## SudoCue - Sudoku Solving Guide |

Last update: January 7, 2007 (Thanks to VK for spotting an incorrect puzzle string for the hidden triple example). This page is protected by copyright
If you want to solve Our user forum is the best place to express your comments about this solving guide, but you can also use the contact form. Do not forget to include your email address if you want a personal answer. Your opinion is highly valued, and many of the strategies in this guide are the products of giants, whose shoulders have helped me see farther. ## Table of Contents- A brief Sudoku History
- The Grid
- Da Rule
- The Givens
- Symmetry
- Minimal Puzzles
- Using Logic
- Sudoku Controversies
- Difficulty with Ratings
- Solving Terms
- Naked Singles
- Last Digits
- Hidden Singles
- Squeezing
- Cross-Hatching
- Locked Candidates
- Disjoint Subsets
- Row/Column Subsets
- Connected Pairs and Coloring
- Empty Rectangle or Hinge
- Remote Pair
- Finned Fish
- XY-Wing
- XYZ-Wing
- Aligned Pair Exclusions (APE)
- Almost Locked Sets - XZ Pattern
- Subset Counting
- Advanced 3D Medusa Coloring
- Bivalue Universal Grave (BUG)
- Uniqueness Test
## A brief Sudoku History
The basis of
The modern version started as a game called “Number Place”, in all probability designed by the late
The puzzle was introduced in Japan by the publisher Nikoli in 1984,
using the name “Sudoku”,
which is an abbreviation for the original Japanese name “Suuji wa dokushin ni kagiru”,
meaning “The Numbers must be Single”.
Not only did
By the end of 2004, Sudoku appeared in ## The Grid
## Rows
## Columns
## Boxes
## CoordinatesRows and columns are used as the coordinates for the cells in the grid. A cell at row 3, column 5 will be addressed as R3C5. You may encounter Sudoku descriptions where the cells are addressed (X,Y), where X is the column and Y is the row. This is probably a more formal way to define cells, but may lead to confusion. Mathematicians also tend to number the rows from bottom to top, as in graphs. Attempts have been made to use a coordinate system with a combination of a letter and a digit as used in the game of chess. Again, confusion led to systems where the letters were assigned to the rows, and other systems with the letters assigned to the columns.
The vast majority of the sudoku community has adopted the ## Da RuleSudoku is a simple game with a single rule:
This rule is often accompanied by the following statement:
## The GivensA number of digits are already in place. These are the “givens”. These givens are placed in such a way that there is only a single solution to the grid, otherwise, you are not playing Sudoku. You are not allowed to remove or change them, because they are part of the puzzle. Alternative names for givens are “clues” or “fixed numbers”.
There is a search going on in the sudoku community for the minimum number of givens that a sudoku puzzle can have.
Currently, 17 is the minimum for 9x9 Sudoku. More than 36000
puzzles with 17 givens have been
collected by
The number of givens does not reflect the difficulty of the puzzle. There are puzzles with 17 givens that can be solved
easily, while there are puzzles with 28 clues amongst the toughest known. The placement of the givens and the information
that you can extract from them is more important than the sheer quantity of givens. The Japanese publisher ## Symmetry
There are people in the Sudoku community who think that a non-symmetrical Sudoku does not deserve the name.
Others believe that symmetry is irrelevant and it is better to have a difficult asymmetrical puzzle than an easy symmetrical
one. This debate will probably continue into the next century. It is, however, true that to enforce symmetry, it may be
neccesary to add givens that would not have been required, had the symmetry demand not been there. It will still be possible
to make very hard Sudokus with symmetry. Formally, there are seven classes of symmetry. Here is an example from each of these classes:
Sometimes you can discover alternative forms of symmetry in hand-crafted sudokus.
180 Degrees Rotational symmetry is the most popular class. Most Japanese sudokus and many provided by
For solving the puzzle, the symmetry serves no purpose. Only the location of the givens show symmetry, not the digits themselves. It is simply impossible to place givens on row 5 and column 5 that have symmetrical digits, because that would require 2 of the same digit in a single row or column. If you like to read more about this topic, I advice you to read the
sudoku symmetry paper written by ## Minimal Puzzles
A valid sudoku has a unique solution. When you remove a given number, either the resulting puzzle still has a
unique solution or the resulting puzzle has multiple solutions. When it is possible to create a valid sudoku
There is nothing special about minimal sudokus. It is simply not possible to make them more difficult by removing a clue. You can make them easier by adding clues. There is a field of tension between symmetry and minimal puzzles. To make a puzzle minimal, it is often not possible to maintain symmetry. A puzzle is said to be “Symmetrically Minimal” when it is not possible to remove a set of givens that keeps the symmetry intact, while maintaining a valid sudoku. ## Using LogicIt has been stated many times: This Sudoku can be solved by logic alone. So where do you start? What exactly is the logic that can be applied to a Sudoku puzzle?
This guide will tell you all about it. And we will start with ## Sudoku ControversiesIf you’re only interested in solving sudokus, you might as well skip this section. There are a few subjects in the world of sudoku that always result in fierce debates. I will not go into too much detail in this solving guide, but most of these controversies are about solving techniques. The feedback shows that these are indeed sensitive issues. To avoid negative sentiments, I have removed my personal views on these issues. Please use the forum if you like to discuss them with me. ## GuessingThe question is: “Are we allowed to guess when solving a sudoku?”
Many people say “No.”, but there are also people who believe guessing is a
valid solving strategy. Almost any sudoku, even the most difficult, has a number of ## Trial & Error
The subject of debate is whether T&E is the same as Outside the sudoku community, T&E is generally accepted as a problem solving strategy. When you solve a jigsaw puzzle (not the sudoku kind), you try to fit the pieces together and reject those that do not fit. In sudoku, you try a candidate and reject those that cause an inconsistency in the puzzle. Same strategy.
The question is: What to do when the piece does fit? Do we continue and try all the remaining pieces to
make sure that this is the ## Bifurcation
This is a continuation of the Many generally accepted techniques, like XY-Wing and Color Wing are limited implementations of bifurcation. Other techniques like Forcing Chains and Tabling do not impose many limitations. It all depend on what a human player accepts as a useful technique. Opinions differ. ## UniquenessThis subject is the most controversial of them all. Many heated debates have been held on uniqueness. The question is: “Are we allowed to use the fact that the sudoku has a unique solution in our solving strategies?” In mathematics, it is not uncommon for a problem to have multiple solutions. Even when a problem has a single solution, proving this fact is part of the problem. However, sudoku may look like a math problem, but it is a merely a puzzle for entertainment. When the maker claims that this puzzle has a unique solution, why should we question this claim, and not use it to our advantage?
A dilemma arises for the supporters of uniqueness based solving techniques, because the arguments to support these methods
can also be used to defend ## Difficulty with RatingsThis subject could also have been placed under the Sudoku Controversies heading. The difficulty level of a puzzle is in the eye of the beholder. Players have different skills, even when they have learned the same techniques. The relative difficulty of solving strategies is subject to personal preferences. Some cannot see naked singles without pencilmarks. Others find hidden subsets so difficult to spot, that they are not even trying to find them. There are many different rating systems. Most of them use simple terms, others use numbers, “stars” or other symbols. ## Rating system examples- Mild - Modest - Mean - Maniac - Nightmare (Ruud)
- Gentle - Moderate - Tough - Diabolical (Mike Mepham)
- Easy - Moderate - Hard - Very Hard - Fiendish - Extreme (Gaby VanHegan)
- Easy - Mild - Difficult - Fiendish (Times)
- Gentle - Moderate - Tricky - Tough - Deadly (Times Killers)
- Easy - Cool - Thinker - Brain - IQ - Insane (DJ Ape)
- Super Easy - Very Easy - Easy - Medium - Hard - Harder - Very Hard - Super Hard (Menneske)
- Very Easy - Easy - Medium - Hard - Harder - Very Hard - Super Hard (Brain Bashers)
- Easy - Medium - Hard - Evil (Websudoku)
- Easy - Moderate - Hard - Extreme - Outrageous - Mind Bending (Killer Sudoku Online)
It is very difficult to classify the wide variety of puzzles in a few named categories. A scoring mechanism like I use in SudoCue is more accurate. Various measurements can be made, each adding to the overall score. ## VisibilityHow easy is it to spot a situation that you can use? Singles are easy to find. Patterns in the candidates can be found when you are able to filter your view to a single digit. Helper programs make pattern recognition easier with a filter tool. Subsets and chains are more difficult to spot, but with pencilmarks they are very visible. ## Alternatives
When there are many tricks to be found in a grid, the chance to spot one of them greatly increases. Even completely
different techniques can compete for attention. A human solver is simply scanning the grid for anything that will work,
unlike a computer program that scans the grid one technique at a time. Alternatives only come into play when the
same search style is used. When you check the pattern for a digit, you may spot a set of locked candidates, alongside
an X-Wing. Chances are slim that you stumble upon a naked triple during this search. When you scan pencilmarks for
naked singles, you may also find the occasional pairs, triples and even unique rectangles. ## Technology
Each solving strategy requires time to learn. Some players are willing to invest time in learning all kinds of
techniques, others are satisfied with a basic “toolkit”. This takes the shape of a pyramid, with
only a few players at the top with the full arsenal of techniques. The thresholds are often set by the major
puzzle manufacturers. A ## Flow
This aspect of difficulty is heavily underestimated. Solving steps that naturally follow each other can cause a
significant drop in difficulty. This is often seen in handmade sudokus, where the maker deliberately builds a
consistent flow in the puzzle, with the occasional twists and turns. ## Solving TermsThere are a few concepts that you need to learn before we continue. Without these, it would be impossible to write a sudoku solving guide. ## House
You now have read the phrase “row, column or box” a couple of times.
This is because a lot of logic deals with all types of grid divisions.
From now on, we will use the term “house” when refering to any group of 9 cells that must
all have different digits. Other terms for house are “group”, “set” and “unit”.
Mathematicians, who look as Sudoku as a
There are two practical rules that you can derive from - When a digit is
*not*present in a house,*one*of the empty cells in that house*must*contain that digit - When a digit is
*already*present in a house,*none*of the empty cells in that house can contain that digit
As a result, it is clear that there is a strong relationship between the cells in a house. This is so important, that we have given a name to that relationship. ## Peers
However, when the cell is empty, we can reverse the logic. We start with the empty cell and scan the peers to determine what digits it cannot contain. Whatever remains can be placed in the cell. You will have to learn looking at the grid in both directions. It is the most important tool for a Sudoku solver. You will be able to practice this again and again, so don’t worry if you are not really familiar with it yet. When a cell contains a digit, there is yet another conclusion to be drawn, so obvious, that many people overlook it. The cell can no longer contain any of the other 8 digits. This conclusion is useless for the filled cell itself, but very useful for each of the houses the cell belongs to. ## CandidateThere is one more word that you need in your vocabulary before we start with the first solving technique.
## Naked SinglesSome of the naming in sudoku solving will trigger the parental controls of your browser, but there is nothing indecent with these naked singles. They’re only digits.
There is a famous line by Sherlock Holmes: That, in essence, is the logic you use to solve a Sudoku. Eliminate the impossible digits until there is a single candidate remaining. You begin each cell with 9 candidates. When peers are filled, these candidates are eliminated one by one. At some point, 8 out of 9 digits are eliminated. The last candidate must be the correct digit. You can place it in the cell. This in turn will eliminate that digit in other peers.
## Full House
## Hidden Singles
Now the cells that I marked in red come into play. These are the conclusions that we can draw from them:
Because
Because
Because
The last remaining candidate in box 2 is
## SqueezingA simple and effective technique to search hidden singles is known by the name “squeezing”. This technique requires you to focus your attention to a row or column of 3 boxes, also known as a “chute”. We are only looking for chutes that have exactly 2 placements for a digit. These 2 placements eliminate 22 of the 25 remaining candidates in the chute, efficiently narrowing down the remaining options. Here is an example:
Squeezing is a limited form of cross-hatching, which will be explained next. Because it requires this specific pattern,
you will only be able to use this technique
in a simple sudoku or in an advanced stage of solving. However, it is a very fast and easy to spot technique.
When you have learned about ## Cross-Hatching
You can use cross-hatching to check rows and columns, but many players use it mainly to check the boxes.
## Locked Candidates
So far, we have only discussed The number strings above each image represent the initial state of the puzzle under discussion. You can copy this string and paste it into your sudoku program for some hands-on practice of the techniques explained in this guide. In most cases, you must do some basic solving to reach the point shown in the diagram. ## Locked Candidates (type 1)000023000004000100050084090001070902093006000000010760000000000800000004060000587
## Unlocked Single050007690000040000009000000000100004000230008008000150000400003006080209002005000
## Locked Candidates (type 2)500200010001900730000000800050020008062039000000004300000000000080467900007300000
Experienced 000042059450090008000000000000004080000000703000836002369200001700000000005001600 ## Disjoint Subsets
So far, we have only looked at single digits. When a single candidate is left in a cell, we call it a
In this section, you will learn how to recognize “subsets”. The term
The isolated parts are sometimes referred to as “disjoint subset”. A disjoint subset contains a certain
number of digits and A disjoint subset shows 2 distinct characteristics: - The cells
*inside*the subset only allow candidates from the digits in the subset - The digits in the subset
*cannot*be placed in cells in the same house*outside*the subset
When both these conditions are met, a true disjoint subset exist and there is nothing you can do with it.
The fun starts when When only the first condition is met, you have spotted a “naked subset”. You can then complete the subset by removing all candidates for the selected digits from the cells outside the subset. When only the second condition is met, you have found a “hidden subset”. Now you can complete the subset by removing all candidates for other digits from the cells within the subset.
Now when you can find a Let’s examine some real examples after this piece of dry theory. ## Naked Pair
To spot subsets, you need 005100000600003000300000706000030601009050400802090000401000005000500008000007200
Naked pairs can often be found at the intersections of lines (rows or columns) and boxes. Sometimes you have the
choice to handle the situation as either ## Naked Triple
When 3 cells in a house only have candidates for 3 digits, then there is a 200000007015000020080200006000000064890000000000050910040815000061030000000006400
As with pairs, naked triples can also occur in the intersection of a line and a box. Here is an example. 700605009000800030001000004007000205090000080308000700200000600010009000800506007
## Naked Quad070050900102000085098000000800060000007405300000080002000000290280000506001070030
So, what’s next? A quin? A 6-tuplet?
There is no need to worry about subsets larger than size 4. Each naked subset has a ## Hidden Pair800070009010000050700608003380105074000207000200030005008409500100000006090000040
## Hidden Triple605301204703000509000000000100936007000000000900457001000000000807000402309108706
## Hidden Quad053100004060005008008200003000000000025800090700001060032000000004780006000000081
If you need more practice finding hidden subsets, try the following puzzles. The first puzzle contains a hidden pair, a hidden triple and a hidden quad. The second puzzle solves very easily until you bump into a well concealed hidden quad. 100000700900050000050000016009036120200009003003004050638005000000070000010300000 ## Row/Column Subsets
The objective of this technique is to isolate a number of rows and columns from the rest of the grid. Row/column
subsets is a This is the formal definition:
“When all candidates for a certain digit within
This technique also works when you swap the words There seems to be no general name for this class of techniques, only names for each of its members of different size: - X-Wing
- Swordfish
- Jellyfish
- Squirmbag
The name “X-Wing” was coined by Wayne Gould
of ## X-Wing028700050054003980000000007001090000006300000090004300000050000502000006600170009
A different point of view.
When you are only solving sudokus from newspapers, you can probably stop reading here. Most sudoku puzzles published
in print can be solved with basic techniques, and rarely require more advanced techniques like subsets and
X-Wing. Only a few go beyond that level. The sudoku program by ## Swordfish200000005000010300780004000900003000605040902000600008000400081006020000500000007
There are 2 sides to every fish.
## Jellyfish003100005850400000006050009025006080000000000070800430500020600000008014100003500
## Fishing with LinesNow that you know what to expect in the pond, there is a little fishing tip that I would like to share with you.
## Connected Pairs and Coloring
“Connected Pair” is a concept that you know already, but now it is time for you to learn its name.
Many solving techniques depend on
connected pairs. In earlier techniques, you have come across the phrase “at least one of these cells must contain this
digit” a couple of times. In
A “conjugate pair” exists when there are only 2 candidates in a house for a digit.
We also call this a “strong pair”.
In this solving guide, I will use the name
To identify the two states that can exist for a conjugate pair, we can use colors, letters or other symbols.
This is also known as the “parity” of the cells within the pair. Many use Time for a little practice.
Now we will look at
The parity of the cells within the cluster.
We now return to the grid with the 7-cell cluster parities assigned.
## Color Trap
## Color Wrap
## Color WingThis pattern for digit 5 is known by many names. You do not need to memorize them all.
You may wonder why I am so strongly focused on these negative aspects. This is because both colored candidates in column
2 are connected to another candidate. I have drawn the usual green connector lines in the diagram. The linked candidates
are
Here comes the result: Because at least one of
This may be the right time to introduce you to the term
“shared peer”. When we examine two cells, all cells that are a ## Empty Rectangle or Hinge
In Science and Sudoku, great discoveries are sometimes made simultaneously and independently by different people. The
The key feature of an Empty Rectangle is a 3x3 box in which 4 cells in a rectangular formation do not contain a candidate for the
digit we are inspecting, leaving an Here is a diagram that shows the complete ER pattern:
With the assistance of the ER/Hinge,
The 4 cells that form the ER do not need to be so tightly packed as in the diagram, as long as all remaining candidates in the box are limited to
one horizontal line and one vertical line. Also, you do not need all 5 remaining candidates in the ER box to be present. A box with only 2
candidates on different rows and columns can already be used as an empty rectangle. Please note that when all candidates
are confined to a single row or column within the box, you missed an easy Time for a reality check.
## Remote Pair
Credit for the discovery of
## Finned Fish
By the end of 2005, it looked like most solving strategies for
The best solving techniques are those that use patterns. The player can learn how to recognize such a pattern, and then apply the
effects (eliminations and/or placements) caused by that pattern. Patterns that are limited to a single digit are better recognized
when we use the digit filters in our Sudoku programs. Most
With ## Finned X-Wing
## XY-Wing
The discovery of this technique can be traced back to June 2005, when We will start with an abstract schema depicting an XY wing.
Here is an XY-Wing with 3 victims.
## XYZ-Wing
The discovery of the XYZ-Wing dates back to the beginning of August 2005. Sudoku historians should read
This is the abstract schema depicting an
Here is a beautiful example of an XYZ-Wing.
After I claimed not to have seen an XYZ-Wing with 2 eliminations, a helpful forum member pointed out that one exists in the Nightmare of January 27. This puzzle will give you lots of practice in XYZ-Wings. ## Aligned Pair Exclusions (APE)
An
This is a hard to spot, laborious and complex technique, which has a large overlap with easier techniques like XY-Wing and
XYZ-Wing. Still, it deserves a place in your technique book. It introduces you into the world of Here is a beautiful example of an Aligned Pair Exclusion. You may have seen this picture before.
## Almost Locked Sets - XZ Pattern
This technique has been added to the arsenal of
To know about
A single
Here is a beautiful example of an
By now, you have learned 3 different techniques, which eliminate the same candidate in 4 different ways using exactly the same cells
as evidence. There must definitely be a common background for these techniques. ## Subset Counting
This technique is not implemented in
The technique uses the following strategy: - Select any number of empty cells, and examine them as a set.
- For each digit that appears as a candidate in the set, count the maximum number of times it can theoretically be placed.
This is the
*multiplicity*of the digit. You can usually perform this count manually. You may also use templates. - Add all multiplicities together. This is the total multiplicity of the set.
- Eliminate all moves that would reduce the total multiplicity
*below*the number of cells in the set
I told you this wasn’t gonna be an easy technique. Maybe an example will help you understand how it works.
Here is a beautiful example of a
Subset counting is very powerful, but it is not easy to pick the cells that can be combined into a set. You must check all possible
placements, which gives it a ## Advanced 3D Medusa Coloring
Colors can be used to mark candidates that act as a group. There are alternative marking systems, like The Medusa techniques are also very useful to find Double Implication Chains (DIC’s) in the grid. You can always find a path from one end of the chain to the other using the alternating colors in a cluster. In the more advanced versions, you use the bridges to expand the chains to neighboring clusters.
This is a first draft of the Medusa section. I am looking for some good examples to illustrate the technique. They will be added soon.
## Medusa Trap
This is the 3D version of the ## Medusa Wrap
This is the 3D version of the
Each Medusa step can also be replicated by a Double Implication Chain. The chain in Eureka Notation for this step is: (6-3)r1c3=(3)r2c3-(3=5)r2c5-(5)r3c5=(5)r3c7-(5=6)r1c7-(6)r1c3 => r1c3<>6 ## Medusa Bridge
This is the 3D version of the ## Medusa Complex
This is a more complex version of the ## Bivalue Universal Grave (BUG)This technique assumes that the Sudoku you are solving has a unique solution. When you are not sure about this fact, you should avoid this technique. Assuming uniqueness is a controversial issue, as I pointed out earlier in this guide. When you prefer to prove the existence of a single solution, there are alternative techniques that can be used. Usually, a grid with a BUG pattern also contains an XY-Wing or an XY-Chain. A contradiction can also be found with a 3D Medusa cluster. Recognizing a BUG may help you focus your attention to these techniques, so that you can still solve the puzzle without assuming uniqueness. The remaining players can follow me into the wondrous world of uniqueness-based logic. The definition of BUG: A situation in which every unsolved cell has 2 candidates and every row, column and box have 2 candidates for each unsolved digit.
Here is an example of a
However, it is possible that you encounter a pattern that can be transformed into a BUG by a single move. These are interesting, because
we can prove that 000047000000100002030000410000200063005008009070000500400070006006802000050900300
A BUG+1 is a pattern that is easy to recognize, because you only need to spot that there is a single cell with 3 candidates, while the remainder of the unsolved cells have 2 candidates each. The alternative XY-Wings or XY-Chains are more difficult to find. The simplicity of the technique is sufficient compensation for many players, who use it on a regular basis. ## Uniqueness Test
In July 2005, some players were reporting an Unmentioned
Logic Technique which triggered a debate that is still raging today. The new technique was soon known as
## Unique Corner
The easiest and most common 080000000205003600300820700000600001004138500900007000002065009001700205000000040
## Unique Side
This technique is rare, although it is not difficult to spot. In this case, the 020000930907001400004720000650000000000807000000000089000074300001600208065000090
Here is an example that you can try for yourself. The unique side occupies 2 boxes and you can only eliminate a single candidate from that row. 020000930907001400004720000650000000000807000000000089000074300001600208065000090 ## Unique SubsetThis technique is probably the most difficult of the UR series. It combines a UR with a naked subset for the surplus candidates. It is extremely rare and very difficult to spot. 601300020004000056070000400000740000030001000008009007050900014703080200080000005
## Unique PairCompared to the Unique Subset, this technique is relatively easy, but it is equally rare. With a Unique Pair, one of the digits that belongs to the Deadly Pattern is strongly linked in the 2 cells with surplus candidates. As a result, we can safely eliminate the other digit from these 2 cells. 930000504080400000002000700000040827000090000621050000009000200000003010407000065
The rectangle is not the only shape you can use for this series of techniques. Here are some examples of other patterns which are equally deadly when they contain only 2 digits:
These patterns are rare and not too easy to locate in the grid.
(Stay tuned for more solving techniques)Thanks for reading this solving guide all the way to the end. If you have found mistakes, plain lies, spelling or grammar errors or other inaccuracies, please report them on the contact form or on the user forum. I have invested a lot of time in this guide and it would be nice to keep it correct and comprehensive. This guide happens to be one of the most popular pages on my website. I am still learning a lot from scouring the forums and the other sudoku resources on the Net. My work on this solving guide always lags a little behind the work that I do on SudoCue. I hope you can forgive me for that. Writing a piece on a solving technique is easier when I have implemented it in the program first. |

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