Unique Rectangle

This page discusses ways of exploiting Unique Rectangles, but it assumes that you know the difference between a Deadly Rectangle and a Unique Rectangle, and that you know what a Hero is. Read the page on Deadly Patterns before starting to work your way through the material below.

UR &mdash this abbreviation for Unique Rectangle will be used throughout.

In all the Grids below, the Heroes are shown in green and written last in a Cell's list of candies — this is done to make the Heroes stand out for purposes of discussion.

To keep track of what you're doing when you exploit a UR, you'll find it extremely helpful to make use of a set of coloured pencils (available at any stationery store). Border-colour the four Cells of the UR, and circle all its Heroes in colour — that will allow you to not lose track of the candies that are the focus of all your efforts. If you're constructing a long Chain and you want to label the Chain Cells (for purposes of verification), label them with coloured letters — those will stand out, but they won't obscure the primary markup (the candies themselves).

 

UR, 1-corner

A 1-corner UR has one or more Heroes in (only) one of its four corners. The Hero candies are shown in green; the deadly candies are left in black.

1-corner URs (two of them). The orange UR has a 3 Hero in one of its corners. In that corner, you can kill the deadly 2 and 7 (leaving only the 3), which will prevent this UR from becoming a Deadly Rectangle. Similarly, the blue UR has 4, 6, and 8 Heroes in one of its corners. In that corner, you can kill the deadly 1 and 5 (leaving the 4, 6, and 8), which will prevent this UR from becoming a Deadly Rectangle. What's on the rest of the Grid (in the white Cells) is irrelevant to the logic in the 1-corner case.

1-corner URs are obviously very easy to exploit. And, surprisingly, they frequently give rise to a very productive cascade of moves, so you should always take advantage of a 1-corner UR as soon as you notice it.

Multiple-corner URs require a lot more thought (but then, the reason you do Sudokus is that you like to think). Also, they don't usually give rise to an immediate long cascade of moves, but they help to clean out the Grid.

The rest of this page will deal with URs that have Heroes in 2, 3, or 4 corners.

 

UR, 2-side-corner

The orange UR below has a 7 Hero in one corner and a 4 Hero in another corner on the same side of the rectangle (i.e., not diagonally across from each other).  It's a 2-side-corner UR.

It could happen that you might find an example like this discussed as follows:

2-side-corner UR (less than optimal exploitation). The orange UR has a 7 Hero in one corner and a 4 Hero in another. At least one of those two Heroes must be true in order for this UR not to be a Deadly Rectangle. If the 4 Hero is true, that makes the purple Cell = 5, which would kill all the other 5s in its Column. If the 7 Hero is true, that makes the yellow Cell = 5, which would kill all the other 5s in the Column. In either case, the 5s in the orange Cells in that Column are killed, so we conclude that we can in fact kill the 5s in the two orange Cells in Column 6.

But this conclusion, although true, is a weak result, as it does not give rise to a Crowning in any Cell. We can often do better than this, as indicated immediately below.

2-side-corner UR:  same Grid as above, better exploitation. If the 4 Hero were true, it would obviously kill the 4 in the purple Cell. So, what if the 7 Hero were true? Assume it is; then follow the Chain ABCDE, and you will see that Cell E = 4, which would again kill the 4 in the purple Cell. Well, one of those two Heroes has to be true, and in either case the 4 in the purple Cell gets killed, so our conclusion is that we can in fact kill the 4 in the purple Cell (which will then be crowned as a 5).

This is a much stronger result because it results in a Crowning. That newly crowned 5 will produce a significant number of Kills and Crownings, and that's the kind of thing we want.

All of the remaining examples will be oriented towards exploiting a multi-corner UR in such a way as to result in the Crowning of a candie in some Cell. Roughly speaking, about 30% of the time you can succeed in doing this without having a heart attack.

The argument given just above is called a Consensus argument:  assuming either Hero to be true gives rise to the same conclusion, so by Consensus we know that that conclusion must be true (since at least one of the Heroes must be true to prevent the occurrence of a Deadly Rectangle).

There is an alternative approach called a Troublemaker argument. In the example below, both kinds of argument will be presented.

2-side-corner UR. The orange UR has a 1 Hero and a 3 Hero. At least one of those two Heroes must be true. Consensus:  if the 3 Hero is true, the 3 in the purple Cell gets killed; but if the 1 Hero is true, then look at the yellow Cell — the 3 in the purple Cell will still be killed. So we have agreement, and we can in fact kill the purple Cell's 3. Now we'll start all over with a different argument. Troublemaker:  what if the 3 in the purple Cell were true? It would kill the 3 Hero. And in fact, because of the yellow Cell, it would also kill the 1 Hero. But you can't kill all the Heroes, because that would make the UR a Deadly Rectangle. So we eliminate the Troublemaker — we kill the 3 in the purple Cell.

You can see that the Consensus approach and the Troublemaker approach are logically equivalent. In simple cases, both approaches are obvious. But in more complex situations, one of the two approaches may be easier than the other.

In practice, Consensus is usually harder, since you don't know what candie all the Heroes would kill, and you have to keep adjusting your sights to find it.

For the Troublemaker approach, of course you don't know which candie will be a good Troublemaker, but you know what your objective is — find a Troublemaker that kills all the Heroes — so you can try likely Troublemakers and quickly discard the ones that get you nowhere.

We'll do one more example using both approaches:

2-side-corner UR. The orange UR has an 8 Hero in one corner and 3 and 8 Heroes in another corner. At least one of these three Heroes must be true. Consensus:  if the 8 Hero is true, the 8 in the purple Cell gets killed; but if the 3-or-8 Hero is true, then it forms a Locked Pair with the 3-or-8 in the yellow Cell, which will still kill the 8 in the purple Cell. We have agreement, so we can in fact kill the purple Cell's 8. The other argument: Troublemaker:  what if the 8 in the purple Cell were true? It would kill the 8 Heroes in both orange Cells. And in fact, because of the yellow Cell, it would also kill the 3 Hero. But you can't kill all the Heroes, since that would make the UR a Deadly Rectangle. So we eliminate the Troublemaker — we kill the 8 in the purple Cell.

In the rest of the examples on this page, I will use only a Consensus argument or only a Troublemaker argument. (You will see that I have a predilection for the Troublemaker approach.)

And I will cut the verbiage a bit, as I assume you will remember the underlying arguments:

• Consensus:  if the assumed truth of every Hero (one by one) implies that one specific candie in one specific Cell is killed, then that candie must be killed (since at least one of the Heroes must be true).

• Troublemaker:  if some candie's being true would mean that all the Heroes were killed, then that candie must be false (since at least one of the Heroes must be true).

In the following UR, there are a total of three Heroes — a 5, a 6, and an 8. There are a number of Cells in the vicinity containing a 5, 6, or 8 candie that might conceivably be a Troublemaker, but most of them don't work. So you have to keep trying in the hope of finding one that does.

2-side-corner UR. The orange UR contains a 5 Hero in one corner, and a 6-or-8 Hero in another corner. Troublemaker:  the 8 in the purple Cell kills the 8 Hero, but it also creates a 6 and a 5 in the yellow Cells which will kill the 6 and 5 Heroes. So the 8 in the purple Cell must be false, and we can kill it.

The 8 in the {7 8} Cell at R5C4 would also qualify as a Troublemaker, but the argument is more complicated; can you figure it out? (Admittedly, of course, once you've found one Troublemaker, you don't need to look for another one.)

Here's another example where there are many potential Troublemakers, and you have to find one that actually works.

2-side-corner UR. The orange UR contains a 7 Hero in one corner and a 3 Hero in another corner. Troublemaker:  the 7 in the purple Cell kills the 7 Hero.  And if you follow the Chain ABCD with Cell A = 7, you'll find Cell D = 3, which kills the 3 Hero. So the 7 in the purple Cell must be false, and we can kill it.

Once I've told you that the 7 in the purple Cell above is a verifiable Troublemaker, you can easily follow my logic proving that it is. But finding a Troublemaker can take a lot of looking. Try it on this Grid: there are other candies here that can be proved to be Troublemakers for this UR (but the argument Chains are fairly long), and there are a lot of dead ends (candies that you can't prove to be Troublemakers).

The more you look for Troublemakers, the better you'll get at it, but sometimes one just can't be found. You'll have to develop a sense of when to give up and go on to other Tactics (there's no reason that you have to exploit a UR).

I will work the following example twice, to show a contrast. The first argument is fairly difficult.

2-side-corner UR. The orange UR has 7 and 9 Heroes in two different corners. Troublemaker:  the 9 in the purple Cell kills the 9 Hero. And if you follow the Chain ABBBCDE with Cell A = 9, you'll find that Cell E = 7, which kills the 7 Hero. So the 9 in the purple Cell must be false, and we can kill it. (Cell A = 9 changes Cells BBB into a 5,6,7 Locked Trio, which kills the 5 in Cell C.)

If you think that was easy, good for you. But maybe you found it tricky, that business of turning the Almost Locked Set in BBB into a genuine Locked Set (via the application of Cell A) — if so, you will be interested in reading through more examples of this gimmick in the page on Chains.

There's a different Troublemaker in this Grid that's a little easier to verify, as shown immediately below.

2-side-corner UR:  same Grid as above, slightly easier argument. Troublemaker:  the 7 in the purple Cell kills the 7 Hero. And if you follow the Chain ABBC with Cell A = 7, you'll find that Cell C = 9, which kills the 9 Hero. So the 7 in the purple Cell must be false, and we can kill it. (Cell A = 7 changes Cells BB into a 5,8 Locked Pair, which kills the 5 in Cell C.)

I will point out again that once you have found a Troublemaker, there's no reason to look for another one. The point of the above example is simply that when there are multiple Troublemakers that work, some of them will be harder to verify than others.

Now:  up until this point, I have always looked for a Troublemaker that lies in a Pair Cell, which is exactly what I want:  killing the Troublemaker then results in a Crowning, which may generate a series of moves.

So the following example will seem strange. I didn't find a Troublemaker in a Pair Cell, so I picked one lying in a Trio. We'll see why.

2-side-corner UR:  Troublemaker not in a Pair Cell. The orange UR contains 4 and 7 Heroes in two different corners. Troublemaker:  the 4 in the purple Cell kills the 4 Hero. And if you follow the Chain ABCDE with Cell A = 4, you'll find that Cell E =7, which kills the 7 Hero. So the 4 in the purple Cell must be false, and we can kill it. Is that fairly useless?  No.  Now Cells A and C contain a 3,8 Locked Pair, which kills the 8 in Cell B. The 4 in Cell B gets crowned, and that gives rise to a series of moves.

So when you're desperate enough to look for a Troublemaker lying in a Cell that's not a Pair Cell, be sure that killing that Troublemaker would actually do you some good (by producing a Crowning one way or another):  it isn't worth the time and effort to locate a Troublemaker that won't give you a big payoff when you kill it.

 

UR, 2-far-corner

The orange UR below has a 6 Hero in one corner and a 7 Hero in another corner. The two corners are diagonally across from each other; I will refer to this as a 2-far-corner UR.

2-far-corner UR. The orange UR has 6 and 7 Heroes in two different corners. Troublemaker:  the 7 in the purple Cell kills the 7 Hero. And if you follow the Chain ABCDEF with Cell A = 7, you'll find that Cell F = 6, which kills the 6 Hero. So the 7 in the purple Cell must be false, and we can kill it.

That was a long Chain. That's the way it is with 2-far-corner URs.

Here's one that's even worse:

2-far-corner UR. The orange UR has a 1 Hero in one corner, and 2 and 5 Heroes in another corner. Troublemaker:  the 1 in the purple Cell kills the 1 Hero. And if you follow the ABCDEEFGH Chain with Cell A =1, you'll find that one of the E Cells = 5 and Cell H = 2, so that the 5 and 2 Heroes are also killed. So the 1 in the purple Cell must be false, and we can kill it. The hypothetically killed candies are grayed out in the Chain so you can see what's going on. (A and C kill the 1 and 4 in D.) (D creates the 1 and 5 in EE, which then kill the 1 and 5 in F.)

How long did it take me to find that Chain?  Forever.  Was it worth it?  Definitely not; this Sudoku would have succombed to other Chain Tactics independent of the UR.

If you work on a really awful UR like this, you will of course sharpen your skills at constructing Chains. You will also sharpen your sense of when you should give up on a UR and proceed to a Chain or a Smart Fork Tactic.

2-far-corner URs tend to be noticeably harder to exploit than 2-side-corner URs.

 

UR, 3-corner

A 3-corner UR has one or more Heroes in each of three different corners.

They're usually not easy, but this one is:

3-corner UR. The orange UR has 6, 6, and 8 Heroes in three of its corners. Consensus:  if the either of the 6 Heroes is true, the 6 in the purple Cell gets killed; but if the 8 Hero is true, then look at the yellow Cells — the 6 in the purple Cell will still be killed. So we have agreement, and we can kill the 6 in the purple Cell. Troublemaker:  same result; you should be able to figure it out for yourself.

3-corner URs are rarely that easy.

The following one is a little harder, but it's still easier than the typical case.

3-corner UR. The orange UR has 4, 9, and 2 Heroes in three of its corners. Troublemaker:  the 9 in the purple Cell kills the 9 Hero, and with the aid of the yellow Cell, it also kills the 4 Hero. And if you follow theChain ABCDE with A = 9, you'll find that Cell E = 2, which kills the 2 Hero. So the 9 in the purple Cell must be false, and we can kill it.

The killing of two Heroes simultaneously by the combined effect of the purple and yellow Cells above is the kind of thing you want to keep your eye out for when trying to find a Troublemaker for a multiple-corner UR.

That was still relatively easy. The following example is more typical of the difficulty of exploiting a 3-corner UR.

3-corner UR. The orange UR has 1, 4, and 7 Heroes in three of its corners. Troublemaker:  the 4 in the purple Cell kills the 4 Hero. With the aid of the yellow Cells, it also kills the 7 Hero. And if you follow the Chain ABCD with Cell A = 4, you will find that Cell D = 1, which kills the 1 Hero. So the 4 in the purple Cell must be false, and we can kill it.

Well, that's all very glib, isn't it?  Again, of course, it took me a long, long time to reach that solution, and it wasn't worth it.  This Sudoku is fairly difficult, but it would have succombed to a Chain Tactic independent of the UR.

Don't spend too much time on 3-corner URs; most of them are intractable.

 

UR, 4-corner

Well, I shouldn't put this example in here, as my goal in general is to present Tactics that apply to frequently occurring patterns. But I couldn't resist the temptation to include one 4-corner UR.

The following is an extremely special case; I've seen it occur precisely three times after doing several hundred Sudokus. Each time, amazingly, the 4-corner UR was exploitable.

The orange and blue Unique Rectangle below contains the candies 2,3,8 in each of its four corners. Which are the deadly candies and which are the Heroes? We don't care. If some Troublemaker were to kill all the 8s in this rectangle, then it would be a Deadly Rectangle in 2,3. (A similar remark applies if a Troublemaker were to kill all four 2s or all four 3s.)

4-corner UR, special case. The orange and blue UR contains candies 2,3,8 in each of its four corners. We could look for a Troublemaker 2-killer or 3-killer or 8-killer. In fact, there exists an 8-killer. Troublemaker: the 8 in the purple Cell kills the 8s in the blue UR Cells. And if you follow the Chain ABCD with Cell A = 8, you will find that Cell D = 8, which kills the 8s in the orange UR Cells. But that would be an impossible situation, because we would wind up with a Deadly Rectangle in 2,3. Therefore the 8 in the purple Cell can't be true, and we can kill it.

If you ever do see this special case arise, it's easy enough to see if you can find a Troublemaker like the one above.

But in general, 4-corner URs are absolutely intractable, and you shouldn't waste any time on them at all.

 

Minimal-result URs

Related external ideas:  type 4 UR; hidden UR.

You recall what Twins are:

• If a Row has only two 3-candies, they are Row-Twin 3s: they may be tagged with an underline (3).

• If a Column has only two 4-candies, they are Column-Twin 4s: they may be tagged with a vertical bar (4).

• If a House contains Twin 9s then one of those 9s has to be true.

There are certain cases where one of the deadly candies in a UR is a Row and/or Column Twin in some corners, and a logical argument can be made that the other deadly candie can be killed in certain corners.

I know that remark is pretty opaque, but we'll look at three examples and you'll see how it works.

This Tactic is frequently mentioned on Sudoku Web sites (partly because the logic is interesting), so it's good to be familiar with the idea. But it does have a distinct disadvantage — it doesn't produce a cascade of Kills and Crowns, so its payoff is often negligible:  it gives a minimal result.

Because of that, I will indicate, in the three examples below, better moves that exist on the Grid. (But of course there isn't always a better move.)

A 2-side-corner example:

2-side-corner minimal-result UR. This blue and pink UR in 3,4 has a Hero in each of its blue Cells. At least one of those two Heroes has to be true. The deadly 4s in the blue Cells happen to be Row-Twins. So one of those two 4s has to be true. Thus one of the blue Cells has to have its 4 true, and one of them has to have its Hero true. Result:  that means that neither of the blue Cells can have its 3 true. So we can kill the deadly 3s in both blue Cells. (I don't care about the Twin status of the 4s in the pink Cells.)

Killing those 3s looks like a weak result — it gives no Crownings, so it produces no immediate moves.

If you can't do better, then you settle for that result. But before you do, it pays to check first to see if there's an approach with a better payoff.

In the case above, there is a Troublemaker: the 1 in R2C8 — that 1 would kill both Heroes (the Chain needs only the Cells in Row 2). Eliminating that 1 causes a new 2,3,5 Locked Trio to spring into existence, and that Locked Trio produces a Kill and a Crowning that gives rise to a cascade of moves that cracks the Sudoku.

So you might say that if there's a Troublemaker, then using it would be better than using the minimal-result Twin-based approach. Very often that's true.

But of course every Grid is different, and I've neatly glossed over something here. When that Twin-based argument killed the 3s in the blue Cells, what was left on the Grid? A BUG+2, and a really easy one to exploit at that. (See the BUG+n page.)

The moral of the story is that the minimal-result Twin-based approach usually won't do much for you, but when the Grid's empty Cells are mostly Pairs, the minimal result may in fact be useful.

Side remark:  all four of the deadly 4s in this example are Twins — they lie in a 4-Cell Twin Loop. This is a very common occurrence, so it's not surprising. Nevertheless, only the Row-Twin status of the 4s in the blue Cells is important to the argument. This situation will arise in the two remaining examples too, but I won't mention it again.

Here's a 2-far-corner example:

2-far-corner minimal-result UR. This orange, blue, and pink UR in 4,8 has a Hero in each of its orange Cells. At least one of those two Heroes has to be true. The deadly 8s happen to be Row-Twins in Row 1 and Column-Twins in Column 8. Suppose the blue-Cell 8 is false:  then its Twin 8s in the orange Cells are both true — which kills all the Heroes (which can't happen). Result:  that means the blue-Cell 8 can't be false, so we can crown it.

In this Grid, that's definitely a weak result. The 8s get crowned in the blue and pink Cells and killed in the orange Cells, but that's it. No more moves.

So it would be worth looking for a Troublemaker, and there is one: the 3 in R1C4 would kill both Heroes (the Chain needs only R1C4 and the Cells in Row 4). Killing that Troublemaker unleashes a cascade of moves that cracks the Sudoku.

And last, a 3-corner example:

3-corner minimal-result UR. This orange, blue, and pink UR in 2,6 has various Heroes in three places, in its blue and orange corners. At least one of those Heroes has to be true. The deadly 6s happen to be Row-Twins in Row 1 and Column-Twins in Column 8. There are only two possibilities: The blue-Cell 6 is true (which kills the blue-Cell 2). Or the blue-Cell 6 is false, so its Twin 6s in the orange Cells are both true, which kills the orange-Cell Heroes, so one of the remaining Heroes (in the blue Cell) must be true (which again kills the blue-Cell 2). Result:  that means the blue-Cell 2 can't be true, so we can kill it.

And that is a really weak result, especially in this particular Grid.

And with the scarcity of Pair Cells on the Grid, I can't find a Troublemaker for this UR.

As a rule of thumb, when the Grid looks this bad and when you can't get anywhere with URs, it's a good idea to see if you've missed anything fundamental.

In this particular case, there's an unused Sashimi in 8 in Rows 5 and 9. Applying this Sashimi has the effect of obliterating the UR we were working on. It also causes a couple of other (simpler) URs to spring into existence, and both of these are tractable. But it does not crack the Sudoku. Finally I had to use a Gotcha Chain (see the Chains page).

The point of all this blah-blah is that this Grid was very complicated, and the minimal result available from the Twin-based argument for the UR was not going to be of much help at all.

However, many Grids aren't this bad, so the bottom line is this:  if you're hard up for moves and you see a UR for which this Twin-based argument works, go ahead and take advantage of it if no Troublemaker is readily available.

 

6-Cell Flip-flop+n

OK, OK, a 6-Cell Flip-flop+n isn't a rectangle. But I don't want to devote an entire page to these, and the ideas involved in handling them are the same as those for Unique Rectangles.

The discussion below assumes that you know the difference between a Flip-flop and a Flip-flop+n, and that you know what the various shapes of a 6-Cell Flip-flop+n look like. Read the page on Deadly Patterns before going through the following material.

A Unique Rectangle is of course just a 4-Cell Flip-flop+n where n is the number of Hero Cells (and for a UR we called the Hero Cells corners, for obvious reasons). Similarly, a 6-Cell Flip-flop+n is composed of 6 Cells, n of which contain one or more Heroes. There are basically four different possible shapes that a 6-Cell Flip-flop+n can take on, and we'll look at them all.

It is occasionally convenient to refer to a Unique Rectangle as an Almost Deadly Rectangle. Similarly, we'll find it helpful to use the term Almost Flip-flop to designate a Flip-flop+n:  a Flip-flop+n is "almost" a Flip-flop in the sense that it contains n Hero Cells which prevent it from degenerating into a Flip-flop.

You recall from the Deadly Patterns page that a Flip-flop is a Deadly Pattern, so it cannot appear in a Sudoku. But a Flip-flop+n (an Almost Flip-flop) is not a Deadly Pattern, so it can appear in a Sudoku (and often does).

The first shape:  this is a 6-Cell Almost Flip-flop having 2 deadly candies and occupying 3 Blocks aligned straight. This particular one has two Hero Cells (it's a Flip-flop+2).

6-Cell Flip-flop+2. First concentrate on only the orange Cells, and note that the underlying {57} Deadly Pattern satisfies the definition of a Flip-flop. This Pattern is saved by its two Heroes, a 1 and an 8. Troublemaker:  what if the 1 in the purple Cell were true? It would kill the 1 Hero. And in fact, because of the yellow Cell, it would also kill the 8 Hero. But you can't kill all the Heroes, because that would make this Flip-flop+2 degenerate into a pure Flip-flop, which is a Deadly Pattern. So we eliminate the Troublemaker — we kill the 1 in the purple Cell.

The second shape:  this is a 6-Cell Almost Flip-flop having 2 deadly candies and occupying 3 Blocks, but it's bent. This particular one has three Hero Cells (it's a Flip-flop+3).

6-Cell Flip-flop+3. First concentrate on only the orange Cells, and note that the underlying {24} Deadly Pattern satisfies the definition of a Flip-flop. This Pattern is saved by its three Heroes, a 6, a 3, and a 9. Troublemaker:  what if the 2 in the purple Cell were true? Because of the yellow Cells, it would kill the 6 Hero. And with the help of the green Cells, it would kill the 3 Hero. Finally, with the aid of the blue Cell, it would kill the 9 Hero. But you can't kill all the Heroes, because that would make this Flip-flop+3 degenerate into a pure Flip-flop, which is a Deadly Pattern. So we eliminate the Troublemaker — we kill the 2 in the purple Cell.

There's something unusual about the above example:  the Troublemaker is a candie that doesn't equal any of the Heroes, so none of its Kills is direct. It took me awhile to find it, but not as long as you might think, because I wanted a Troublemaker lying in a Pair Cell, and there weren't very many promising-looking Pair Cells available.

The third shape:  this is a 6-Cell Almost Flip-flop having 3 deadly candies and occupying 2 Blocks. This particular one has only one Hero Cell (it's a Flip-flop+1).

6-Cell Flip-flop+1. First look at the orange Cells, and note that the underlying {13}, {17}, {37} Deadly Pattern satisfies the definition of a Flip-flop. This Pattern is saved by its lone Hero, a 5. In the one single Hero Cell, you can kill the deadly 3 and 7 (leaving the 5), which will prevent this Flip-flop+1 from degenerating into a pure Flip-flop, which is a Deadly Pattern. What's on the rest of the Grid (in the white Cells) is irrelevant to the logic in the single-Hero-Cell case.

The above Pattern is the first one we've seen that involves more than just two deadly candies. This has the perverse effect that, on a Grid loaded with numbers, a Pattern like this turns out to be hard to notice, so in fact I don't actively look for it. There are enough other Tactics that will lead me to a Solution.

The fourth shape:  this is a 6-Cell Almost Flip-flop having 3 deadly candies and occupying 3 Blocks. This particular one has three Hero Cells (it's a Flip-flop+3).

6-Cell Flip-flop+3. First concentrate on only the orange Cells, and note that the underlying {56}, {58}, {68} Deadly Pattern satisfies the definition of a Flip-flop. This Pattern is saved by its three Heroes, a 4, a 7, and a 7. Troublemaker:  what if the 7 in the purple Cell were true? It would kill both the 7 Heroes. And with the help of the yellow Cell, it would also kill the 4 Hero. But you can't kill all the Heroes, because that would make this Flip-flop+3 degenerate into a pure Flip-flop, which is a Deadly Pattern. So we eliminate the Troublemaker — we kill the 7 in the purple Cell.

Because this Pattern involves three deadly candies, in practice it's hard to notice.

That's the last of the possible shapes for a 6-Cell Flip-flop+n; there are no more 6-Cell shapes satisfying the heavy requirements for a Flip-flop.

Are there larger Flip-flops on which larger Almost Flip-flops are based? Yes, you can find some of them at

• www.sudopedia.org/wiki/Deadly_pattern

but most people don't bother with these, partly because some of them are hard to recognize, but mostly because an Almost Flip-flop consisting of lots of Cells will tend to have multiple Heroes in many of them, and finding a Troublemaker that would kill all the Heroes becomes impossible.

 

OK, So What Do We Do with All This?

• At any time whatsoever that you notice a 1-corner UR, exploit it immediately.

• Carry out Local Tactics and sweep for Fish before you actively scan for URs. (Reason:  you want URs that have as few Heroes as possible so that you stand some chance of finding a Troublemaker.)

• A UR lies entirely within one Band or one Tower, so scan each Band and each Tower looking for a UR; this doesn't take long.

• If you see a 1-corner UR, exploit it.

• If you see a 2-side-corner UR that has only a single Hero in each of its two corners, try to find a Troublemaker:  you stand a reasonable chance.

• You almost always want to find a Troublemaker lying in a Pair Cell: that way when you kill it, you will get a Crowning, which will probably generate a series of moves.

• For other multi-corner URs, look for a Troublemaker if this Tactic generally appeals to you; but don't spend forever on it.

• You can learn to notice the first two shapes of 6-Cell Almost Flip-flops while you're scanning for URs.

• The last two shapes of 6-Cell Almost Flip-flops are harder to recognize; most people won't look for them.

• If you do notice a 6-Cell Almost Flip-flop that has Heroes in only one of its Cells, exploit it. (You will probably never see this.)

• If you do notice a 6-Cell Almost Flip-flop that has only a single Hero in two of its Cells, try to find a Troublemaker:  you stand a reasonable chance.

• Most of the 6-Cell Almost Flip-flops you see will have way too many Heroes; don't waste your time looking for a Troublemaker for these.