With these Rules alone, you will be able to solve over 99% of all published Sudoku puzzles. Underlined Rules are Links to pictorial examples in the Tutorials. Sudoku-Help+ Solving Rules |
© Copyright Greg Shalless 2006. I, Greg Shalless, claim "Coining Rights" to the following terms for their use in relation to Sudoku puzzles: "Constraint Region", "nxn Gridlock", "XY-Zap", "Cluedoku", "SHD Rating", the classification of puzzles according to the most complex Solving Rule required to solve them and the names
Novice, Player, Expert, & Master for those classes. A picture is worth a thousand words! So after studying these Rules you might like to check out the Worked Examples in the following Tutorials: 1. Featuring all Novice Class rules, Locked Pairs & Triples, XY-Zap and the nxn Gridlock rules, 2. Featuring Multi Value Chains, and 3. Featuring Intersect Reject and the Single & Multi Value Chains rules. Description (Note: "Constraint Region" is my generic term for a 3x3 Box, Row or Column) |
Novice Class | Certain Placement |
Only Spot Boxes | This is the Only Spot (cell) in this Box that this Value can go without it being placed twice in a Row or Column. |
Only Spot Rows | This is the Only Spot (cell) in this Row that this Value can go without it being placed twice in a Box or Column. |
Only Spot Cols | This is the Only Spot (cell) in this Column that this Value can go without it being placed twice in a Box or Row. |
Only Value | This is the Only Value that can go in this Cell, because all other values are already present in one or more of the Constraint Regions of which this cell is a member. |
Player Class | Basic Candidate Elimination |
Locked Pairs | 2 Cells in the same Constraint Region have their candidates drawn from a set of 2 values, each of which is represented at least once, therefore those 2 values are restricted to those 2 cells and can be eliminated as candidates from the other unsolved cells of that Constraint Region. |
Locked Triples | 3 Cells in the same Constraint Region have their candidates drawn from a set of 3 values, each of which is represented at least once, therefore those 3 values are restricted to those 3 cells and can be eliminated as candidates from the other unsolved cells of that Constraint Region. |
Locked Quads | 4 Cells in the same Constraint Region have their candidates drawn from a set of 4 values, each of which is represented at least once, therefore those 4 values are restricted to those 4 cells and can be eliminated as candidates from the other unsolved cells of that Constraint Region. |
Intersect Reject | If the 3-Cell Intersection of a Box with a Line (either a Row or a Column) is such that it contains a candidate value where that Intersection is the only place in either the Box or the Line where that value can go, it can be Rejected as a candidate from all cells not in the Intersection in both the Line and the Box. |
Expert Class | Complex Candidate Elimination (difficult to explain without pictures) |
2x2 Gridlock (X-Wings) |
If you can find a candidate value appearing twice in each of 2 Rows, such that the cells where the candidate value appears also happen to share the same 2 Columns, the candidate value is Gridlocked to the 4 (2x2) cells forming the points of intersection of the 2 Rows and 2 Columns and can therefore be eliminated as a candidate from all other cells in the 2 Columns. Alternatively you can exchange the words Rows and Columns in the preceding sentence for an equivalent set of Row eliminations. Similar eliminations can be done with other intersecting Constraint Regions (Boxes with Rows & Boxes with Columns and vice-versa) and the Sudoku-Help+Solver's 2x2 Gridlock Rule will detect them all, however the much simpler to identify Intersect-Reject rule will make the same eliminations. |
3x3 Gridlock (Swordfish) |
If you can find a candidate value appearing no more than 3 times in each of (exactly) 3 Rows, such that the cells where the candidate value appears also happen to share the same (exactly) 3 Columns, the candidate value is Gridlocked to the 9 (3x3) cells forming the points of intersection of the 3 Rows and 3 Columns and can therefore be eliminated as a candidate from all other cells in the 3 Columns. Alternatively you can exchange the words Rows and Columns in the preceding sentence for an equivalent set of Row eliminations. Similar eliminations may be able to be done with other intersecting Constraint Regions (Boxes with Rows & Boxes with Columns and vice-versa), however the much simpler to identify Intersect-Reject rule will make the same eliminations. |
4x4 Gridlock (Jellyfish) |
If you can find a candidate value appearing no more than 4 times in each of (exactly) 4 Rows, such that the cells where the candidate value appears also happen to share the same (exactly) 4 Columns, the candidate value is Gridlocked to the 16 (4x4) cells forming the points of intersection of the 4 Rows and 4 Columns and can therefore be eliminated as a candidate from all other cells in the 4 Columns. Alternatively you can exchange the words Rows and Columns in the preceding sentence for an equivalent set of Row eliminations. Similar eliminations involving Boxes are not possible because a Box cannot intersect with more than 3 Rows or Columns. |
XY-Zap (XY-Wings) |
If you can find a cell with candidates {xy} and another with candidates {xz} in one of the Constraint Regions of which the {xy}-cell is a member, and then a third cell with candidates {yz} in another (different) Constraint Region of which the {xy}-cell is a member, then regardless which of the values (x or y) the {xy}-cell ultimately takes, either the {xz)-cell or the {yz}-cell must contain the value z. Therefore the value z can be Zapped as a candidate in any cell that is a member of a Constraint Region in common with both the {xz}-cell and the {yz}-cell. |
Pair Chains (Remote Pairs) |
Two Locked Pairs for the same two values, which have a cell in common, can be thought of as a two link Chain of Locked Pairs, the two end-points of which must be the same value. Conversely the two end-points of a linked Chain of Locked Pairs must hold the opposite values of the Locked Pair if the Chain is of odd length (eg. a single link chain is just the Locked Pair itself and its end-points must hold different values). If you can find a Locked Pair Chain of 3, 5 or even 7 links, both values of the pair can be eliminated as candidates in any cell that is a member of a Constraint Region in common with both end-points of the Chain. |
Single Value Chains | A Single-Value-Pair is two cells in a Constraint Region that are the only two cells in that Constraint Region that can take a specific Value. A Single-Value-Chain of odd length is formed when a series of Single-Value-Pairs are linked (in all the even links in the Chain) by the fact that one cell from each linked Single-Value-Pair is in the same Constraint Region. If you can find a Single-Value-Chain of 3, 5 or even 7 links for a specific Value, that Value can be eliminated as a candidate in any cell that is a member of a Constraint Region in common with both end-points of the Chain. |
Multi Value Chains (XY Chains) |
This is the killer rule for solving difficult Sudoku Puzzles. To find it you only need to look at bi-value cells so those solvers who don't like to do full mark-up can deploy it. We are looking for a Chain of bi-value cells where each link in the chain is two cells in the same Constraint Region with a value in common. You start with a bi-value cell, say {az}, and look for another bi-value cell in a common Constraint Region with a value in common with it, say {ab}. After that we look for {bc}, then {cd} and so on. The aim is to find a cell whose other value is z, for example {dz}. Now {az} might be z but if it's a, then {ab} is b, {bc} is c, {cd} is d and {dz} is z, so any cell in the same Constraint Region as both {az} and {dz} cannot be z since one of them must be. A one link Multi-Value Chain is a Locked Pair. A two link Multi-Value Chain is a Locked Triple if all three cells {az}, {ab} and {bz} are in the same Constraint Region, and if {az} and {bz} aren't in the same Constraint Region then it is an XY-Zap. Longer Chains will often enable eliminations not discernible with any other Solving Rule. |
Hidden Pairs | If 2 cells in the same Constraint Region contain (Hidden amongst others) candidates drawn from a set of 2 values which do not appear in any other cells in that Constraint Region, all other candidate values in those 2 cells can be eliminated. A Hidden Group of 2 in a Constraint Region where n cells remain unsolved is equivalent to a Locked Group of n-2, which may be easier to spot. |
Hidden Triples | If 3 cells in the same Constraint Region contain (Hidden amongst others) candidates drawn from a set of 3 values which do not appear in any other cells in that Constraint Region, all other candidate values in those 3 cells can be eliminated. A Hidden Group of 3 in a Constraint Region where n cells remain unsolved is equivalent to a Locked Group of n-3, which may be easier to spot. |
Hidden Quads | If 4 cells in the same Constraint Region contain (Hidden amongst others) candidates drawn from a set of 4 values which do not appear in any other cells in that Constraint Region, all other candidate values in those 4 cells can be eliminated. A Hidden Group of 4 in a Constraint Region where n cells remain unsolved is equivalent to a Locked Group of n-4, which may be easier to spot. |
Master Class | Puzzles having a single solution but not solvable without the use of some solving strategy other than the ones described above. |
XYZ-Wings | If you can find a cell with candidates {xyz} and another with candidates {xz} in one of the Constraint Regions of which the {xyz}-cell is a member, and then a third cell with candidates {yz} in another (different) Constraint Region of which the {xyz}-cell is a member, then regardless which of the values (x, y or z) the {xyz}-cell ultimately takes, either it, or the {xz)-cell or the {yz}-cell must contain the value z. Therefore the value z can be eliminated as a candidate in any cell that is a member of a Constraint Region in common with all three cells forming the pattern. |
Hinge (Empty Rectangle) |
Any cell that can "see" one end of a Single-Value-Pair in the value x directly, and can "see" the other end of it, indirectly through a "Hinge" formed by an "Empty Rectangle" in x, cannot be x. |
nxn Blocked Gridlock (Finned Fish) |
When a potential nxn Gridlock pattern in the value x is blocked by the presence of an extra candidate cell (or two) for x, on one of the pattern's would-be defining lines in a Box containing one of the Gridlock points, x can still be eliminated from the cells within that Box from which it would have been eliminated had the extra x (or two) not been there. |
Blocked Single Value Chains | If a potential Single-Value-Chain pattern in the value x is blocked by the presence of an extra candidate cell (or two) for x, in the same Constraint Region as one of the Chain's would-be end-points and the other cell in that end-point's potential Single-Value-Pair, x can be eliminated from those cells from which it would have been eliminated had the extra x (or two) not been there, if they're also in a common Constraint Region with the pattern blocking cells. |
Cluedoku Puzzles | "Cluedoku" puzzles are Sudoku puzzles using letters rather than digits as the symbols in the puzzle and which technically have more than one solution (usually 2 valid solutions but no more than 4) and where a clue is provided. The answer to the clue will be apparent in the puzzle when the correct solution is found whereas the other (technically valid) solutions will not bear any relationship to this clue. Whilst I am not the first to introduce the concept of Sudoku puzzles with letters, I believe I may be the first to introduce puzzles where the use of those letters actually adds some value, in that the puzzle cannot be solved without finding the answer to the clue, hence the name. Sudoku-Help+ has 10 Cluedoku puzzles in three different classes. You can see some example Cluedoku Puzzles here. Print them out and give them a try. |
Coining Rights | I have coined the term "Coining Rights" to mean that whilst I do not wish to discourage the use of my new terminology by others, I would like to be acknowledged as the person who first "coined" the terms, by those who choose to use this terminology in relation to Sudoku puzzles. |