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 auto-fill 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 Auto-fill feature is turned on or off by clicking the Auto-Fill 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, auto-filling 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 in-progress 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 in-progress 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 fill-in 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 step-by-step 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 on-screen 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 Sudoku-solving 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 follow-on 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 9-cell 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 9-cell square shaded box, find two sets of matching 2-digit numbers, then you have a naked pair within a box. Those two 2-digit 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 3-cell rows of each box. When any possible digit occurs 2 or 3 times in a box's 3-cell 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 two-digit set occurs twice in a row but one or both occurrences of the set is accompanied by other digits; in other words, the two-digit set does not stand alone.
 
Rule 10: Naked Pair -- When a stand-alone 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 stand-alone threesome of identical digits are found within 3 cells of a row they are a naked triple. A 3-digit 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 stand-alone foursome of identical digits are found within 4 cells of a row they are a naked quadruple. A 4-digit 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: X-Wing -- An X-Wing 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 X-Wing -- Similar to the regular X-Wing 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 X-Wing Hybrid -- Discovered by Pritt Galford, a Finned X-Wing 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 X-Wing 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 X-Wing 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 X-Wing 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: Y-Wing -- A Y-Wing 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 non-pivot cells.
 
Rule 19A: XYZ-Wing -- An XYZ-Wing 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: XYZ-Wing Hybrid -- Discovered by Pritt Galford, the requirements of Rule 19A XYZ-Wing Hybrid begins with an understanding of Rule 19A XYZ-Wing and includes two additional requirements regardless of whether Rule 19A XYZ-Wing gains anything toward a puzzle solution.  The two additional conditions are: 1) Discover whether pincer 2 of Rule 19A XYZ-Wing 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 XYZ-Wing Hybrid is activated; and therefore, the common candidate digit among Rule 19A XYZ-Wing'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: WXYZ-Wing -- A WXYZ-Wing 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 two-digit (founding digits) possible values in three of the four corners and the remaining corner contains the same two-digit 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 two-digit (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 2-digit possible candidates.  The top corners contain the same 2-digit 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 two-digit (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 2-digit possible candidates.  The top corners must contain the same 2-digit 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 two-digit (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 2-digit possible candidates.  The top and bottom corners must be in different boxes.  The top corners must contain the same 2-digit 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 two-digit (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 2-digit possible candidates.  The top corners must be in the same box as their bottom counterpart and contain the same 2-digit 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 stand-alone fivesome of identical digits are found within 5 cells of a row they are a naked quintuple.  A 5-digit 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 stand-alone set of 6 identical digits are found within 6 cells of a row they are a naked sextuple.  A 6-digit combination must occur as a 6-digit 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 stand-alone set of 7 identical digits are found within 7 cells of a row they are a naked septet.  A 7-digit combination must occur as a 7-digit 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 bi-value 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 bi-value 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 bi-value 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.
 
Ultra-Extremely Complex Rules:
 
Rule 31: Multi-value X-Wing -- 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 multi-value X-Wing 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 multi-value X-Wing (multi-value because of the different key values in identifying the rule as opposed to like values in determining ordinary X-Wings; 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 link-digit may be removed from the possible candidates.
 
Rule 32A: 3-Link Hybrid Chains -- Similar to a 3-link 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 3-link chain, the common point of both links can be either of a certain three cells as the intersection within the intersecting box.
 
Rule 32B: 3-Link Hybrid Chains -- Similar to a 3-link chain of the Single Chains Rule 32A.  See Rule 32B's detailed explanation and example in the software.
 
Rule 33: XY-Chains -- An XY-Chain is a series of links of 2-digit possible candidates in cells having a common possible digit from the end of one link to the beginning of another link.  An XY-Chain must be an odd number of links long with a minimum of at least three links and the non-common 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 XY-Chain, thereby preventing link end overlap in all founding cells.  When all these conditions are met, an XY-Chain has been found.  The logic behind the XY-Chain 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 XY-Chain is discovered, both ends of the chain point to two row/column cells where the pincer-digit 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 pincer-digit 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 mirror-imaged 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 twice-occurring 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 bi-value possible candidates by row, column or box.  From any starting point in the network designate alternating properties (such as green and yellow) to the bi-value beginning link/ending-link 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 property-designated 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: Sue-De-Coq -- Named after the person who discovered the logical process of almost locked candidate sets enabling removal of certain candidates.  Finding a Sue-De-Coq begins by finding in a box, a two- or three-cell 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 Sue-De-Coq 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 two-path-resolved 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.
 
 

 
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