



Ultra Sudoku Solver
Software Frequently Asked Questions 


Is There a Help Feature to Assist Me to
Get Going? 

Click the Help button that leads to 23 help subjects. In total, they explain completely in English how Ultra Sudoku Solver software works. This feature
alone should equip you with all the information you need to enjoy the challenges and successes of Sudoku puzzles
using our Ultra Sudoku Solver. 


Does Ultra Sudoku
Solver Let Me Create My Own Sudoku Puzzles? 

Yes! Just click the Create a
Sudoku button to instantly create your own Sudoku puzzle selecting
from six difficulty levels. 


How Does Ultra Sudoku Solver Work? 

Begin the puzzle by entering your starting numbers.
Or click the Create a Sudoku button to instantly create your own Sudoku puzzle
 this option lets you chose from six levels of difficulty. Either way,
click the Save Original Grid button to save your starting puzzle givens.
Decide whether you want the Ultra Sudoku Solver software to
display the vacant cell possible candidate values to assist you in the Sudoku
solving process, or whether you want to tackle the puzzle all on your own.
You make the choice by a click of the Show Possibles button. If you decide
to go without the possible cell candidate values shown you can change your mind
at any time later.
Decide whether you want the computer to autofill unsolved cells
at any time during the solving process
when you solve a cell that causes any remaining cells to only have one possible
candidate. Why have to key in a solution digit for a cell when there is
only possible choice? With this feature turned on, the software does it
for you. The Autofill feature is turned on or off by clicking the AutoFill button;
it remains in its set state until you change it.
Then proceed to solve Sudoku using the ease of Ultra Sudoku Solver's assists:
updated cell possible candidates, autofilling cells having only one possible
value and error prevention.
If you become stuck at any point in the solving process, click the 1 Step
button and the software will advance the puzzle one solving step. Then proceed on you own from there.
Use the 1 Step button as many times and as often as you want.
When all else fails, click the Solve It button and Ultra Sudoku Solver will finish the
puzzle for you.
At any point you may save an inprogress puzzle and later recall it to be worked on further.
With this convenience, you can save the puzzle, proceed and then check you
progress by the Solve It button. If the Sudoku is solved, you know you
were on the right trail; then Restore the Board puts you back to before you
clicked the Solve It button; do a Save, and proceed. If the point you
clicked Solve It failed to solve the puzzle, then you know you were on the wrong
solution path; so just do a Restore to get you back to last good point.
An inprogress puzzle may be printed at any point. Maybe you want to take it with you on the bus or train. 


Want to Learn How to Solve Sudoku Puzzles? 

Sudoku puzzles are an arrangement of nine rows of nine cells; therefore, it
is a nine by nine layout. The puzzle is further divided into nine 3 by 3
cell groups that are shaded called boxes. The creator of the puzzle always
gives the values of some of the puzzle cells as a starting point. There is
only one solution to a valid Sudoku puzzle. Your task is to fillin the
empty cells of the Sudoku in a manner so that:
A) Each row contains the digits from 1 to 9.
B) Each column contains the digits from 1 to 9.
C) Each shaded box of 9 cells contains all the digits from 1 to 9.
Needless to say, these requirements preclude duplicate or missing numbers in
any row, column, or box.
Sudoku puzzles are solved by using a series of logical If/Then
thought processes on a stepbystep basis to finish it. Think of each
If/Then logical process as a rule when true causes a specific result action to
be taken. If the rule Condition is True, Then perform a certain
action. The discovery of a certain condition in a Sudoku means a
consequential action can be taken to either solve an unsolved cell or to reduce
the possible candidate digits for that unsolved cell. One of these If/Then
processes is a step toward total puzzle solution. One step's action taken
leads to another step with its action and so on until the puzzle is finished.
There are 54 rules built into Ultra Sudoku Solver.
Some of the puzzle solving rules are very simple to find in Sudokus. The
rules become more complex as increasingly more difficult techniques are used to discover their
rule presence in the puzzles. Some rules are very complex. Some are
super difficult to find occurring in a Sudoku, and some are extremely complex to
grasp and apply to puzzle solving. Fortunately, most Sudoku puzzles do not
require the real mind bender rules for a person to be able to complete a puzzle.
But nevertheless, Ultra Sudoku Solver has a Rules Explanation
button and by clicking it you will get a description of each of the 54 rules: 1)
Detailed definition in English of each rule, 2) How to find and recognize each
rule's presence in a puzzle, 3) Exact action to be taken when you discover each
rule's occurrence, and 4) an onscreen example of each rule together with the
action consequence to be taken to move toward completing Sudoku.
Therefore, with Ultra Sudoku Solver's Rules Explained button, you have an
excellent means to learn all of the software's Sudokusolving rules firsthand. No books or
other references are needed. 


Explain the 54 Sudoku Solving Rules 

General descriptions of Ultra Sudoku Solver’s rules are presented below and
are meant to be a summary of the rules  complete rule definitions and the
followon actions are covered fully in detail in the software. The easier to find and apply rules tend to be lower
rule numbers. As the rule number becomes higher you can expect the rule to be tougher to identify its presence in a Sudoku.
Usually, when a rule is described in terms of applying to rows, it also applies to columns, and in some cases to
the 9cell boxes as well. Even though the difficulty to recognize a
rule's presence in a puzzle generally rises with the rule number, usually the action to be
taken when the rule requirements are true (met) is usually pretty straightforward. 

Simple Rules: 

Rule 1: Only One Possibility is the method that discovers when only one possible digit may be placed in a cell.
This one is the simplest of all! 

Rule 2: Box with a Naked Pair  Within a 9cell square shaded box, find two sets of matching 2digit numbers, then you have a naked pair within a box. Those two 2digit sets are called naked because they have no other digits accompanying them. 

Rule 3: Box with a Number in One Row or Column  When a digit occurs only in one row within a box. 

Rule 4: Box with a Digit Occurring Only Once  When a digit is found to occur only once among the possible
candidates of the cells in a box. 

Rule 5: Row with a Digit Occurring Only Once  When a digit is found to occur among the possible
candidates in a row only once. 

Rule 6: Column with a Digit Occurring Only Once  When a digit is found to occur among the possible
candidates in a column only once. 

Rule 7: Digit in a Row/Column of a Box But Not Outside the Box  Observe and count each digit's possible occurrences within the three 3cell rows of each box. When any possible digit occurs 2 or 3 times in a box's 3cell row and does not exist in the same row outside the box, you have found a Rule 7. 

Medium Rules: 

Rule 8:
Hidden Pair  A hidden pair exists when a twodigit set occurs twice in a row but one or both occurrences of the set is accompanied by other digits; in other words, the twodigit set does not stand alone.


Rule 10: Naked Pair  When a standalone pair of identical digits are found within two cells of a row they are a naked pair. However, naked pairs occurring within a box and are neither oriented in the same row, or column, are handled by Rule 2. 

Rule 11: Naked Triple  When a standalone threesome of identical digits are found within 3 cells of a row they are a naked triple. A 3digit combination must occur in a threesome set to be a naked triple; each cell must have at least two of the defining three digits; and since it is a naked triple, there may be no accompanying digits in any of the three cells. 

Rule 12: Naked Quadruple  When a standalone foursome of identical digits are found within 4 cells of a row they are a naked quadruple. A 4digit combination must occur in a foursome set among 4 cells to be a naked quadruple; each of the founding cells must have at least two (but as many as four) of the defining four digits; and since it is a naked quadruple, there may be no accompanying possible candidates in any of the four cells. 

Difficult Rules: 

Rule 13: Hidden Triple  If three cells in a row contain all the occurrences of 3 possible candidate digits, even though they are accompanied by other candidates, then a hidden triple exists. Each of the three founding cells must hold at least two of the three founding possible
candidates. 

Rule 14:
Hidden Quadruple  If four cells in a row contain all the occurrences of 4 possible candidate digits, even though they are accompanied by other candidates, then a hidden quadruple exists. Each of the founding cells must hold at least two of the four founding possible
candidates.


Rule 15: XWing  An XWing is the discovery of: 1) a row in which a candidate digit is found only in two cells AND 2) only one additional row is found with the same candidate founding digit in only the same two column cells as the former. 

Rule 15A:
Finned XWing  Similar to the regular
XWing in that: 1) find a row in which a candidate digit is
found only in two cells AND 2) one additional row is found
with the same candidate founding digit in the same two
column cells as the former but 3) with an additional
occurrence of the founding digit in only one of the two
boxes of the founding cells in the same row as the founding
cells, AND 4) no other occurrences of the founding digit in
the row's box that does not contain the founding digit
corners. 

Rule 15B:
Finned XWing Hybrid  Discovered by Pritt
Galford, a Finned XWing
Hybrid begins with finding the only two occurrences in a
puzzle row of a candidate digit existing in different boxes.
Call these the primary corners. Then find occurrences
of the same candidate digit in the same columns of a row in
different boxes from the primary corners. Call these
newfound cells the secondary corners. When the
secondary corners are identified, further analysis proceeds
to determine whether a Finned XWing Hybrid actually exists
as follows: 1) cells of a secondary corner's box that are
not in the same column and not in the same row as the
secondary corner are noted to be in the deadzone, 2) cells
of a secondary corner's box that are not in the same column
and but are in the same row as the secondary corner are
called the pzone, and 3) the cells in the same row as the
secondary corners but in neither of the secondary corners'
boxes are designated to the nzone. The three zones
analyzed to finally determine the presence, or not, of a
Finned XWing Hybrid formation as follows. 1) The
deadzone must not contain any occurrences of the candidate
digit from the basis of the primary corners. 2) The
pzone must contain at least one occurrence of the candidate
digit from the basis of the primary corners. 3) The
nzone must not contain any occurrences of the candidate
digit of the primary corners. Having reached this
point with these three conditions met, a Finned XWing
Hybrid is found and this action can be taken: the primary
corner candidate digit can be eliminated from the opposite
secondary corner that was analyzed. 

Rule 16: Swordfish  A swordfish is the discovery of three rows containing no more than three cells with the possible candidate (founding digit) and also sharing the same three columns as well. 

Rule 17: Jellyfish  A jellyfish formation of possible candidate values is exactly like as detailed in the explanation for Rule 16 for swordfish except that we are dealing with four rows and four columns in the candidate arrangement instead of three rows and three columns applicable to a swordfish
rule.


Rule 18: Squirmbag  A squirmbag formation of possible candidate
values is exactly like as detailed in the explanation for Rule 16 for
swordfish (and Rule 17 for jellyfish) except that it always involves five
rows and five columns of the possible candidates instead of three rows and three columns relating to
the swordfish rule, and four rows and four columns required by the jellyfish
rule configuration. 

Really Difficult Rules: 

Rule 19: YWing  A YWing is the discovery of three cells within two horizontally or vertically aligned boxes that meet all the following conditions: 1) only three different digits populate the three cells, 2) each of the three cells contains only two possible
candidates, 3) two of the cells must be in the same row, 4) the three cells must exist in only two boxes, 5) the cell in the box containing two of the founding cells and is in the same row as the other founding cell becomes the pivot cell, and 6) the pivot cell contains one matching digit from the two nonpivot cells. 

Rule 19A:
XYZWing  An XYZWing occurs when all of
the following conditions are met: 1) a cell with a three
candidates is found in a row  this cell becomes the
hinge; and 2) within the hinge's box is found a cell with
two candidates of which both candidates occur among the
candidates of the hinge  this cell becomes known as
pincer # 1; and 3) searching in the puzzle grid's row of the
hinge cell, except cells in the box of the hinge, is found a
cell with two candidates, both of which occur among the
hinge's candidates  this cell becomes known as pincer #
2; and 4) only one of the two candidates of pincer # 2
occurs in pincer # 1. When these four conditions are
met, the common digit among the three cells of the hinge and
two pincers may be deleted from the candidates of the cells
of the hinge's box in the hinge cell's row, except the hinge
cell itself. This is true because the formation of the
hinge and two pincers is a configuration where all three
cells can "see" the two target cells subject to candidate
removal. 

Rule
19A: XYZWing Hybrid  Discovered by Pritt
Galford, the requirements of Rule 19A XYZWing Hybrid begins
with an understanding of Rule 19A XYZWing and includes two additional requirements regardless
of whether Rule 19A XYZWing gains anything toward a puzzle
solution.
The two additional conditions are: 1) Discover whether
pincer 2 of Rule 19A XYZWing is part of a naked pair in
pincer 2's box which exists in pincer 1's row, and if so,
that cell becomes pincer 3; and 2) Does the pincer 3 share
exactly one candidate digit with pincer 1? If all
these conditions are met, then Rule 19A XYZWing Hybrid is
activated; and therefore, the common candidate digit among Rule
19A XYZWing's hinge and its pincer 1 and pincer 2 may be
removed from any cells in the hinge's row except those in
the hinge's box and in pincer 2's box. 

Rule 19B:
WXYZWing  A WXYZWing is present when all
of the following conditions are met: 1) a cell with a four
candidates is found in a row  this cell becomes the
hinge; and 2) within the hinge's box is found a cell with
two candidates of which both candidates occur among the
candidates of the hinge  this cell is then known as
pincer # 1; and 3) searching in the puzzle grid's row
of the hinge cell, except cells in the box of the hinge, is
found a cell with two candidates, both of which occur among
the hinge's candidates  this cell becomes pincer # 2; 4)
only one of the two candidates of pincer # 2 occurs in
pincer # 1; 5) searching in the puzzle grid's row of the
hinge cell beginning in the next cell after pincer # 2,
except cells in the box of the hinge, is found a cell with
two candidates, both of which occur among the hinge's
candidates  this cell becomes pincer # 3; 6) only one of
the two candidates of pincer # 3 occurs in pincers # 1 and
2. When these six conditions are met, the common digit
among the four cells of the hinge and three pincers may be
deleted from the candidates of the cells of the hinge's box
in the hinge cell's row, except the hinge cell itself.
This is true because the formation of the hinge and three
pincers is a configuration where all four cells can "see"
the two target cells subject to candidate removal. 

Rule 20: Unique Rectangle Type 1  When there are four cells of a puzzle in a rectangular formation involving only two boxes with the same twodigit (founding digits) possible values in three of the four corners and the remaining corner contains the same twodigit possible
candidates plus one or more other candidates, then the configuration is a Unique Rectangle Type 1. 

Rule 21: Unique Rectangle Type 2  A Type 2 Unique Rectangle occurs when there are four cells of a puzzle in a rectangular formation involving only two boxes with the same twodigit (founding digits) possible values in all four cells plus the following additional qualifications. Call the lower 2 corners the bottom of the rectangle and the top two corners the top. The top two corners must be in the same box; obviously, the same applies to the bottom corners although the top and bottom corners must be in different boxes. The bottom corners contain the 2digit
possible candidates. The top corners contain the same 2digit possible
candidates plus one other possible digit (the target digit) present in each top corner. In order to avoid the impossible situation of two solutions to the puzzle, then the target digit will occur in the solution in one of the 2 top cells. And since that is true, then the target digit must be removed from: A) all the cells of the top box except the top's corners, and B) from all cells in the same row, that contains the top corners, that are outside the top box. Further, it must be remembered that this type unique rectangle may be inverted. 

Rule 22:
Unique Rectangle Type 2B  A Type 2B Unique Rectangle occurs when there are four cells of a puzzle in a rectangular formation involving only two boxes with the same twodigit (founding digits) possible values in all four cells plus the following additional qualifications. Call the lower 2 corners the bottom of the rectangle and the top two corners the top. The bottom corners must not be in the same box and must hold corners containing the 2digit possible candidates. The top corners must contain the same 2digit possible
candidates as the bottom cells plus one other same possible digit (the target digit) present in each top corner.
In order to avoid the impossible situation of two solutions to the puzzle,
then the target digit will occur in the solution in one of the 2 top cells.
And since that is true, then the target digit must be removed from all cells
in the same row containing the top corners, except the top corner cells of
the rectangle. Remember that this type unique rectangle may be
inverted. 

Rule 23: Unique Rectangle Type 4  A Type 4 Unique Rectangle occurs when there are four cells of a puzzle in a rectangular formation involving only two boxes with the same twodigit (founding digits) possible values in all four cells plus the following additional qualifications. Call the lower 2 corners the bottom of the rectangle and the top two corners the top. The bottom corners must be in the same box and must hold corners containing the 2digit possible candidates. The top and bottom corners must be in different boxes. The top corners must contain the same 2digit possible
candidates as the bottom cells plus one other same possible digit present in each top corner. In order to avoid the impossible situation of two solutions to the puzzle, then one or other of the founding digits must be discovered to exist as possible
candidates in only the top corner cells and nowhere else in the top box, and when this is the case, then a Unique Rectangle Type 4 has been uncovered. Whichever one of the founding digits is found to only occur in the upper corners of the top box, it is the other founding digit of the duo that becomes the target digit. The target digit must be removed from the top corner cells.
This type unique rectangle may be inverted. 

Rule 24: Unique Rectangle Type 4B  A Type 4B Unique Rectangle occurs when there are four cells of a puzzle in a rectangular formation involving only two boxes with the same twodigit (founding digits) possible values in all four cells plus the following additional qualifications. Call the lower 2 corners the bottom of the rectangle and the top two corners the top. The bottom corner cells must be in different boxes and must contain the 2digit possible candidates. The top corners must be in the same box as their bottom counterpart and contain the same 2digit possible
candidates as the bottom cells plus at least one other possible digit present in each top corner. One of the founding digits, of the two in the top corners, must occur only in the cells of the top corners and nowhere else in that row. That being true, then the other founding digit becomes the target digit and a Unique Rectangle Type 4B has been discovered. In order to avoid the impossible situation of two solutions to the puzzle, the target digit must therefore be removed from the top corner cells.
This type unique rectangle may be inverted. 

Rule 34:
Avoided Rectangles  When solving a Sudoku a
condition may be found where three of the four cells have
been solved that are in a rectangular (or square) formation
involving two boxes and in which the two diagonally opposite
corners were solved to the same value. When this is
true, the odd corner's resolved digit (diagonally from the
unsolved cell) must be deleted from the candidates of the
unsolved corner cell to prevent the puzzle from
having two solutions. 

Extremely Difficult Rules: 

Rule 25: Naked Quintuple  When a standalone fivesome of identical digits are found within 5 cells of a row they are a naked quintuple. A 5digit combination must occur in a fivesome set among 5 cells to be a naked quintuple; each of the founding cells must have at least two (but as many as five) of the defining five digits; and since it is a naked quintuple, there may be no accompanying possible candidates in any of the five founding cells. 

Rule 26: Hidden Quintuple  If five cells in a row contain all the occurrences of 5 possible candidate digits, even though they are accompanied by other candidates, then a hidden quintuple exists. Each of the founding cells must hold at least two of the five founding possible
candidates. 

Rule 27: Naked Sextuple  When a standalone set of 6 identical digits are found within 6 cells of a row they are a naked sextuple. A 6digit combination must occur as a 6digit set among 6 cells to be a naked sextuple; each of the founding cells must have at least two (but as many as six) of the defining six digits; and since it is a naked sextuple, there may be no accompanying possible candidates in any of the six founding cells. 

Rule 28:
Hidden Sextuple  If six cells in a row contain all the occurrences of 6 possible candidate digits, even though they are accompanied by other candidates, then a hidden sextuple exists. Each of the founding cells must hold at least two of the six founding possible
candidates. 

Rule 29: Naked Septet  When a standalone set of 7 identical digits are found within 7 cells of a row they are a naked septet. A 7digit combination must occur as a 7digit set among 7 cells to be a naked septet; each of the founding cells must have at least two (but as many as seven) of the defining seven digits; and since it is a naked septet, there may be no accompanying possible candidates in any of the seven founding cells.


Rule 30: Hidden Septet  If seven cells in a row contain all the occurrences of 7 possible candidate digits, even though they are accompanied by other candidates, then a hidden septet exists. Each of the founding cells must hold at least two of the seven founding possible
candidates.


Rule 42:
Aligned Pairs Exclusion  This pattern of
candidate digits occurs when these four conditions are met:
1) two cells in the same row or column of a box, and 2) each
of which has at least two candidate digits, and 3) and there
are other bivalue candidate cells within the same
row/column outside the box of the founding cells having the
same value combinations as the founding cells and 4) there
are bivalue cells within the box of the founding cells also
having the same combination of candidate digits as the
founding cells. When all four conditions apply, it is
possible the outlying bivalue cells (either in the founding
cells' row/column or box) prevent certain of the founding
cells' candidates from occurring in the puzzle solution.
When this condition occurs, the precluded candidates may be
eliminated from the possible solution in the founding cells.


UltraExtremely Complex Rules: 

Rule 31: Multivalue XWing  When a possible digit occurs only twice in a row and some other possible digit occurs only twice in a different row then further analysis is required to learn whether a multivalue XWing is present. The first columns of the two rows must have only one other accompanying possible digit and they must be identical. Likewise, the second columns of the two rows must have only one other accompanying possible digit and they also must be identical although different from the set of accompanying digits in the first column. Then recognize that the intersections of the two rows and the two columns thus detected represent four corners. Having met the requirements so far, there are five more complex tests,
beyond the rule summary presented here, that determine the multivalue XWing (multivalue because of the different key values in identifying the rule as opposed to
like values in determining ordinary XWings; see Rule 15) possible digit elimination processes. 

Rule 32: Single's Chains  A single's chain is a chain of three or more links. Each link represents the discovery of a digit that occurs only twice in a row, column, or box. Links that join together are links that the end cell of a link matches the beginning cell of another link ... like a chain.
When a chain of three or more links involving the same link digit (a
possible candidate digit which occurs exactly two times in a row, column or
box, and in which the total links of the chain are an odd number, and the
link type (row, column, or box) is not the same in any two consecutive
links, e.g., a link that is a row link must join with a column link or a box
link, and a column link must join with box or row links at each of its ends. And of course, box links have to join with row and/or column links. When all these conditions are met, a single's chain has been found. Once found, both ends of the chain point
to two row/column cells where the linkdigit may be removed from the possible candidates.


Rule 32A:
3Link Hybrid Chains  Similar to
a 3link chain of the Single Chains Rule 32 except that
between the end of the first link and the second link
instead of the common point being an individual puzzle cell,
in this hybrid version of a 3link chain, the common point
of both links can be either of a certain three cells as the
intersection within the intersecting box. 

Rule 32B:
3Link Hybrid Chains  Similar to
a 3link chain of the Single Chains Rule 32A. See Rule
32B's detailed explanation and example in the software. 

Rule 33: XYChains  An XYChain is a series of links of 2digit possible
candidates in cells having a common possible digit from the end of one link to the beginning of another link. An XYChain must be an odd number of links long with a minimum of at least three links and the noncommon linking (pincer) digit in the chain's two ends must be equal. The end of one link must be the same founding cell (in the row, column, or box) as the beginning of the next link to which it joins. The cell ends of all the links must not be cell ends of any other link in an XYChain, thereby preventing link end overlap in all founding cells. When all these conditions are met, an XYChain has been found. The logic behind the XYChain is that with an odd number of links
with the same pincer digit at both ends of the chain, then no matter which end of the chain the pincer digit occurs in the puzzle solution, the pincer digit is blocked from occurring at the two intersection coordinates of the two ends and, in addition, also cannot occur in other puzzle cells commonly seen by both chain ends. Once an XYChain is discovered, both ends of the chain point to two row/column cells where the pincerdigit may be removed, if present, from the possible candidates. And any cells which may be seen (by commonality of row, column, or box) from the chain's ends may also have the pincerdigit removed from their possible candidates. 

Rule 35:
Hidden Rectangles  Hidden Rectangles are
formations of four cells (corners) that comprise a square or
rectangle and involve two boxes only. By definition, a
Hidden Rectangle always has accompanying candidate digits in
its corners, other than Corner 1, in up to 3 of its other
corners. The discovery process begins by finding a
cell having just two possible candidates  call them a
twosome  and call it Corner 1. Then proceed to find
another cell in the same row having the same twosome but
perhaps with additional candidates  it becomes Corner 2.
Then search the column of Corner 1 for a cell containing the
twosome candidates (and maybe additional others) of Corner 1
 which becomes Corner 3. At that point, the
coordinates of Corner 4 can be determined. If Corner 4
also contains the candidates of Corner 1 as well as others,
then a possible Hidden Rectangle is found pending further
analysis. If Corners 1 & 2 have exactly two candidates
each (implicitly both the same twosome) or if Corners 1 & 3
have the same relationship, then it is a Type 2 Hidden
Rectangle; otherwise, it's a Type 1 Hidden Rectangle.
It is worthwhile to note that the corner relationships may
be oriented either horizontally or vertically and also in
mirrorimaged configurations. 

In a Type 1 Hidden Rectangle,
analyze the row of Corner 3 to determine whether one of
Corner 1's candidates occurs only twice in the row; if so,
then analyze the column of Corner 1 to find out whether the
same twiceoccurring candidate also occurs only twice in the
column. If so, then the other of Corner 1's twosome
candidate digits can be removed from Corner 4. The
Type 2 Hidden Rectangle further analysis is similar: If
Corner 1 & 3 are exact candidate matches, then check the row
of Corner 3 to see if one of Corner 1's candidates occurs
only twice, if so, the other candidate of Corner 1 can be
eliminated from Corner 2. If it is Corner 2 that
matches Corner 1, then analyze the column of Corner 1 to
discover that one of its candidates occurs only twice, and
if so, the other candidate of Corner 1 can be eliminated
from Corner 4. 

Rule
36: Medusa Rule Type 1  Begin by creating a linked
network of from/to puzzle cells of bivalue possible
candidates by row, column or box. From any starting
point in the network designate alternating properties (such
as green and yellow) to the bivalue beginning
link/endinglink cells throughout the linked network.
Then, if any network cell is found having two linked
candidates with the same property, i.e., both green or both
yellow, a contradiction exists and neither can be correct
cell solutions. When found, all candidate digits of
the same contradiction property may be removed throughout
the networked cells. 

Rule
37: Medusa Rule Type 2  Create a network of linked
cells as described above in the forepart of Rule 36.
Then, if any row, column or box is found having the same
candidate digit with the same property, i.e., both green or
both yellow, a contradiction exists and neither can be
correct cell solutions. When found, all candidate
digits of the same contradiction property may be removed
throughout the networked cells. 

Rule
38: Medusa Rule Type 3  Begin with a network of
linked cells as described above in the forepart of Rule 36.
When any networked cell is found having both properties,
i.e., one green and one yellow designated to candidate
digits together with additional candidates, either
propertydesignated digit could be the correct cell
solution. However, all other candidate digits in the
same networked cell may be removed. 

Rule
39: Medusa Rule Type 4  Begin with a network of
linked cells as described above in the forepart of Rule 36.
When any cell candidate is found that can "see" (occurs in
the same row, column or box) with networked candidates of
the same digit having both properties, i.e., one green and
one yellow, both of which could be the correct cell
solutions, then that candidate digit may be removed from the
"seeing" cells. 

Rule
40: Medusa Rule Type 5  Start with a network of
linked cells as described above in the forepart of Rule 36.
When any cell's candidate is found that can "see" with the
same networked candidate digit, other than in the same row,
column or box and having both properties, i.e., one green
and one yellow, both of which could be the correct cell
solutions, then that candidate digit may be removed from the
"seeing" cells. 

Rule
41: Medusa Rule Type 6  From a network of linked
cells as described above in the forepart of Rule 36, analyze
as follows: Look for a cell candidate, having no
designated property, green or yellow, but can "see" itself
in the same row, column or box with a designated property
and additionally is accompanied by another candidate digit
in its cell having a property opposite to the property of
the "seen" other candidate. When this occurs, the
"seeing" candidate may be removed.. 

Rule 43:
Alternating Inference Chains  Search the
entire Sudoku analyzing every candidate digit in every
unresolved cell looking for a candidate that if it were the
solution to that cell would force another cell in the same
row, column or box to be the same value as the analyzed
digit. In that case, it would introduce an
impossibility into the Sudoku. When such a digit is
found, it may be deleted as a candidate from its cell since
it cannot be that cell's solution. 

Rule 44:
Empty Rectangles  Occurs in Sudoku boxes
when the resolved cells of a box happen to be arranged in a
rectangle (or square) formation and there is one
row and one column in the box in which all cells are
unsolved. These conditions together with a naked
pair in another puzzle row or column leads to discovery
of a candidate digit that, if present, may be removed in another cell. 

Rule
45: Loops & Cycles  Among cells of a Sudoku can
be found relationships of one cell to certain others because
of their candidates and positions. The relationships are
called links as in a chain comprised of individual links. A
group of links are a chain. When the two ends of a chain
are connected, its closure becomes a cycle, or loop, of
linked cell relationships against which certain logic rules
may be applied. 

Links are of two types:
strong and weak, and links come in three kinds: row, column,
and box. A separate issue is whether a loop contains a
discontinuity. A discontinuity exists if the links contain
two consecutive strong links or two consecutive weak links.
Then there are three action rules to be applied to loops: 

Action 1 occurs when two
conditions are met: 1) there are an even number of links,
and 2) no discontinuity. The conditions met, the weak
links' digit can be removed from the candidates of all cells
in the link's unit (row, column or box) except the founding
cells. Action 2 logic applies under three conditions: 1)
the number of links in a loop is odd, 2) there is one
discontinuity, and 3) the cell at the discontinuity is
between two strong links. When met, the link digit solves
the cell in question. Action 3 is operative when three
conditions occur: 1) the number of links in a loop is odd,
2) there is one discontinuity, and 3) the cell at the
discontinuity is between two weak links. Fulfilled, the
link digit may be removed from the candidates of the cell at
the discontinuity. 

Rule 46:
SueDeCoq  Named after the person
who discovered the logical process of almost locked
candidate sets enabling removal of certain candidates.
Finding a SueDeCoq begins by finding in a box, a two or
threecell set of candidates in the same row, in which the
total number of different candidates are at least two
greater than the number of cells that comprise them. Call
these cells Group A. Secondly, in the same row as Group A,
but outside the box, must be found a cell of bivalue, or
greater, set of candidates all of which must occur in the
cells of Group A. This cell becomes Group B. Thirdly, in
the box that contains Group A, find one or two bivalue, or
greater, cells that are replicated among the candidates of
Group A. Call this cell or cells Group C. Fourth, all of
the candidates in Group B must be different from Group C.
Every candidate among the three groups must not occur only
once. When a SueDeCoq has been found meeting all the
requirements, then certain candidates may be eliminated from
other cells. 

Rule 47:
Nice Loops  An extension of the logic of
Rule 45, this rule analyzes all links (whether strong or
weak) but subject to the link propagation rules seeking to
find loop closure. When closure is found, a process
checks to determine whether the loop is without
discontinuity the end of the last link and beginning of the
first link. Up to one discontinuity is permitted.
Loops without a discontinuity enables two possible actions:
1) If any cell has two strong links of different candidate
digits, then all other candidates can be removed from that
cell except the digits of the two strong links, and 2) When
a link is weak, the link's digit can be eliminated from all
cells the two cells can see, except the two founding cells.
If the loop has a discontinuity, then three possible actions
are possible: 1) When the discontinuity has two weak links
of the same digit, that candidate digit can be eliminated
from the discontinuity cell, 2) When the discontinuity is
between two strong links of the same digit, the digit solves
the cell in the discontinuity cell, and 3) When a
discontinuity has a strong and weak link of different link
digits, the weak link's link digit can be eliminated from
the candidates of the discontinuity cell. 

Rule 50: Forced Chains  When all else fails to produce a puzzle solution, one remaining strategy may be the answer. It's the Forced Chains analyses. The process of beginning a chain chase involves starting with a cell that has only two possible
candidates, call it an origin cell. Arbitrarily select one of the origin cell's two digits (calling it the origin digit) and assume it is the correct digit for the cell. To be of any value, opting for the choice just made, the origin digit will unlock other
puzzle cells that may be solved using any and all of the other rules
(except Rule 50). This, in turn, unlocks still more cells using
all the rules in a chain link fashion. 

As one progresses from cell to cell in the solution process, pay particular attention to frequent occurrences when one cell unlocks
(reduces another cell's candidates to just one) another and that cell unlocks more than one other cell.
All of the newly unlocked cells must be chased to determine the effect they
have. Eventually, an outcome from each chain chase will be apparent (they are described below). 

When one has almost gotten a puzzle completely solved but there remain a few unresolved cells or when an impasse has occurred when the way is not clear to proceed, that is the best time to resort to chasing chains. There are possible four outcomes from a single chain chase: 

1) A solution is reached because chasing the chain solves every unresolved puzzle cell. This is the best possible outcome since the puzzle is solved ... it's over, you solved it. 

2) The chasing process may reach a point where there are remaining unresolved puzzle cells having no candidates. But even this is not a total loss. Any chain chase that leads to a dead end with cells having no remaining candidates is proof that the origin cell's origin digit selected
for the chase and assumed to be correct is actually wrong. Therefore, that incorrect origin digit may be removed from the origin cell leaving only one possibility for the origin cell, so at least, you've solved that cell. 

3) The process of resolving one cell after another in chain fashion may lead to a dead end with no further progress to be made, even when employing all of the rules after each new cell is resolved; however, unresolved cells remain and they all have candidates. This outcome yields no information as to the validity of the selected chase starting digit in the origin cell. 

4) The chasing process by two
different paths in the chain may lead to the same
resolution for a specific cell. In this case, you
may NOT necessarily assume a cell resolved by two paths to be the same
value is correct  what is required is to chase the
puzzle forward assuming the twopathresolved cell is
correct to determine whether Sudoku is solvable.
It may be solvable or it may not be. 

When the chain chases of
the two digits of a twosome yields no puzzle solution,
then move
on to other chain chasing opportunities. 





Last Updated
07/13/2015 
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