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Difference between revisions of "Sudoku/Puzzles/Inkala's Maze/Solution process"

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Inkala's Maze ([https://www.sudokuwiki.org/sudoku.htm?bd=800000000003600000070090200050007000000045700000100030001000068008500010090000400 in SW Solver]) has relatively few bifurcation seeds, but more than some "unsolvables." There is a bivalue cell display in SW Solver, and it shows seven chains and one cell pair. One chain connects three cells, so for a Bivalue Ariadne's Thread (BAT) process, a pair in that is an optimal choice, it would appear. We can choose the next bifurcation pair as a reasonable choice among what has become available.  
+
Inkala's Maze ([https://www.sudokuwiki.org/sudoku.htm?bd=800000000003600000070090200050007000000045700000100030001000068008500010090000400 in SW Solver]) has relatively few bifurcation seeds, but more than some "unsolvables." There is a bivalue cell display in SW Solver, and it shows seven chains and one cell pair. One chain connects three cells, so for a Bivalue Ariadne's Thread (BAT) process, a pair in that is an optimal choice, it would appear. We can choose the next bifurcation pair as a reasonable choice among what has become available. Before implementing a choice, the decision cell is marked, the candidate chosen is marked, and the savepoint created which also contains the choice description. With a regional pair choice, only the first possibility is in the name; if returning to this savepoint for the alternate trial, it will be visible in the puzzle when the savepoint is restored.
  
*1. 61=7 [https://www.sudokuwiki.org/sudoku.htm?bd=800000000003600000070090200050007000000045700700100030001000068008500010090000400 link]
+
Choice is not crucial. However, better choices (with more proposed resolutions) will generally result in faster determination of choice status. The notation below is:
 +
 
 +
*At the beginning of each like is a savepoint number. The savepoint is given a sequence by Hodoku. The Nishio is then additionally named as the ##=#, where ## is the Gordonian cell number (i.e. XY = rXcy), and # is the first candidate tested. When a savepoint is reloaded because of later findings, an asterisk is added to the second-choice cell number. So whenever a savepoint leads to contradiction, and if the Nishio has not been tested as the opposite, that savepoint number returns, with the alternate choice (in this study, only bivalue nishios were selected), and the asterisk. So if it comes to a contradiction as well, the process backs up to the previous uncontradicted choice, and backs up recursively if it was also on its last leg.
 +
 
 +
*Generally the study process within each choice is not shown, except that if Simultaneous Bivalue Nishio was used, seed cell and results are mentioned. As well, at one point, a set of individual cell Nishios were used.
 +
 
 +
*One may use any strategy within the study of a leg; however, for speed, spending a lot of time searching for complex patterns is probably not efficient. It is very fast and easy to recuse another level. The deeper the recursion, the simpler the puzzle will be to analyze.
 +
 
 +
*The procedure is to pick a seed pair for a bifurcation Nishio. The first seed pairs were the only three-cell strong chains in the puzzle. Subsequent choices were not formally determined. Which candidate of a pair is chosen for the first nishio is of little importance. In the end, for a complete proof, the reverse choice will always be examined as well.
 +
 
 +
*recursion depth is shown by the level of indent.
 +
 
 +
*1 61=7 [https://www.sudokuwiki.org/sudoku.htm?bd=800000000003600000070090200050007000000045700700100030001000068008500010090000400 link]
 
*-2 77=5  
 
*-2 77=5  
 
*--3 27=1
 
*--3 27=1
Line 49: Line 61:
 
*-----22 22=4* contrad.
 
*-----22 22=4* contrad.
 
*----21 27=8* SBN 65={28} => 8 => contrad.
 
*----21 27=8* SBN 65={28} => 8 => contrad.
*---7 71=5
+
*7 71=5*
*----26 77=3
+
*-26 77=3
*-----27 75=2 SBN 89={27} => 2 => contrad.
+
*--27 75=2 SBN 89={27} => 2 => contrad.
*-----27 75=7*
+
*--27 75=7*
*------28 93=2
+
*---28 93=2
*-------29 94=3 contrad.
+
*----29 94=3 contrad.
*-------29 94=8* SBN 38={58} => 8 => SBN 36={13} => 3; SBN 22={12} => 2 => SBN 12={16} => 6 => contrad.
+
*----29 94=8* SBN 38={58} => 8 => SBN 36={13} => 3; SBN 22={12} => 2 => SBN 12={16} => 6 => contrad.
*------28 93=6*
+
*---28 93=6*
*-------30 33=4 SBN 31={16} => 6 => contrad.
+
*----30 33=4 SBN 31={16} => 6 => contrad.
*----26 77=9* SBN 72={23} => 3  
+
*-26 77=9* SBN 72={23} => 3  
*-----31 82=2 contrad.
+
*--31 82=2 contrad.
*-----31 82=4*
+
*--31 82=4*
*------32 81=2
+
*---32 81=2
*-------33 65=2 SBN 45={38} => contrad.
+
*----33 65=2 SBN 45={38} => contrad.
*-------33 65=8 SBN 45={23} => 2 => contrad.
+
*----33 65=8 SBN 45={23} => 2 => contrad.
*------32 81=7*
+
*---32 81=7*
*-------34 65=2 SBN 38={58} => 8 => contrad.
+
*----34 65=2 SBN 38={58} => 8 => contrad.
*-------34 65=8*
+
*----34 65=8*
*--------35 75=2 contrad.
+
*-----35 75=2 contrad.
*--------35 75=7* contrad.
+
*-----35 75=7* contrad.
*---7 71=7*contrad. (check this!) incorrect savepoint 7, should have been
+
*---7 71=7* contrad. (check this!) incorrect savepoint 7, should have been
 
*7 71=5* [resolved]
 
*7 71=5* [resolved]
 
*-36 77=3
 
*-36 77=3
Line 81: Line 93:
 
*----40 89=7* contrad.
 
*----40 89=7* contrad.
 
*---39 76=9*
 
*---39 76=9*
 +
*----41 85=3 SBN on 82={26} => 6 SBN 43={26} => contrad in mutual.
 +
*-36 77=9*
 +
*--42 89=2
 +
*---43=2 contrad.
 +
*---43= -savepoint 43 incorrect.
 +
*---44 72=2 contrad.
 +
*---44 72=3*
 +
*----45 75=2 contrad.
 +
*----45 75=7
 +
*-----46 65=2
 +
*------47 45=3 SBN 95={18} => 8 BSN 22={12} => 2 contrad.
 +
*------47 45=8* contrad.
 +
*-----46 65=8* contrad.
 +
*--42 89=7* BSN 65={28} => 2 => nishios 59<>6,49<>4,19<>4,41<>4. BSN 61={49} => 4 contrad.
 +
all paths other than this resolution sequence are shown to lead to contradictions.
 +
# 63=7
 +
# 71=5
 +
# 77=3
 +
 +
This process should crack any proper sudoku and even show alternate solutions, if there are only a few -- or if there is no solution. However, of course, people may prefer to use a computer solver like the SudokuWiki Solver to check for the number of solutions.
 +
 +
The solution found here is a valid solution, it must be if it satisfies the rules. However, it is possible that some forks were eliminated through a faulty analysis, which would remain faulty even if the conclusion (contradiction) was correct.
 +
 +
Comment, questions, and corrections are welcome.

Latest revision as of 18:47, 3 December 2019

Inkala's Maze (in SW Solver) has relatively few bifurcation seeds, but more than some "unsolvables." There is a bivalue cell display in SW Solver, and it shows seven chains and one cell pair. One chain connects three cells, so for a Bivalue Ariadne's Thread (BAT) process, a pair in that is an optimal choice, it would appear. We can choose the next bifurcation pair as a reasonable choice among what has become available. Before implementing a choice, the decision cell is marked, the candidate chosen is marked, and the savepoint created which also contains the choice description. With a regional pair choice, only the first possibility is in the name; if returning to this savepoint for the alternate trial, it will be visible in the puzzle when the savepoint is restored.

Choice is not crucial. However, better choices (with more proposed resolutions) will generally result in faster determination of choice status. The notation below is:

  • At the beginning of each like is a savepoint number. The savepoint is given a sequence by Hodoku. The Nishio is then additionally named as the ##=#, where ## is the Gordonian cell number (i.e. XY = rXcy), and # is the first candidate tested. When a savepoint is reloaded because of later findings, an asterisk is added to the second-choice cell number. So whenever a savepoint leads to contradiction, and if the Nishio has not been tested as the opposite, that savepoint number returns, with the alternate choice (in this study, only bivalue nishios were selected), and the asterisk. So if it comes to a contradiction as well, the process backs up to the previous uncontradicted choice, and backs up recursively if it was also on its last leg.
  • Generally the study process within each choice is not shown, except that if Simultaneous Bivalue Nishio was used, seed cell and results are mentioned. As well, at one point, a set of individual cell Nishios were used.
  • One may use any strategy within the study of a leg; however, for speed, spending a lot of time searching for complex patterns is probably not efficient. It is very fast and easy to recuse another level. The deeper the recursion, the simpler the puzzle will be to analyze.
  • The procedure is to pick a seed pair for a bifurcation Nishio. The first seed pairs were the only three-cell strong chains in the puzzle. Subsequent choices were not formally determined. Which candidate of a pair is chosen for the first nishio is of little importance. In the end, for a complete proof, the reverse choice will always be examined as well.
  • recursion depth is shown by the level of indent.
  • 1 61=7 link
  • -2 77=5
  • --3 27=1
  • ---4 31=1 contrad.
  • ---4 36=1* contrad.
  • --3 27=8* contrad.
  • -2 71=5*
  • --5 91=3
  • ---6 72=2 contrad.
  • ---6 72=4* contrad.
  • 1 63=7* [resolved]
  • -7 77=5
  • --8 87=3
  • ---9 93=5
  • ----10 33=4 contrad.
  • ----10 33=6*
  • -----11 91=3
  • ------12 72=2
  • ------12 72=4* contrad.
  • -----11 91=6*
  • ------13 82=2 contrad.
  • ------13 82=4*
  • -------14 13=4
  • --------15 27=1
  • ---------16 74=4 contrad.
  • ---------16 74=9* contrad.
  • --------15 27=8*
  • ---------17 12=1 contrad.
  • ---------17 12=2* contrad.
  • -------14 13=9*
  • --------18 74=4 contrad.
  • --------18 74=9* contrad.
  • ---9 93=6*
  • ----19 82=2
  • -----20 72=3 contrad.
  • -----20 72=4* contrad.
  • ----19 82=4* SBNs 72={23}, <>2 72=3 etc. contrad.
  • ---8 87=9*
  • ----21 27=1
  • -----22 22=2
  • ------23 26=4 contrad.
  • ------23 26=8*
  • -------24 72=3
  • --------25 82=4 SBN 12={16} => 6 => contrad.
  • --------25 82=6* SBN 12={14) => 1 => contrad.
  • -------24 72=4* SBN 12={16} => 1 => contrad.
  • -----22 22=4* contrad.
  • ----21 27=8* SBN 65={28} => 8 => contrad.
  • 7 71=5*
  • -26 77=3
  • --27 75=2 SBN 89={27} => 2 => contrad.
  • --27 75=7*
  • ---28 93=2
  • ----29 94=3 contrad.
  • ----29 94=8* SBN 38={58} => 8 => SBN 36={13} => 3; SBN 22={12} => 2 => SBN 12={16} => 6 => contrad.
  • ---28 93=6*
  • ----30 33=4 SBN 31={16} => 6 => contrad.
  • -26 77=9* SBN 72={23} => 3
  • --31 82=2 contrad.
  • --31 82=4*
  • ---32 81=2
  • ----33 65=2 SBN 45={38} => contrad.
  • ----33 65=8 SBN 45={23} => 2 => contrad.
  • ---32 81=7*
  • ----34 65=2 SBN 38={58} => 8 => contrad.
  • ----34 65=8*
  • -----35 75=2 contrad.
  • -----35 75=7* contrad.
  • ---7 71=7* contrad. (check this!) incorrect savepoint 7, should have been
  • 7 71=5* [resolved]
  • -36 77=3
  • --37 72=2
  • ---38 89=2 contrad.
  • ---38 89=7* SBN 62={68} => 8 => [full solution]

uniqueness proof:

  • --37 72=4*
  • ---39 76=2
  • ----40 89=2 contrad.
  • ----40 89=7* contrad.
  • ---39 76=9*
  • ----41 85=3 SBN on 82={26} => 6 SBN 43={26} => contrad in mutual.
  • -36 77=9*
  • --42 89=2
  • ---43=2 contrad.
  • ---43= -savepoint 43 incorrect.
  • ---44 72=2 contrad.
  • ---44 72=3*
  • ----45 75=2 contrad.
  • ----45 75=7
  • -----46 65=2
  • ------47 45=3 SBN 95={18} => 8 BSN 22={12} => 2 contrad.
  • ------47 45=8* contrad.
  • -----46 65=8* contrad.
  • --42 89=7* BSN 65={28} => 2 => nishios 59<>6,49<>4,19<>4,41<>4. BSN 61={49} => 4 contrad.

all paths other than this resolution sequence are shown to lead to contradictions.

  1. 63=7
  2. 71=5
  3. 77=3

This process should crack any proper sudoku and even show alternate solutions, if there are only a few -- or if there is no solution. However, of course, people may prefer to use a computer solver like the SudokuWiki Solver to check for the number of solutions.

The solution found here is a valid solution, it must be if it satisfies the rules. However, it is possible that some forks were eliminated through a faulty analysis, which would remain faulty even if the conclusion (contradiction) was correct.

Comment, questions, and corrections are welcome.