The Unique Rectangle

The solution of a valid Sudoku puzzle is unique, meaning there is only one. Yet puzzles are made and published that have multiple solutions. Reviewing published Sudoku collections, I have encountered many. It’s not just that multiple solution Sudoku are more difficult. It’s not that they embarrass their publishers. Multiple solution puzzles are invalid. They can have solutions that human solvers cannot find, because they violate logic that solvers use to solve them.

It is an inhuman problem to determine whether or not a puzzle has a unique solution. It can be done by computer, using a systematic trial of every possible solution. In fact, a multiple solution can be attributed to a faulty algorithm or not running it long enough. It is not a challenge for the solver. It is an embarrassment for the composer and the publisher.

For these reasons, uniqueness of solution is assumed by every solver on every puzzle. The customary title  “unique rectangle” has led some to argue that the UR methods assume the puzzle has a unique solution. Nonsense. Rather, the UR is based on an assumption about assumed unique solution, that it will not have a flaming badge of multiplicity on its back and front. It will not have a so called “deadly rectangle” of two numbers, matching on opposite corners, which can be interchanged on one side to create another solution.

The Unique Rectangle, or UR, is one of the first two methods to look for in Sysudoku Advanced. The other is the remote pair. We don’t start these pages with the remote pair, however, because it is a special case of an XY-chain, which is a special case of an alternating inference chain. 

Scanning for a UR, look for identical bv on two corners of a rectangle, with the same two numbers appearing in the other two corners.

Here is a case from Frank Longo’s Absolute Nasty IV, # 71. To have a deadly rectangle, should eliminations duplicate the bv on the other corners, all corners have to be contained in two, not four, boxes.  So the removal of 5r2c2 would not make the red dashed rectangle a possible UR.

The grey rectangle qualifies, and we’ll keep it in mind as we look at available means of UR attack.

There are two ways to go about it. Perhaps the most fun is to do your own search for a removal or confirmation that avoids the deadly rectangle. The other is to diagnose the UR case as one of about six known types, which come with an avoidance prescription. As to the fun choice, let’s just leave it as a challenge. We’ll go through the types, as I did in the review, and find a single removal that does it.

The UR type table resides on the Tools page.  It was compiled from Stuart’s Logic of Sudoku, and the site, and modified, based on review experiences. Let’s go through the types, and compare with our Nasty case above.

Type 1

Type 1, or unique corner, has extra’s 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 one with the added confirmation of 4r4c5 that ends Longo’s Nastiest Ever 607.


Type 2

Next try for the 71 UR is the unique side, or Type 2. Single extras of the name number on a side allows an outside candidate on the same line to sweep both, so every one of them must go. Not a fit for 71, but a frequently encountered one. It also comes with a likely bonus. Here in A. D. Ardson Very Hard, v.2 318, 7r7c4 is removed, because in the solution it comes with a deadly rectangle. The removal generates a Nc4 boxline, removing two more 7’s from the N box.

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 5r7c2.

Did you find it? 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.

To find this kind of DIY UR removal, and to document it once found, I use alternating chains as introduced on the Sysudoku Basic page. In this version of the UR, 5r6c3 would remove one of the extra candidates by wink and the other, by AIC wink.

Looking back, this can also be considered an AIC enabled Type 2 UR. We’ll call it a Type i2, the “i” standing for irregular. Looking forward in this page, it’s also an AIC enabled Type i5.

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 the page. 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.

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.






Type 5

Next in the UR table is Type 5, which had up to now, the restriction that each extra be the same number. 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

Type 6 is another rare bird, and we again turn to Hodoku 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.

Hodoku examples are composed to arrive at the technique to be illustrated immediately after candidates are identified. Sysudoku Basic on these examples can be quite untypical.

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 one extra and the slinked candidate, making both slink partners true, and all extras are removed.

As you see above, finding a rare UR and its removal doesn’t necessarily have much impact. This DIY AIC wink version packs more punch.  The outside candidate 4r7c4 is removed, because it see all of the extras, the toxic set. The subset r7s59 removes the Type 6 victim, and another corner 9, making the deadly rectangle impossible.

However, this action doesn’t make it an irregular hidden UR, because that would imply the same logic, except for substituting an AIC wink for a unit wink.

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. Some Sudoku writers have taken it as significant that puzzle composers have to avoid forcing deadly rectangles as they remove clues to form puzzles. This belief leads them to imply that such opportunities exist only if the corner clues are not givens.

Actually, it’s less complicated than that.  As stated at the beginning of this page, solvers always assume uniqueness of the puzzle, not a method. It is a rule of Sudoku, because otherwise, solving logic is unreliable and solving is a waste of time.  In using UR’s, solvers continue to assume uniqueness, but additionally, that even if that fails, a published Sudoku solution will not contain rectangles with interchangeable adjacent corners. It doesn’t matter how a deadly rectangle occurs.  It doesn’t matter if the rectangle contains givens, or does not. If acting to prevent a deadly rectangle leads to a dead end, it is an invalid puzzle.

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!

Deadly Loops

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 to keep them from forming. As of now, I have no case, but perhaps a sysudokie reader will send one in.

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 case 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 double solution.



In general the pairs have to be along two lines through the same boxes. Additional candidates of the four numbers on those lines must be paired to be indifferent to the two solutions.