid. To do this, you must click on

the grid squares to open them up. If you open a square that contains

a mine, it will blow up and you lose. However, if you open a square

that does not contain a mine, you will be told how many mines are in

the squares adjacent to the square you opened. Using this information

it is usually possible to pinpoint the mines' locations more

accurately than you would be able to do by guesswork alone.

The game provides three difficulty levels (Beginner, Intermediate,

Expert), plus the option of creating your own. On each of the main

levels, the game will record your best time, and so the ultimate

object is to get this as low as possible.

You can mark a square as mined by right-clicking it. This will make

it easier for you to work out which squares are mined and which are

free, as you do not have to hold the mines' locations in your head.

Right-clicking a square twice will mark it with a question-mark,

which you can use to indicate uncertainty about a square. Right-

clicking a square three times returns it to its unmarked state.

The following example demonstrates the game's method:

Key: ? unopened square

- blank square

X square that contains a mine

???? Step 1. You have no information and must choose a starting

???? square. Useful fact: the FIRST square you click on

???? is always "free", so you can get started. We'll try

???? the square in the top-left corner.

---- Step 2. The top-left square was blank, i.e. not mined and

-111 not adjacent to any mines. Hence all surrounding

12?? squares were opened automatically for us. Now, the

???? square to the right of the "2" must contain a mine.

---- Step 3. Notice that the "1" inside the nook is only adjacent

-111 to one unopened square, so we have marked this as

12X? mined. The "1" above the marked square now touches a

???? known mine, so the other squares it touches are free.

---- Step 4. We have revealed a "1" next to the known mine, so,

-111 by repeating the process of Step 3, two further

12X1 squares on the bottom row are known to be free.

????

---- Step 5. Now, the "2" on the bottom row is adjacent to one

-111 known mine, and there is only one square its other

12X1 mine could be on. Once this square is marked, we can

??21 deduce that the bottom-left corner square is free.

- - -

How to become a moderately good player

First, look for places where you know there must be mines.

Here are the most common formations:

1. --1??

-11??

11X??

?????

To deduce where a mine is from a "1" square, there must be only one

unopened square touching it, so the only possible formation is the

"nook". In the diagram above, the square X must contain a mine.

2. ----1 ----1

-1221 11211

12XX? ?X3X?

????? ?????

The most common formation with only two unopened squares adjacent to

a "2" occurs when two mines are together just next to a nook. Another

possiblity is a straight edge, if the square orthogonally adjacent to

the "2" has for some reason already been opened. Both "X"s are mines.

3. ---1? --11?

2322? -13X?

XXX?? 12XX?

????? ?????

The most common formation with a "3" occurs when three mines are

touching along a straight edge, as in the first diagram. Also

important is the case where a "3" sits inside a corner, and so is

only adjacent to three squares, as in the second diagram.

4. ---1?

--12?

124X?

?XXX?

The most common formation with a "4" (which is, however, not very

common) requires a nook just one square away from a straight edge,

and one mine in the nook and three next to it along the edge.

5. --1?? ----1

--2X? 11211

125X? ?X5X?

?XXX? ?XXX?

With a "5", two important formations are the outside corner, and the

straight edge with a "5" jutting out from it.

Once you have marked all known mines in this manner, the next step is

to look for numbered squares whose quota of mines is already filled.

For example, in the left-hand "5" diagram above, once you have marked

all five mines, the "2" just to the left of the "5" is adjacent to

two known mines, and so the bottom-left corner square is free.

By opening all squares you find that are known to be free, and then

marking any new mines that you can definitely locate using your new

information, and repeating the process, you should be able to

regularly complete the Beginner and Intermediate levels. Sometimes,

of course, you will just be unlucky and not be able to find out

enough information. Just don't get upset if this happens!

- - -

How to become an amazingly good player

PART ONE

With an enormous amount of luck, the Expert level as well can be

completed with just the techniques described above. However, just one

addition to these will make a huge difference, and you should be able

to complete Expert with at least a reasonable success rate - although

you will still often have bad luck or reach an impossible position.

--1??

111??

ABC??

?????

Imagine you get this formation in the top-left corner of a level. The

squares labelled "A", "B", and "C" are all unopened; I have labelled

them merely for convenience.

The "1" immediately above square A is adjacent to exactly one mine;

in other words, exactly one of the squares A and B contains a mine.

But the "1" immediately above square B is also adjacent to this mine,

whether it turns out to be in A or B! Therefore, without knowing

whether it is square A or B that is mined, we can deduce that square

C is free!

Of course, if we had the identical formation with a "2" instead of a

"1" above B, we would know instead that square C contained a mine.

Before reading on, read through this carefully until you understand

completely the logical steps taken in the last four paragraphs.

The core idea is that we can make a deduction of the following type:

"exactly one of these two specific squares contains a mine". There

are, in fact, enormously many more formations than the above in which

this type of deduction can be made - and can prove useful.

PART TWO

The following technique is a more elaborate version of that described

above.

----1

11221

ABCD?

?????

Again, the labels "A", "B", "C", and "D" are merely for convenience.

The "2" immediately above square C tells us that exactly two of

squares B, C, D contain mines. But it cannot be the case that both

B and C contain mines: if they did, the "1" above B would be adjacent

to two mines, which is impossible. Therefore the two mines are in

square D and one of squares B and C. This in turn tells us that

square A is free - because the one mine adjacent to the "1" is in

either square B or square C.

Alternatively, the "1" immediately above square B tells us that

exactly one of squares A, B, C contains a mine, so the other two

must be free. But squares B and C cannot both be free: if they were,

there would be no room to place two mines adjacent to the "2". As

before, we deduce that square A is free and square D is mined.

PART THREE

Once these techiques have been grasped, there is no limit to the

complexity of the situations in which they may apply.

-1???

12???

??A??

?????

This adds merely one extra level of complexity. If this formation

occurs in a corner of the grid, each "1" is adjacent to one mine;

this accounts for both mines adjacent to the "2", so A is free.

---1?

1113?

?ABCD

?????

Here, the top "1" is adjacent to one mine, and there is one mine in

squares A, B, and C, hence at most one in B and C. Therefore square

D is mined, from which we can deduce that square A is free. In

general, something useful can be deduced when a "2" occurs next to a

"1" along an edge, or a "3" next to two "1"s around a corner.

--1?? --1?? --1??

--1?? --1A? --1A?

112?? 1121? 1121?

????? ?B1C? ?B2C?

In the first diagram, we can use the methods already described to

show that the squares immediately to the right of and below the "2"

are free. Suppose that we uncover these to reveal "1"s, as in the

second diagram. Now, only three uncovered squares (A, B, and C) are

left touching the "2", so two of these are mines. But A and C cannot

both be mines because of the "1" between them; neither can B and C,

for the same reason. Therefore A and B are the mines.

(It is to be noted that the reasoning exactly parallels that in the

case where a "1" and a "2" are adjacent along an edge. Also, if we

slightly modify the example, as in the third diagram, we can still

deduce by the same reasoning that the square B contains a mine and

that the three squares to the right of the rightmost "1" are free.)

X2-2X X2-2X

X414X X424X

X???X X???X

????? ?????

There are two special cases to be considered when just three squares

in a row are adjacent to a row of numbered squares. In the first

diagram, one of the squares next to each "4" is a mine, and the "1"

therefore means that the squares immediately below each "4" are free,

so that the mine is in the square below the "1". Conversely, in the

second diagram there are two mines, below the "4"s, while the square

below the "1" is free.

--1?? --1A?

112?? 112B?

????? ?C1D?

????? ?????

Again, in the first diagram (assuming this formation is in the corner

of the grid) we know that the square under the "2" is free. Suppose

that opening this reveals a "1". Now, exactly one of A and B contains

a mine; so exactly one of C and D contains a mine; therefore square

B and the three squares under the "1" are all free (which means that

it is A that contains the mine).

This is an example of a two-stage deduction, which is not at all

unusual; much longer chains of deduction do occur, but more rarely.

I will not give examples here, because it should be clear from the

examples already given how the methods of deduction work.

PART FOUR

Having learned these techniques, it helps to commit to memory the

following common formations in which they occur.

----1

11111

??A??

?????

(Any situtation in which two "1"s next to each other come out from

one of the edge walls.) The third square in the row below the "1"s

must be free.

----1

12111

??X??

?????

(As above, with the second square changed to a "2".) The third square

in the row below the "1"s and "2" must contain a mine.

----1

11111

??1??

?????

(A common follow-up to the first example above, if clicking on

square A revealed a "1".) The three squares below the "1" are free.

----1

11111

??1??

?122?

?????

(And a common follow-up to the follow-up.) The squares to the left

of, under, and diagonally under the lowest "1" are all free.

----1

11111

??1??

1122?

--1X?

--1??

The square marked X contains a mine. (Make sure that you can follow

this example and the previous two - the logic is the same for all of

them. Which three squares in the above diagram are certainly free?)

-----

11222

A??X?

?????

(Any situation in which a "1" and a "2" are adjacent along a straight

edge.) The square diagonally below the "1" (square A) must be free,

and the square diagonally below the "2" must contain a mine.

-----

11221

??XX?

?????

(Any situation with the combination "1221" occurring along a straight

edge.) By the reasoning above applied twice, the squares directly

under both "2"s must contain mines.

-----

11211

?X?X?

?????

(Any situation with the combination "121" occurring along a straight

edge.) Again by the reasoning above applied twice, the squares

diagonally below the "2" on both sides must contain mines.

???X?

?14X?

???X?

?????

(Any situation in which a "1" is adjacent to a "4".) To fit four

mines around the "4", only one of them being adjacent to the "1", all

three squares on the side opposite the "1" must be mined. Likewise,

all three squares touching the "1" on the side opposite the "4" are

free.

112??

1X2??

233A?

??B??

The "2"s and "3"s with a known mine (the X) already adjacent to them

function like "1"s and "2"s - because they have one and two of the

remaining squares next to them mined, respectively. So, using what we

already know about two "1"s or a "1" followed by a "2" along an edge,

we deduce that square A is free and that B contains a mine.

- - -

Dealing with closed cells

?11??

?32??

XX2??

221??

A closed cell is an area, like the two unopened squares in the upper-

left corner in the diagram above, about which no further information

can ever be gained; and yet the information you have does not allow

you to find the remaining mines. In this case, it is obvious that

exactly one of the two squares contains a mine, but there is no way

to find out which other than by guessing.

Closed cells are, of course, extremely annoying. Once you've got

started on a grid, you can usually almost complete it by pure logic;

but if you get a two-square closed cell then you MUST guess where the

mine is. My advice is: as soon as the cell comes up, guess. It will

only be more annoying if you complete the rest of the grid and then

return to the cell only to make the wrong guess.

Four-square closed cells are another matter entirely.

X22X1

2??21

2??21

X22X1

Imagine that this formation occurs in the middle of a large grid,

and that we do know that the four "X"s are all mines. The closed

cell must contain two mines, one in each row and one in each column,

and we cannot tell where they are without guessing. Like the two-

square closed cell, it is a precisely fifty-fifty chance.

??11?

??22?

12X1?

1111?

However, four-square closed cells much more commonly occur in the

corner of the grid. Here, we know that the bottom row and the right-

hand column of the cell each contain exactly one mine, but we know

nothing about the top row or the left-hand column. Therefore, it is

correct to solve the rest of the grid before dealing with the cell;

this allows us to be armed with the knowledge of the total number

of mines the cell contains. If it contains ONE mine, this must be in

the bottom-right corner; if it contains THREE, these must be in all

the squares except that corner; only if the cell contains exactly TWO

mines are we necessitated to guess.

??11?

??22?

23X1?

1111?

Don't fall into the common trap of seeing something that looks like

a closed cell and assuming it is one. Here, we do have enough

information to solve the cell: the lower two squares are mined, the

top-right square is free, and the number in this square will tell us

whether the remaining square is mined or free.

11111

X22X1

3??21

???1-

Some closed cells are much more complex. The one above occurred in a

real game; this formation was in the bottom-left corner of a grid,

and I knew that it contained exactly THREE mines.

Probability is the key to this one. If the square just to the left of

the "1" is a mine, then so is the square to the right of the "3" and

exactly one of the two squares under the "3". Conversely, if the

square to the left of the "1" is free, then the square to the left

of the "2" and both squares under the "3" are mined. The former of

these is more probable, simply because there are more arrangements

of the mines that fit it; therefore I clicked in the square to the

left of the "2", knowing that, whether this square turned out to be

a "3" or a "4", I would be able to find all the mines.

Sometimes, as in the above example, going for the guess that has the

greatest probability of being right is the correct method; more

often, though, you need to go for the guess that will provide the

most new information if you are right.

122?2

2X4?X

3X??X

2X33X

This one is also a position from a real game. There is exactly one

mine among the top two unopened squares, and exactly one among the

bottom two. If you click on one of the top two and are right, then

whichever one you clicked, it will provide no new information about

which of the bottom two is mined, so you will have a 25% chance of

solving the cell completely. However, if you open one the bottom two

squares and are right, then you will certainly be able to solve the

top half of the cell. This is therefore the right tactic, giving you

a 50% chance overall.

- - -

Minesweeper Challenge

This interesting variant of the game is played by choosing a Custom

grid, 16x30 (the same size as Expert) but with 100 mines instead of

99. Each time you complete it, add one more mine! See how far you can

get...

(Contributed by INSANE)

Playing this Challenge is a REALLY good way to improve at Minesweeper.

Because the positions and deductions you will meet tend to be more

complex than those in the Expert game, you will get more used to

making these deductions under pressure. I highly recommend a crash

diet of Minesweeper Challenge only, preferably on a machine with a

slow mouse, for a couple of months, and then returning to the normal

Minesweeper modes on a good machine, as an excellent way to improve

your best times.

The furthest I have got is 127 mines.

- - -

Speed tips

1. There's an option to turn off the "question-marks" feature, so

that right-clicking on a marked square unmarks it immediately.

Hooray. Use this option. Question-marks are useless anyway, and you

will lose less time if you mark the wrong square by accident.

(Thanks to the FAQs by AlaskaFox and KJobst for drawing this

option to my attention.)

2. If you have already marked a mine next to a "1", there are two

ways to open all the unopened squares next to the "1": either click

on the "1" with both mouse buttons simultaneously, or shift-click it

with the left button only. Obviously, being able to open many squares

at once will increase your speed. My advice is to use the shift-click

method, so that you do not get confused and click a square with the

wrong button - which could easily have fatal consequences!

Of course, this also applies when you have marked exactly two mines

next to a "2" and similarly with the other numbers (except "8"s).

3. Think ahead. Here's an example of what I mean:

--1A?

111B?

?????

?????

You get this formation when you click the top-left corner of a grid.

Well, exactly one of A and B contains a mine, so the three squares

underneath the corner "1" are all free. But, if you think ahead,

instead of opening them by clicking on them, you will mark a mine

directly under the leftmost "1" (think about it) and use the shift-

click method to open these squares much more quickly.

4. Don't waste time marking mines and opening single squares where

this will not lead to any new information; instead, try to open up

another area of the grid. If you have multiple areas to work on,

switching between them will give you time to think each time you turn

up some new information.

5. Learn to recognise the patterns of mines around a square, so that

if you have a "4" (for example) in the middle of a cluster of marked

mines and other numbers, with one unopened square adjacent to the

"4", you will be able to tell at a glance whether that square is free

or contains a mine.

6. Start with a corner square. Corners are adjacent to fewer squares,

so you are more likely to find a blank square, which will open up an

area of the grid quickly. If you start in the centre, you are likely

to waste valuable time clicking around until you can get started.

7. Don't always mark mines. You complete the grid not by marking all

mines, but by opening all unmined squares - which I still think is

rather odd. If you can, visualise where the mines are and work out

from that which squares are free without marking the mines. This tip

is especially important for getting good times on the Beginner and

Intermediate levels.

8. You can stop the timer by holding down both mouse buttons over it

and pressing Escape, then releasing both buttons. Using this, you can

get a time of 1 second on all three levels! Except that it won't be

very satisfying, and it will wipe out your genuine best times. Oh,

and the "best times" box will record your time as "1 seconds".

- - -

Other stuff

The colours. Just in case you wanted to know, the colours of the

eight possible numbers are: 1 blue; 2 green; 3 red; 4 dark

blue/purple; 5 reddy brown; 6 light greeny blue. Some versions of the

game have 7 dark brown/black and 8 grey; others have 7 magenta and

8 black.

My best times so far, again just in case you really want to know:

Beginner - 3 seconds

Intermediate - 28 seconds

Expert - 94 seconds

The sizes of the different grids:

Beginner - 9x9 (81 squares) with 10 mines. This is also the smallest

grid you are allowed to choose under Custom, and the fewest mines.

Intermediate - 16x16 (256 squares) with 40 mines.

Expert - 16x30 (480 squares) with 99 mines.

Custom - up to a maximum of 24x30 (720 squares) with 667 mines.

There is a glitch in choosing a Custom grid size. If you click on

Custom to choose a grid size, then change your mind and click

"Cancel", you will keep the grid size, but the program will consider

it as a custom grid size and will not record your best times. To

avoid this problem, re-choose the grid size from the menu.

Interesting fact: the clock moves from 0 seconds to 1 at the instant

you open the first square, which means that, even if your first click

opens all the squares (as it might on a maximum-sized custom grid

with only ten mines), your cannot record a time of zero seconds. It

also follows that a final time of (for example) 60 seconds actually

means "more than 59 seconds, up to and including 60 seconds exactly".

And that's all I know about Minesweeper. If you wish to e-mail me

about the subject, or to suggest improvements to this FAQ, you are

welcome to do so.

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