This guide lists the different general reasoning techniques I'm aware of for solving Tametsi puzzles. I'm writing this mostly because I'm interested to find out if other people use completely different methods, but the ones listed here have been enough to get me through all the puzzles.

Semi-spolier alert: Note that part of the fun of the game is the "Aha!" moments where you discover a new technique, and reading this might deprive you of that enjoyment.

In the startup window, I recommend turning on the following settings:

• Darken finished tiles
  
• Revealed tiles count down
  
• Column hints count down
  
• Treat grey like other colors
  
• Allow drawing

Let the computer handle the basic arithmetic for you so you can save your brain power for brilliant feats of logical deduction. I can't count how many times I misclicked a tile before turning these on, especially in the puzzles with less regular geometry, because I was moving too fast and miscounted a tile's neighbors.

Unless you have a very good memory, it can be easy to forget information you've gleaned about one part of the puzzle if you focus on another. The drawing tools are also a good way to double-check your logic. In particular, I use lines to mark groups of tiles that I know contain exactly one mine between them. (If I know they contain two mines, I mark them with a double line, and so on. I also use a + or – to indicate tiles that contain at least one or at most one tile.)

For example, here I marked the tile locations for the 1 clues on the edges. That lets me see that to complete the 2, one of the two tiles marked in the middle image must be occupied:


But, that means that the tiles I just marked complete the 1, which means that every other tile surrounding the 1 except for those two can be revealed.

In time, you'll get so used to recognizing patterns like this that you won't need to draw them out, but the lines can still be useful, especially if you're trying to figure out if you know where all the tiles are for a given column hint.

This is an obvious one, but note how, in the example above, the 2 tile had four unrevealed neighbors. From another clue I was able to mark two of those neighbors to indicate there's one mine in them - but I don't know which one.

That means that all the neighbors I didn't mark (in this case, two of them) contain all the mines not accounted for by that marking (in this case, 2 – 1 = 1 mine).

If you can't make progress any other way, start with the hypothesis that one tile is occupied (or unoccupied) and work out what other tiles around it would be forced into a certain state if that were the case. (I use \ to mark occupied tiles and · for unoccupied). To keep track of my starting hypothesis I mark it with a doubled symbol, so × for occupied and ·· for unoccupied.


In this case, I picked a hypothesis for which of the three neighbors of the circled 2 is unoccupied, and worked my way along the top of the puzzle filling in the implications of that hypothesis. Then I reached the 2 circled on the right. It already has 2 neighbors filled in, so all its other neighbors should be empty. But that leaves me with no way to complete the 3. The hypothesis is disproven, so I can mark the ·· tile (and, sadly, only that tile) as a mine.

If the hypothesis does not lead to a contradiction, that does not mean it is correct; there may be multiple ways to fill in the tiles. This makes brute a time-consuming and inefficient approach. It can also be hard to guess which hypothesis to test, although you should steer away from sections of the puzzle that you already have "locked in" in a lattice of either-or connections.

If your initial brute force search yielded no contradiction, you can try reversing the hypothesis and working that one out, as well. Pick a different color and leave the other marks in place. Then, look for any tiles that you marked the same way for both hypotheses. You have proven that those tiles must have the state you marked.

Instead of testing just one tile on or off you might have three tiles, exactly one of which is occupied. You can do the same thing with three colors, and mark any tiles that have the same state in all three.


In the example above, I tested all three possibilities for which neighbor of the middle 1 is occupied. That only led to one tile that would have the same state in all three. I revealed that tile and found that it has two mines next to it. The orange marks only place one mine next to that tile, so I can rule out the orange hypothesis. The remaining two agree on a lot of tiles, which I can now reveal. (This is a simplified example; you could probably solve it without working it out like this.)

Just keep in mind that you have only proven the state of a given tile if the set of hypotheses you've tested cover all possible states: that is, make sure there's no way that all your hypotheses could be false.

Sometimes you will notice that one clue covers all the same tiles as another, plus some extras. In that case you know that the number of mines in the extra tiles is equal to the difference between the two clues. This shows up frequently in single rows of revealed tiles cutting in from an "edge":

--image--

This pattern shows up in less obvious ways as well:

If the difference between two clues is N, and the difference between the two subsets they cover is also exactly N, that means that when "moving" from one set to the other N unoccupied tiles were replaced with N occupied ones. The simplest place this shows up is shown below:


The 2 and the 1 have two tiles in common, and each have one tile that's not in the other's range. That means the tile that's only in the 2's range must be occupied, and the tile that's only in the 1's range can be revealed. (This is another pattern you'll learn to automatically fill in.)

Again, this pattern can show up in less obvious ways:

--image--

It's common in square and hexagonal puzzles to have rows of revealed tiles in between hidden tiles. In the example below, the three tiles on the very left must have 1 or 2 mines, and the three on the very right must have 0 or 1. I started with each of those counts as a hypothesis and worked my way inward, figuring out how many mines must be in each added par of tiles to reach the sum in between them:


On the left, the 1 hypothesis failed: at one point I would have had to add 3 mines in just 2 tiles. That means the only valid numbering is the top one, starting with 2. I can fill in pairs of 2 mines and reveal pairs (or, at the end, a triple) with 0 mines, and mark the rest with lines to indicate that if one is occupied, the other is not. On the right, both hypotheses led to valid numberings, with every number different, so I can't reach any conclusions.


The same approach works for square grids. In the example below the chain has to start with either 0 or 1 tile occupied in the pair on the left. The 0 hypothesis leads to a contradiction on the very right, so I'm left with the (unfortunately rather boring) 1 hypothesis.


In some cases on a square grid, the chain has unrevealed tiles at both ends. In that case you need to check all possible values for the first two columns instead of just the first one:


In this case I checked all the solutions on the right side first, then moved to the left after discovering so many options that none of them agreed. I should have started on the left, as it turns out: that chain has only one solution (beginning with 0, 0) and that solution tells me the chain ends with 2. Because there's a 4 in the "hallway" between the chains, that means the chain on the right has to start with 2, so I can throw out all those other options I painstakingly explored (including the otherwise valid one starting with 3):

What I call "chains" are a powerful technique because they let you reason about blocks of tiles (i.e. the pairs above and below each revealed tile in a row) instead of individual tiles. But what about some of the wonkier geometries - can you do the same thing there? Sometimes! For example, note the highlighted intersections below (in a puzzle with corner connectivity instead of just edges):


That's a chain, alternating between groups of two and three (except at the midpoint, which is trickier. I hypothesize that the 1's mine is not on the left and follow the chain up - and get a contradiction. I won't have enough mines to complete the 4. So, the initial mine is on the left. I follow the chain up with that knowledge.

When I get to the midpoint I don't know if the square (which is not part of either chain intersection) is occupied or not. So instead of fixed numbers I switch to marking with + (meaning 1-2 mines) until I get to the top and find that the last intersection needs 2 mines to complete the 4. I can then propagate that information back down the chain. In a minute of work I've found three tiles I can reveal, three I can mark as mines, and exact counts for other intersections.


Here's another complex example. The highlighted hexagons form a chain of tiles linked by two neighbors on each side. The 1 tells me at most one tile at the start is occupied, so I mark it with a –, and continue up, alternating + and – because everything is a 2. That tells me at most 1 (and therefore exactly 1) of the lower neighbors of the 3 at the end of the chain are occupied - so I can fill in the other two. Then I propagate the exact count that I've now established back down the chain.

Remember that column hints and color counts can help you in three ways:

• proving that the unknown tiles must all be negative,
  
• proving that the unknown tiles must all be positive, or
  
• letting you place constraints on the tiles.

In the example above, the hint tells me there are 2 more mines in that column - in which only 3 tiles are unrevealed. Only one of the mines can be placed next to the 1, so the other must go above it. (If the hint had only been a 1, I'd have no way to be certain that either of the spaces to the right of the 1 is occupied; that mine could have been above the 1 instead.)

Sometimes, you can also add up two columns (or two colors) at the same time, if you have a lot of possible mines that are split between the two of them. This happens often with chains that resolved into a long row of either-ors.

Especially near the end of the puzzle or when you've found most of the mines of a given color, there are situations where you need to look at a group of tiles and ask what the minimum or maximum number of mines is that they could contain. (For example, you may have previous determined, "There's either a mine here, or a mine in both these locations.")

If you're stuck in one part of the puzzle, look somewhere else (or put the game away for a while) to let your brain reset. This is where marking up the puzzle with drawings is particularly helpful, because when you come back your existing inferences are still there. Go through all the information (column hints, color counts, etc.); one of them is often the key. Also check that you've extracted all the possible information from each revealed tile count.