# The Unique Rectangle

The solution of a valid Sudoku puzzle is unique. There should be only one. Yet puzzles are made and published that have multiple solutions. The Sysudoku reviews of published Sudoku collections have encountered a few. It’s not only that such Sudoku are more difficult, and embarrass their publishers. They can be invalid, having solutions that human solvers cannot find, because these puzzles violate logic that solvers use to solve them.

The customary title “unique rectangle” has led some to argue that the UR methods are invalid because they are based on an assumption that the puzzle has a unique solution. In fact, all logical methods assume that.

Unique rectangle methods are based on an additional assumption about puzzle composers. They will publish no puzzle with a solution like this one.  The 6 and 8 in the rectangle r69c78 can be interchanged for a second solution, without disturbing the placement of any other numbers. Call it a deadly rectangle.

Solvers use the fact that deadly rectangles are not published to eliminate a candidate whose placement would cause one, and to place a candidate necessary to prevent one.

Here is an example, from Frank Longo’s Absolute Nasty IV, # 71.

One signal that there is such an opportunity is an aligned naked pair, and matching values in two opposite corners. The rectangle that could resolve to the deadly one above is r68c78.

The UR action we take is to remove any candidate whose placement would remove  the extra candidates 3 and 5 in the rectangle. Or to place a candidate required to prevent their removal.

Because it is easy to spot, the Unique Rectangle, or UR, is one of the first methods to look for in Sysudoku Advanced. Once you have spotted a possible deadly rectangle, there are two ways to go about it. One is a DIY removal or confirmation that avoids it. The other is to diagnose the UR case as one of about six known types, each of which comes with an avoidance prescription.

“Do It Yourself” is certainly the more entertaining. Let’s look at a DIY on the 71 UR First of all, slinks are solid and winks are dashed. The rectangle is marked in grey, and the diamond signals removal.

A key bv involves both extras, and allows 5r6c3 to force out the 5 and both 3’s. Follow the curves. If 5r6c3 is true, 5r6c8 and 5r6c6 are false.  The internal slink of the bv then makes 3r6c6 true, sweeping the extra 3’s out.

We’ll come to the same conclusion by identifying the UR type, but first, let’s have Absolute Nasty 71 explain a fact about UR that is often cited, but seldom explained. It’s that deadly rectangles can only have corners in two boxes, and never in four.

To illustrate why, here is a grid from Frank Longo’s Absolute Nasty IV, # 71. The corner cells of the red dashed rectangle contain 1 and 6-candidates, but there is no possibility of a deadly rectangle of values 1 and 6.

We can’t remove 5r2c2 even though its placement will eliminate all 5’s from the rectangle.

The grid below shows why four box rectangles lose their interchange privileges.

Imagine this solution on the red dashed rectangle above.  The given N6 teams with the corner clue NW6 to place a 6 in the NE box. An interchange of 1’s and 6’s is prohibited by this placement

Now look at the 6 and 8 grey marked rectangle when the 3 and 5 extra candidates are removed. The interchange is permitted. There are no other 6 or 8 placements in rows 5 and 9, and the 6 and 8 placements in NEc9 are undisturbed by the interchange.

The grey rectangle r69c78 is therefore marked as a possible deadly rectangle, and our earlier DIY analysis exploits it for a removal. A nice, if Nasty illustration.

To classify a UR as one of eight types doesn’t require a photographic memory. We use a table. You can pick it up by following along, at least until we find the type of the Nasty 71 UR.

You can find the Sysudoku UR type table here or on the Tools page.    It was compiled from the Hodoku website, Stuart’s Logic of Sudoku, and the sudocue.net site. This Sysudoku version is modified, based on review experiences.

Let’s go through the types,  and show some review examples. Along the way, we’ll identify our Nasty case above.

Type 1

Type 1, or unique corner, has extra’s only in one corner.  The deadly rectangle occurs unless an extra is true. That’s not so helpful if there’s more than one extra, until we realize that the UR candidates must both be false. Here’s two each with two victims, Castillo’s Only Hard 338. S4 was enough for a collapse.

The smaller rectangle, r78c67 is a placement UR. 4r8c6 must be in the solution.

Type 2

Next  is the unique side, or Type 2. Any candidate seeing both of the single extras on a side of the rectangle must go.  Not a fit for 71, but a frequently encountered type.

Usually, the victim is a candidate in the same line, but here in A. D. Ardson Very Hard, v.2 318, two more 7’s from the N box are removed as well.

Are you thinking that 7r7c4 may not be guilty, because one of the other victims might have done it? Well, no. The other two are a group that had better be false, and after their removal, 7r7c4 is escorted out because the deadly rectangle is still possible.

Type 3

OK, moving down the table, the Type 3, or the unique subset, description fits the 71 UR. It has multiple extras in adjacent corners. Following the prescribed action, we treat the extra’s corner cells as one cell, in either box or line containing them, in this case E or r6. We combine the extra candidates as the contents of that cell, in this case 35, then remove any candidate the subset,  in this case r6np35, would remove, in this case 5r7c3.

Once you are familiar with the UR types, you will at least recognize when you are looking at one of them. In this case, a familiarity with Type 2 might have led you to 3r6c6 as a sweeper, and 5r6c3 which sweeps one of the extras and forces the sweeper of the other extra.

Type 4

Moving on to Type 4, or in the sudocue guide, the unique pair, the spotting description of the UR table says to look for a slink of UR partners on one side, with extas only in the same two corners.

In Longo’s Absolute Nastiest 673, two of Type 4 UR turn up in line marking, the first on marking r5, and the second on c4.

The Type 4 action is to remove the other UR partner in the same two corners.  The rationale doesn’t fitin the table, but it’s this: One slink partner is true, so if either UR partner in the extra corners is true, the two corners are reserved for the UR partners.  The extras are all removed.  Now to prove you’ve been awake and paying attention, decide on the removals before checking at the end of Type 6. A boxline on one of them brings a clue.

Like both of these, Type 4’s often turn up in dead X-wings.  We try to leave the values out of the fill string. A good time to look for a Type 4 UR is, therefore, when the marking of such an X-wing line is completed.

Type 5

Next in the UR table is Type 5, which had up to now, the restriction that each extra be the same value. The extra candidates are a toxic set, because at least one of them must be true to prevent the deadly rectangle. Any candidate seeing them all is removed.

A Type 5 UR of this description is rarely encountered, and if I had one in the reviews, I missed it. So here is the first one from the Hodoku techniques page on the UR.

Hodoku describes this puzzle as “the only published example of a UR Type 5”. It’s rare, yes. But don’t bet on that

Actually, the UR Type i5 is another application of the expanded “seeing” opportunities offered by the AIC weak link.  We are dropping the customary restriction of the UR table, holding UR  extras to a single value. This restriction enabled unit based winks to see them all. However, AIC winks enable a candidate can see candidates of other values. The UR often creates a toxic set of more than one value. Dropping the single value restriction admits many DIY forcing chain UR attacks to the  UR Type 5 and UR Type 2 fight clubs. In fact, three diagrams back is a Type 2 with victim seeing one extra with an XY-chain, one form of AIC.

Type 6

As defined by Hodoku, Type 6 is another rare bird, and we again go there for a full grid example. It’s a matter of all extras being in diagonal corners. If one of the UR candidates is confined to the rectangle, it’s true values will be diagonal. If either is allowed on the extras diagonal, both will be, and all extras are removed.

Sysudoku prefers to drop this restrictive definition of Type 6, in favor of an open Type 5, removing any outside candidate that sees all extras, excepting Type 1 and Type 2. The diagonal corners restriction is just one condition guaranteeing the removal. The wider definition allows us to drop the requirement that extras have the same value, allowing the entirely valid “seeing” by inference chains.

If you’re wondering about the almost solved grid in line marking, Hodoku examples are composed to arrive at the technique to be illustrated immediately after candidates are identified.

Here are the UR partner removals for the two Type 4 UR of Nastiest 673. Going on from here is not so nasty. DIY and look it up in the Nastiest review.

Hidden UR

Another rarity on the UR table is the Hidden UR, with multiple extras, and one corner free of them. Hidden requires one UR partner to slink in both directions away from the opposite corner.

This example is from the Tom Sheldon Master Class review, puzzle 120.  The opposite corner unslinked UR partner, 9r7c6 here, is true, it will remove extra 4r7c6 and slinked  7r7c6, placing 7’s in adjacent corners, and 9 in r8c9. Tom would never allow that.

Did you notice the  the dead 7-wing  in the hidden type 1?  The dead X-wing is a signal to look for a hidden UR.

The Type 2 and related Type 2b hidden UR hide a similar mechanism, until you notice their single UR slink. Requiring only one, they turn up more frequently than Type 1.

Consider this schematic of all three hidden UR a supplement to the UR chart above. Candidates x and y are extras, z is the UR value removed, and u is the other UR value. In Type 1, the offending opposite corner z removes the slink partner of two slinks, forcing two u’s and a z.

In the type 2 and 2b one side is a naked pair, often called “the floor” and extras are confined to the opposite side, “the ceiling”.  A single slink goes ceiling to floor on one side. In both, the unslinked UR candidate on the ceiling across from the slink partner must be removed, because it will force a deadly rectangle. In Type 2, the naked pair is in one box; in Type 2b it spans two boxes. The chain action is a bit different in the two types.

On the left is a Hidden UR Type 2b from KrazyDad Insane v.4, b.8, n.5. On the right another one from KD 4X5. The Type 2 may be more rare.

More DIY Examples

Playing around with possible deadly rectangles is fun, and you might have better results than  prescribed  in the chart. In this version of Sheldon’s Hidden UR Type 1 above, for example, the outside candidate 4r7c4 is removed, because it sees all of the extras, a toxic set. The subset r7s59 removes the HiddenType 1 victim, and much more. It’s a Type 6i, meeting the Hodoku requirement with victims of different value, via inference chain “seeing”.

And here’s another Type 6i with something else for you to look up in the Guide, under AIC building, a subset node.

One extra is seen by 5r1c1, which sees the other by taking 5r5c5 from an ALS to leave a naked triple, deleting 9r5c2 to leave 2 in charge, who promptly removes the other extra to make things deadly.

Avoidable UR

It is possible to have clues placed on UR corners, and still have removals or placements that are the only way to prevent a deadly rectangle.

Several types of avoidable rectangle, from Andrew Stuart’s The Logic of Sudoku, are illustrated in my post of January 22, 2013.  Here is a different case, where a Type 1 UR triggers a Type 1 avoidable UR. It’s from Week 48 of Paul Stephens’ Mastering Sudoku Week by Week. The first Type 1 is grey; the second is maroon. Go, Oxford High!

In the very next post, Beyond the Rectangle, of January 29, 2013 is a set of deadly loops of identical bv from the Sudocue guide.

The schematic illustrates a 10-node deadly loop. There are loops of every even number down to 4, the deadly rectangle.

These loops, being local BUG’s, have the same UR powers of removal and confirmation. Blue and green placements can be interchanged, without affecting any other cells.

No worries about spotting these if they ever arise, but the point is, to identify them when they are almost complete, and take the UR kind of preventative action.

Extended Rectangle

Another uniqueness situation very much related to the UR is the extended unique rectangle, in which three aligned pairs of four matching numbers can be resolved into two different solutions without affecting any other cells of the grid.

The one example I have from the reviews comes from Frank Longo’s The Nastiest Sudoku Book Ever – his title, not my opinion – with the review selected puzzle 651. 9r5c5 must be removed to avoid the deadly loop of 2, 4 and 6.