Advanced Minesweeper Patterns

I. Introduction

This article is aimed at experienced Minesweeper players who want to play fast. You are expected to know the rules of the game and be familiar with basic patterns like 1-2-1 or 1-2-2-1. If you are not, it is strongly recommended that you first read our Introduction to Minesweeper and our guide to Advanced Minesweeper strategy.

This article assumes that you already have a lot of experience playing Minesweeper. Ideally, you can not only apply simple patterns on the fly but also automatically and without any effort subtract already placed flags from the numbers on the board while applying these patterns. Moreover, you should be well-versed in the logical thinking required in Minesweeper because the patterns described here are not always accompanied by an explanation: you should be able to easily fill in such omissions by yourself.

The Minesweeper patterns described in this article do not form an exhaustive list. Rather, these are the patterns players find most useful while playing. Also, some easy patterns are omitted because most players learn them by themselves quickly and with little effort. If you are a genuinely advanced player, you may already be familiar with most of the patterns described here. In such a case, hopefully, this article will at least help you systematize your knowledge.

II. The ones

We start with something simple: patterns based on number 1. Most of these patterns are easy to spot and quite common. Therefore, many advanced players already know at least some of them. We start with them not only because they are very useful while playing, but also because some of them can be used to explain other, more complicated patterns.

The simplest pattern based on ones is the “1-1+” pattern. You encounter it whenever you see a long row of 1s, blocked from at least one side by the board's edge or by already opened fields, as in Figure 1 (top). In that case, every third 1 cannot have a mine at its nearest adjacent field. In Figure 2 (bottom), the fields that cannot contain mines are marked with question marks.

Figure 2 (top-left) shows another example of a 1-1+ pattern. Here, instead of the border, the pattern starts with two flags and, like before, provides us with two mine-free locations. However, that is not all. If the number of ones is of the form 3n+2 (where n = 0, 1, 2…), and there are no other numbers in the row, we can uncover all three fields at the end of that row. This can be seen in Figure 2 (top-right).

Sometimes, there are two 1-1+ patterns close to each other. In Figure 2 (bottom-left), we can first consider the horizontal pattern which indicates one field without a mine. However, the same board also has a vertical pattern consisting of only two fields (n = 0). It is marked with a red ellipse in Figure 2 (bottom-right) and indicates three fields without mines. As you can see, the shortest possible 1-1+ pattern is two fields long.

Two very useful patterns can be easily explained once you know the 1-1+ pattern. The first is the “1-1-1” pattern. It consists of a sequence of three consecutive ones, blocked from both sides. When you apply the 1-1+ pattern to this sequence twice – from both ends – you can easily see that the outer 1s cannot share a border with a mine, and the mine must be in the middle (see Figure 3, top-left). A similar situation is with the “1-1-1-1” pattern, in which there is a sequence of four ones blocked from both sides. Again, applying the 1-1+ pattern to this situation twice, from both ends, tells us that the inner ones cannot share a border with a mine, and the mines must be at both ends of the pattern (Figure 3, top-right and bottom).

If a number discovered in the fields along a 1-1+ pattern is also a 1, then each such 1 has three more adjacent fields that do not contain mines. This is called a “poking 1” pattern and happens quite often. Figure 4 (top) shows what could happen after opening the fields indicated as mine-free in Figure 1. We get two ones, and there are six more fields we can open.

The last pattern in this section is the “dangling 1” pattern. Consider the 2 marked in red in Figure 4 (middle). It has four covered fields around it. Three of these four fields are adjacent to the 1 on the right (they are marked in blue). Because there is exactly one mine in these three fields, the 1 on the right reduces the 2 marked in red to a 1 with the only available covered field marked in green. This field must contain a mine and is therefore marked with a mine in Figure 4 (bottom). On the other hand, the fields adjacent to the 1 and not adjacent to the 2 cannot contain mines and can be safely opened. These fields are marked with question marks.

The idea behind the dangling 1 pattern is to spot a 1 whose neighborhood almost entirely surrounds a higher number (which is a 2 in our example). Then, the other covered fields adjacent to that higher number will all contain mines. This pattern is harder to use than the other 1-based patterns discussed in this section. Unless you memorize all possible combinations, it requires some effort to spot it as well as to decide where the mines are and which fields are mine-free. Moreover, it does not occur as frequently as the previously mentioned patterns, and therefore is appreciated only by the most advanced players.

There are more questions you may have after looking at Figure 4 (bottom). How did we know to open the field with the dangling 1? And how did we know to put a flag to the right of the 4? These questions will be addressed in the next section, where we discuss the 1-2 patterns.

III. The 1-2 patterns

The second class of patterns we consider shares a common theme: a 1 and a 2 next to each other. Although they occur frequently during the game, these patterns are often not immediately apparent. After learning about them, many advanced Minesweeper players are amazed by their usefulness and surprised that they have not noticed them earlier.

The basic structure of the general “1-2” pattern is shown in Figure 5 (left). The 1 and the 2 belonging to the pattern are marked with a red ellipse. The 1 indicates that there is precisely one mine in the three covered adjacent fields, marked with a blue ellipse. This works similarly to the dangling 1 described above: the neighboring 2 is reduced to 1 with only one available covered field, which must contain a mine. As a result, whenever we encounter a 1-2 pattern, we can immediately plant one flag and open one field, as shown in Figure 5 (right).

Now consider the situation in Figure 6 (top). You can see the 1-2 pattern marked in red. This allows us to plant one flag and open one field, which results in the situation shown in Figure 4 (middle and bottom). How about the other neighboring 1 and 2 visible in Figure 6 (bottom)? Unfortunately, this is not a proper 1-2 pattern. For the 1-2 pattern to work, the 2 must have only one covered field available after removing the fields adjacent to the 1. There are three such fields instead of one (marked in blue), so the pattern cannot be applied.

The 1-2 pattern is really powerful. It allows us to quickly explain other very common patterns: the 1-2-1 and 1-2-2-1. Both of them are composed of two 1-2 patterns. In Figure 7, the 1-2 pattern is first applied to the upper 1-2 (left), then to the lower 1-2 (middle). Combining them gives us the traditional solution to the 1-2-1 pattern (top-right) and the 1-2-2-1 pattern (bottom-right).

Other patterns based on the 1-2 pattern also occur frequently. The first example is the “2-3” pattern in which a 2 and a 3 are surrounded from both sides by a flag (Figure 8, top). The neighboring flags reduce these 2 and 3 to 1 and 2, immediately providing us with locations of one mine and one mine-free field. Similarly, the “4 in a corner” pattern (Figure 8, middle) is just another 1-2 pattern we may get after accounting for the two flags next to a 4. Finally, the “2-1-2” pattern (Figure 8, bottom) informs us about the location of three mines and two fields without mines. These are just some of the most common variants of the 1-2 patterns, and a good player will surely find more.

IV. Ellipses

An ellipse is a pair of fields with exactly one mine, but it is uncertain which of the two fields holds the mine. The name comes from a shape often used to mark such pairs on diagrams. Ellipses can be used to infer the presence or absence of a mine by reducing the neighboring number. They are not easy to detect but are very powerful: for example, the 1-1+ pattern is based on an ellipse (see Figure 9: the ellipses are marked in red).

Except for some “easy” patterns like the 1-1+ pattern, patterns involving ellipses do not occur frequently and will be useful only for the most experienced players. Typically, only after playing for a very long time a player has seen them enough to memorize them and use them automatically. For example, consider the “1-2-1 by ellipse” pattern. The ellipse is marked in Figure 10 (top-left). It reduces the bottom 2 to 1, revealing a 1-2-1 pattern, also shown in Figure 10 (top-right). The “1-2-2-1 by ellipse” pattern works similarly: an ellipse reduces a 2 to 1, creating a 1-2-2-1 pattern (Figure 10, bottom).

As shown in the examples above, ellipses can be used to spot simpler patterns in places you wouldn't normally find them. However, this is not their only use. From time to time, you may encounter the “empty corner” pattern, where two ellipses combine to reduce a 2 down to 0, as shown in Figure 11 (top). Finally, you may also use a “free ellipse” simply to reduce a number, as is also shown in Figure 11 (bottom). However, whether such use of ellipses counts as a Minesweeper pattern is debatable.

V. Bonus

Before we conclude this presentation, let us look at one curiosity: a square of ones. If you are a really experienced player, you might have encountered it a few times. This opening is very frustrating and requires a lot of effort to resolve. Although it is impossible to find spots where the mines are, you can deduce which fields do not contain mines and can be opened to discover more numbers, as in Figure 12. Hopefully, this will put you on an easy path forward.