Here is the next group in our Classics Collection. Actually, the next puzzle style was supposed to be a simple Domino Search, but Ulrich convinced me that it would be a bit too dull even for a beginners series, so we decided to do Blackout Dominoes instead. This may sound like a weird choice because they do not come up that often in contests, hence one could wonder if they belong in our series at all.
To be honest, I do not have a lot of experience with Blackout Dominoes. I have solved them a handful of times, and I cannot recall ever designing one. This means, among other things, I am not sure if the puzzles you will find in the PDF file below are “typical” specimens of this style. Anyway, I hope you enjoy the puzzles. Here is today’s set: Blackout Dominoes
Rules: Shade some cells, divide the remaining unshaded cells into dominoes, and enter numbers into the unshaded cells, so that each domino from the given set appears exactly once. Horizontally or vertically adjacent cells that are part of different dominoes must contain equal numbers. Shaded cells cannot be horizontally or vertically adjacent to each other or to the border of the grid (they may touch each other or the border diagonally). Cells with numbers cannot be shaded.
Example and solution:
Blackout Dominoes is one of rather few puzzle types which regularly use an irregular grid, if you know what I mean. (Kakuro is another, but I do not think we have had any others with this property in the Classics Collection.) I have seen puzzle authors design them in a way that the outcome features some creative imagery. Personally, I prefer more mundane, compact shapes.
The next thing I would like to mention is that authors sometimes give shaded cells or walls inside the grid as clues. This is something I usually try to avoid, just as I prefer Skyscrapers with no given entries. Obviously one needs a few given numbers in Blackout Dominoes, since otherwise the solution cannot be unique, but you will find no other clues in today’s puzzles.
When it comes to solving techniques, it turns out that the solution virtually always breaks down into two parts. The first is about the dissection of the grid into dominoes (and shaded cells), the second is about entering the numbers. You will find that there is limited interaction between the two parts; most of the dissection typically follows from the grid shape and the givens. Only very few freedoms remain, especially in smaller grids.
The dissection usually starts at the border of the grid, near “humps”. Since border cells are not allowed to be shaded, a domino position can be inferred if a cell is connected to only one other. For example, in the puzzle BD1 (short for “Blackout Dominoes 1”) from the PDF file, the two leftmost cells instantly lead to dominoes. Such steps can easily lead to more humps, as with the cell containing a 1 in the bottom row.
Humps with a width of two cells are also interesting. The crucial observation is that two dominoes cannot lie together to form a 2×2 square, since they would have to contain the same pair of numbers. As a consequence, dominoes cannot go from such a hump inside the grid. Have a look at BD2: If one of the two leftmost cells was part of a horizontal domino, so would be the other, violating the above principle. Therefore, the only possibility for these two cells is a vertical domino.
Note that this has implications for the cells close to the hump. In our case, there cannot be a second vertical domino next to the first one, so the lower of the adjacent cells – which lies on the border again – must be part of a horizontal domino. This leads to more humps in the bottom row.
Another hump of width 2 can be seen in the top row, which requires a horizontal cut. This time, neither of the two adjacent cells lies on the border. It is easy to make a mistake at this point, trying to draw more dominoes here, but in fact one of them might be shaded. The only thing that is certain is that they cannot form another horizontal domino. I suggest drawing a wall between them, even though it is not clear yet if this wall separates two dominoes or a domino and a shaded cell.
There are other interesting constellations, but I will only mention one of them at this point. Take a look at the second row from the top in BD3, in particular the second cell from the left in said row. (I will refrain from using the RxCy notation here.) The cell in question cannot be connected to the cell above, because that would yield two vertical dominoes in a 2×2 block. Likewise, it cannot be connected to the cell to its left.
Thus we obtain a few more walls. In some cases, they do not seem to do us much good; every now and then they serve to locate a shaded cell (or a domino), though. For instance, the respective cell near the top-right corner of BD1 cannot be connected to any other cell, applying the same technique four times. In general, this is a ueful step for humps of size 3.
One more thing regarding the dissection part. The total number of unshaded cells is always known from the start. (As you can see from the list of available dominoes, exactly 20 cells are needed for the 0-3 set, 30 for the 0-4 set and 42 for the 0-5 set.) It might be a good idea to count the cells in a grid right away, because if can be helpful to know the exact number of shaded cells.
The point is that only the interior cells can be shaded as per the rules. And sometimes there is not much room for as many shaded cells. For example, you can determine that three cells must be shaded in BD1; using the early hump steps, only one possibility remains without any shaded cells being adjacent to each other or the border.
We are approaching the part where the numbers come into play. Before we get to the actual techniques, you may notice that the unshaded cells form “groups” which must be filled with identical numbers – basically every time the cells are separated by walls.
Imagine for a second that all numbers are removed from the solved example grid, and only the walls and the shaded cells remain. The two upper cells in the leftmost column must contain the same number, i.e. they form a group of size 2. The lowest cell in the same column forms another group of size 2 with the cell right next to it. In the second row from the right there is a group of size 3. And so on.
Please observe that, near the top hump, we even have a group of size 5! Larger groups occur every time one domino touches two others in a 2×2 block. Note that this constellation contains in particular a domino with twice the same number – this may not sound like a big revelation, but the study of locations for those specific dominoes can be useful because they need either a large group or two adjacent groups.
The notion of “groups” is pretty much the foundation for all the filling that follows in the second part of the solve. If one entry in such a group is given, the rest can be entered immediately. For instance, it will turn out that there is a large group near the bottom of BD1, with a 1 inside it. This yields a lot of 1’s in this part of the grid.
Although it is primarily relevant for authors, I would like to point out that Blackout Dominoes are surprisingly rigid when it comes to creating them. Each number most occur exactly the same number of times (e.g. five times with the 0-3 domino set). The grid must be designed such that the groups work out accordingly.
Back to the solver’s perspective. Suppose we already have located a number four times (using the 0-3 set again). This means the same entry has to come up exactly once more. This requires a “dead end”, usually in a hump of size 1, but keep in mind that dead ends can also occur inside the grid using shaded cells.
An entry with three known occurrences needs two more, often in a group of size 2. (Two dead ends would be possible, but this is rather rare; one fitting group can be encountered far more often.) In larger puzzles, there are various ways to reach the required number of occurrences. As we have already seen, one single large group is possible.
In practice, large groups are quite common, since it is hard to have a unique solution otherwise. However, the groups cannot be too big. In areas which are known from earlier steps to contain many unshaded cells, shaded cells nearby are often needed to limit the scope of the respective groups.
Look at the the second cell from the right in the second row from the bottom in BD2. The early hump techniques give us a couple of domino positions in this part of the grid. If the cell in question was unshaded, it would create a rather large cluster with more than five equal entries – just try it. (This argument often leads to minor bifurcations.)
Let us continue with a couple of practical steps. If two adjacent numbers are given (as in BD2 or BD4), they must belong to the same domino. Remember, number cells cannot be shaded, and if there was a wall separating them, it would in effect create a group of size at least 2 with two different entries.
Another constellation I like to use consists of two different givens in cells with a common neighbor (as in two diagonally touching cells). The situation can be seen in BD3; look at the cell adjacent to the 1 and the 2. Such a cell can be shaded, or it can be connected to one of the given numbers. However, it cannot be part of a domino in one of the other two directions, or else it would create a group of size 3 with different entries.
As a consequence, a cell adjacent to three different entries must be shaded; it simply cannot accommodate the various groups next to it. This constellation comes up several times in this set, in BD3 as well as some of the larger puzzles.
When you are solving a Blackout Dominoes puzzle, starting the dissection from the humps and working your way down to the center of the grid (exploiting the above number constellations in the process), you will often reach a stage where large portions of the grid are filled and some minor pieces remain. Typically there are a few groups of size 2 and 3, and maybe a couple of dead ends as well.
When that happens, you will need some smaller, less impressive techniques to complete the grid. There is no brilliant strategy for this part of the solve, it is just a fuzzy bunch of tiny steps. Basically, you will have to use the rule that each domino must occur exactly once in the solution.
Suppose a large group has been located and some of its neighbors are already known. That group often has a few more “exits” which must then contain other numbers. For example, you will find a large group of 3’s in the east side of the BD2 grid (including a 3-3 domino), and the 1-3 domino has already been used, too. The cells adjacent to the remaining 3’s must therefore contain a 0 and a 2.
Of course this technique works just as well with several smaller groups containing the same entry, as long as many of its neighboring cells are already filled. Basically, whenever there is a large number of identical entries, it may be a good idea to study which of its potential “partners” have been used so far. Ideally this yields new entries, but even sets of candidates may help.
If a group of cells – in the above sense – is empty, a few deductions about its neighbors are still possible. In particular, two cells touching the same group cannot contain the same number; otherwise it would give rise to two identical dominoes. In the simplest case, the “outer” cells of two touching dominoes must contain two different entries. This technique serves only to eliminate a candidate in a specific position, but it may still be useful to know this step.
Finally, I want to stress once more that dominoes containing the same number twice are a good tool when it comes to the final filling stage. The point is that these dominoes cannot occur in isolation, they need more space. In BD1, for example, you will reach a position with several empty cells in the top-right part of the grid. For dominoes like 0-2 or 0-3 there will be several possible locations left, so you may want to study locations of the dominoes 0-0, 2-2 and 3-3 instead.
So much about solving techniques. Have fun!