MINESWEEPER FAQ version 1.125 by Michael Spencer (michael.spencer@balliol.ox.ac.uk) last updated 15 June 2004 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 1.121-124: minor updates 1.12: Ultimate Minesweeper Challenge added; minor improvements 1.11: 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 1.03: a further example has been added in Section 4; the wording has been slightly improved in some of the tips and examples 1.02: minor update 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 first 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. Naturally, the same applies to any numbers differing by three. 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 do we have 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 of 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) with 100 mines. 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. Ultimate Minesweeper Challenge is the same, but played on a 24x30 grid, the largest size available. (Contributed by mariostar224) I recommend starting with 100 mines and increasing the mine count in fives, so as not to wear your patience out too quickly. I have completed Minesweeper Challenge with 135 mines, and UMC with 200. - - - 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 - 2 seconds Intermediate - 18 seconds Expert - 77 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. Some earlier versions of the game have an 8x8 Beginner grid instead. 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.