# Sudoku from http://www.sudokuwiki.org/Print_Weekly_Sudoku.asp?unsolvable=57

level: medium

position: .........8.13.....37..2..65..98..2...1..9..8...2..76..24..1..97.....45.6......... initial

# Autosolve

position: .........8.13.....374.2..65..98..2...1..9..8..82..76.924..1..97.....45.6......... autosolve

# Pair Reduction Variants

## Pair Reduction Analysis

The following important HDP chains were detected:

```* DIS # A1: 5,6 => CTR => A1: 9
* DIS # B1: 5,6 => CTR => B1: 2,9
* DIS # C5: 5,6 => CTR => C5: 3,7
* DIS # D1: 1,9 => CTR => D1: 4,5,6,7
* DIS # I1: 2,4 => CTR => I1: 1,3,8
* DIS # A4: 4,5 => CTR => A4: 6,7
* DIS # H4: 3,4 => CTR => H4: 1,5,7
* DIS # D9: 5,6 => CTR => D9: 2,7,9
* DIS # F9: 5,6 => CTR => F9: 2,3,8,9
* CNT   9 HDP CHAINS /  55 HYP OPENED
```

See Appendix: Full HDP Chains for full list of HDP chains.

## Pair Reduction

The following important HDP chains were detected:

```* DIS # A1: 5,6 => CTR => A1: 9
* DIS A1: 9 # B1: 5,6 => CTR => B1: 2
* DIS A1: 9 + B1: 2 # C5: 5,6 => CTR => C5: 3,7
* DIS A1: 9 + B1: 2 + C5: 3,7 # D1: 5,6 => CTR => D1: 1,4,7
* DIS A1: 9 + B1: 2 + C5: 3,7 + D1: 1,4,7 # F3: 1,9 => CTR => F3: 8
* PRF A1: 9 + B1: 2 + C5: 3,7 + D1: 1,4,7 + F3: 8 # H2: 7 => SOL
* STA A1: 9 + B1: 2 + C5: 3,7 + D1: 1,4,7 + F3: 8 + H2: 7
* CNT   6 HDP CHAINS /  20 HYP OPENED
```

See Appendix: Full HDP Chains for full list of HDP chains.

# Details

## Positions

 .........8.13.....37..2..65..98..2...1..9..8...2..76..24..1..97.....45.6......... initial .........8.13.....374.2..65..98..2...1..9..8..82..76.924..1..97.....45.6......... autosolve 926475138851369472374128965769841253513296784482537619245613897137984526698752341 solved

level: medium

## Pairing Analysis

```--------------------------------------------------
* PAIRS (8)
C1: 5,6
D3: 1,9
I2: 2,4
A6: 4,5
I5: 3,4
B8: 3,9
D7: 5,6
G7: 3,8

--------------------------------------------------
* CONSTRAINT PAIRS (AUTO SOLVE)
F4,D6: 1.. / F4 = 1  => 18 pairs (_) / D6 = 1  => 11 pairs (_)
A8,A9: 1.. / A8 = 1  =>  9 pairs (_) / A9 = 1  => 11 pairs (_)
D6,H6: 1.. / D6 = 1  => 11 pairs (_) / H6 = 1  => 18 pairs (_)
A8,H8: 1.. / A8 = 1  =>  9 pairs (_) / H8 = 1  => 11 pairs (_)
B1,B2: 2.. / B1 = 2  =>  8 pairs (_) / B2 = 2  =>  0 pairs (X)
D5,F5: 2.. / D5 = 2  =>  0 pairs (*) / F5 = 2  =>  0 pairs (X)
D8,H8: 2.. / D8 = 2  =>  9 pairs (_) / H8 = 2  => 10 pairs (_)
F5,F9: 2.. / F5 = 2  =>  8 pairs (_) / F9 = 2  =>  0 pairs (*)
B4,C5: 3.. / B4 = 3  => 11 pairs (_) / C5 = 3  => 16 pairs (_)
E6,H6: 3.. / E6 = 3  =>  9 pairs (_) / H6 = 3  => 24 pairs (_)
H4,H6: 5.. / H4 = 5  => 11 pairs (_) / H6 = 5  => 17 pairs (_)
H4,G5: 7.. / H4 = 7  => 22 pairs (_) / G5 = 7  => 10 pairs (_)
A4,H4: 7.. / A4 = 7  => 10 pairs (_) / H4 = 7  => 22 pairs (_)
F3,G3: 8.. / F3 = 8  =>  9 pairs (_) / G3 = 8  => 15 pairs (_)
C8,E8: 8.. / C8 = 8  =>  9 pairs (_) / E8 = 8  =>  9 pairs (_)
I1,I9: 8.. / I1 = 8  =>  9 pairs (_) / I9 = 8  => 11 pairs (_)
* DURATION: 0:00:18.636499  START: 04:26:57.393372  END: 04:27:16.029871 2017-05-01
* CP COUNT: (16)
* SOLUTION FOUND

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (A6,B8,C1,D3,D7,G7,I2,I5)
* .........8.13.....374.2..65..98..2...1..9..8..82..76.924..1..97.....45.6.........
* PAIR C1: 5,6 BLK 1
A1: 5,6,9                                # reduction candidate for 5,6
A1: 5,6 => CTR
* .2.47....8913..47237412..657.98412.3.1329678448253761924.6138971379845269.8752341
A1: 9                                    #  9 pairs
B1: 5,6,2,9                              # reduction candidate for 5,6
B1: 5,6 => CTR
* 9........8213..97437492..657398..2.1.1..92783.821376.924..1389719...4536..3.89412
B1: 2,9                                  #  9 pairs
B2: 5,6,2,9                              # reduction candidate for 5,6
B2: 5,6                                  # 10 pairs
* PAIR C1: 5,6 ROW 1
D1: 5,6,1,4,7,9                          # reduction candidate for 5,6
D1: 5,6                                  # 21 pairs
E1: 5,6,4,7,8                            # reduction candidate for 5,6
E1: 5,6                                  # 16 pairs
F1: 5,6,1,8,9                            # reduction candidate for 5,6
F1: 5,6                                  # 11 pairs
* PAIR C1: 5,6 COL C
C5: 5,6,3,7                              # reduction candidate for 5,6
C5: 5,6 => CTR
* 92.7..3418.134.9.237412..65.398.1254.1..92783.82.3761924..1..9719.2.4536...9.3.2.
C5: 3,7                                  # 11 pairs
C7: 5,6,3,8                              # reduction candidate for 5,6
C7: 5,6                                  # 13 pairs
C9: 5,6,3,7,8                            # reduction candidate for 5,6
C9: 5,6                                  # 14 pairs
* PAIR D3: 1,9 BLK 2
D1: 1,9,4,5,6,7                          # reduction candidate for 1,9
D1: 1,9 => CTR
* 92.1....88.13..9.2374928165..98.125..1..9278..82.3761924..13897197284536.38759421
D1: 4,5,6,7                              #  8 pairs
F1: 1,9,5,6,8                            # reduction candidate for 1,9
F1: 1,9                                  # 11 pairs
F3: 1,9,8                                # reduction candidate for 1,9
F3: 1,9                                  # 15 pairs
* PAIR D3: 1,9 ROW 3
G3: 1,9,8                                # reduction candidate for 1,9
G3: 8                                    # 15 pairs
G3: 1,9                                  #  9 pairs
* PAIR I2: 2,4 BLK 3
H1: 2,4,1,3,7                            # reduction candidate for 2,4
H1: 2,4                                  # 13 pairs
I1: 2,4,1,3,8                            # reduction candidate for 2,4
I1: 2,4 => CTR
* .......3.8.13.....37492..65.398..2.1.1..92.83.821376.924..1.397.93.845.6.....3..8
I1: 1,3,8                                #  8 pairs
H2: 2,4,7                                # reduction candidate for 2,4
H2: 2,4                                  # 10 pairs
* PAIR I2: 2,4 COL I
I9: 2,4,1,3,8                            # reduction candidate for 2,4
I9: 2,4                                  # 20 pairs
I9: 1,3,8                                #  9 pairs
* PAIR A6: 4,5 BLK 4
A4: 4,5,6,7                              # reduction candidate for 4,5
A4: 4,5 => CTR
* 9..4.....8.13.59..37492..65539846271.1.592.8.482137659245613897.9...45.6.6..59...
A4: 6,7                                  #  9 pairs
A5: 4,5,6,7                              # reduction candidate for 4,5
A5: 4,5                                  # 12 pairs
* PAIR A6: 4,5 ROW 6
D6: 4,5,1                                # reduction candidate for 4,5
D6: 4,5                                  # 18 pairs
E6: 4,5,3                                # reduction candidate for 4,5
E6: 4,5                                  # 24 pairs
H6: 4,5,1,3                              # reduction candidate for 4,5
H6: 4,5                                  # 17 pairs
* PAIR I5: 3,4 BLK 6
H4: 3,4,1,5,7                            # reduction candidate for 3,4
H4: 3,4 => CTR
* ...4.13.88.13..9423749281657.984.231.13.9278448213765924..13897.3.2.45161..7.9423
H4: 1,5,7                                #  8 pairs
I4: 3,4,1                                # reduction candidate for 3,4
I4: 3,4                                  # 16 pairs
G5: 3,4,7                                # reduction candidate for 3,4
G5: 3,4                                  # 22 pairs
H6: 3,4,1,5                              # reduction candidate for 3,4
H6: 3,4                                  # 20 pairs
* PAIR I5: 3,4 COL I
I1: 3,4,1,2,8                            # reduction candidate for 3,4
I1: 3,4                                  # 24 pairs
I9: 3,4,1,2,8                            # reduction candidate for 3,4
I9: 3,4                                  # 12 pairs
* PAIR B8: 3,9 BLK 7
B9: 3,9,5,6                              # reduction candidate for 3,9
B9: 3,9                                  # 19 pairs
B9: 5,6                                  #  9 pairs
* PAIR D7: 5,6 BLK 8
F7: 5,6,3,8                              # reduction candidate for 5,6
F7: 5,6                                  # 10 pairs
D9: 5,6,2,7,9                            # reduction candidate for 5,6
D9: 5,6 => CTR
* ...7.....8.13.....37412..654.98.12...1.29..8..824.761924..1..971..9.4526.....2...
D9: 2,7,9                                #  8 pairs
E9: 5,6,3,7,8                            # reduction candidate for 5,6
E9: 5,6                                  # 14 pairs
F9: 5,6,2,3,8,9                          # reduction candidate for 5,6
F9: 5,6 => CTR
* .2.74.1388.13..4723741289657.9861254.1..92783.82.3761924.61389713.9.4526...2..341
F9: 2,3,8,9                              #  8 pairs
* PAIR D7: 5,6 ROW 7
C7: 5,6,3,8                              # reduction candidate for 5,6
C7: 5,6                                  # 13 pairs
C7: 3,8                                  # 10 pairs
* PAIR D7: 5,6 COL D
D1: 5,6,1,4,7,9                          # reduction candidate for 5,6
D1: 5,6                                  # 21 pairs
D5: 5,6,2,4                              # reduction candidate for 5,6
D5: 5,6                                  # 10 pairs
* PAIR G7: 3,8 BLK 9
G9: 3,8,1,4                              # reduction candidate for 3,8
G9: 3,8                                  # 13 pairs
I9: 3,8,1,2,4                            # reduction candidate for 3,8
I9: 3,8                                  # 12 pairs
* PAIR G7: 3,8 ROW 7
C7: 3,8,5,6                              # reduction candidate for 3,8
C7: 3,8                                  # 10 pairs
F7: 3,8,5,6                              # reduction candidate for 3,8
F7: 3,8                                  # 13 pairs
* PAIR G7: 3,8 COL G
G1: 3,8,1,4,7,9                          # reduction candidate for 3,8
G1: 3,8                                  # 12 pairs
G1: 1,4,7,9                              #  8 pairs
* INCONCLUSIVE
* SAVE PR GRAPH zz-www.sudokuwiki.org-0057-base-pr-000.dot
* REASONING
* DIS # A1: 5,6 => CTR => A1: 9
* DIS # B1: 5,6 => CTR => B1: 2,9
* DIS # C5: 5,6 => CTR => C5: 3,7
* DIS # D1: 1,9 => CTR => D1: 4,5,6,7
* DIS # I1: 2,4 => CTR => I1: 1,3,8
* DIS # A4: 4,5 => CTR => A4: 6,7
* DIS # H4: 3,4 => CTR => H4: 1,5,7
* DIS # D9: 5,6 => CTR => D9: 2,7,9
* DIS # F9: 5,6 => CTR => F9: 2,3,8,9
* CNT   9 HDP CHAINS /  55 HYP OPENED

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (A6,B8,C1,D3,D7,G7,I2,I5)
* .........8.13.....374.2..65..98..2...1..9..8..82..76.924..1..97.....45.6.........
* PAIR C1: 5,6 BLK 1
A1: 5,6,9                                # reduction candidate for 5,6
A1: 5,6 => CTR
* .2.47....8913..47237412..657.98412.3.1329678448253761924.6138971379845269.8752341
B1: 5,6,2                                # reduction candidate for 5,6
B1: 5,6 => CTR
* 9........8213..97437492..657398..2.1.1..92783.821376.924..1389719...4536..3.89412
* RESTART
* PAIR C1: 5,6 ROW 1
D1: 5,6,1,4,7                            # reduction candidate for 5,6
D1: 5,6                                  # 21 pairs
E1: 5,6,4,7,8                            # reduction candidate for 5,6
E1: 5,6                                  # 16 pairs
F1: 5,6,1,8                              # reduction candidate for 5,6
F1: 5,6                                  # 11 pairs
* PAIR C1: 5,6 COL C
C5: 5,6,3,7                              # reduction candidate for 5,6
C5: 5,6 => CTR
* 92.7..3418.134.9.237412..65.398.1254.1..92783.82.3761924..1..9719.2.4536...9.3.2.
C7: 5,6,3,8                              # reduction candidate for 5,6
C7: 5,6                                  # 15 pairs
C9: 5,6,3,7,8                            # reduction candidate for 5,6
C9: 5,6                                  # 16 pairs
* PAIR RESTART
* PAIR C1: 5,6 ROW 1
D1: 5,6,1,4,7                            # reduction candidate for 5,6
D1: 5,6 => CTR
* 92.......8.13.....37412..654.98.12...1..9..8..824.761924..1..971....45.6.........
E1: 5,6,4,7,8                            # reduction candidate for 5,6
E1: 5,6                                  # 19 pairs
F1: 5,6,1,8                              # reduction candidate for 5,6
F1: 5,6                                  # 14 pairs
* PAIR C1: 5,6 COL C
C7: 5,6,3,8                              # reduction candidate for 5,6
C7: 5,6                                  # 15 pairs
C9: 5,6,3,7,8                            # reduction candidate for 5,6
C9: 5,6                                  # 16 pairs
* PAIR RESTART
* PAIR C1: 5,6 ROW 1
E1: 5,6,4,7,8                            # reduction candidate for 5,6
E1: 5,6                                  # 19 pairs
F1: 5,6,1,8                              # reduction candidate for 5,6
F1: 5,6                                  # 14 pairs
* PAIR C1: 5,6 COL C
C7: 5,6,3,8                              # reduction candidate for 5,6
C7: 5,6                                  # 15 pairs
C9: 5,6,3,7,8                            # reduction candidate for 5,6
C9: 5,6                                  # 16 pairs
* PAIR D3: 1,9 BLK 2
F3: 1,9,8                                # reduction candidate for 1,9
F3: 1,9 => CTR
* 92.......8.13..9.2374.2.8657.984.25..1369278448215763924.51.3971....45266.527...8
* PAIR I2: 2,4 BLK 3
H2: 2,4,7                                # reduction candidate for 2,4
H2: 7 => SOLVED
* 926475138851369472374128965769841253513296784482537619245613897137984526698752341
* DURATION: 0:00:17.202300  START: 04:27:51.603247  END: 04:28:08.805547 2017-05-01
* SOLUTION FOUND
* SAVE PR GRAPH zz-www.sudokuwiki.org-0057-base-pr-001.dot
* REASONING
* DIS # A1: 5,6 => CTR => A1: 9
* DIS A1: 9 # B1: 5,6 => CTR => B1: 2
* DIS A1: 9 + B1: 2 # C5: 5,6 => CTR => C5: 3,7
* DIS A1: 9 + B1: 2 + C5: 3,7 # D1: 5,6 => CTR => D1: 1,4,7
* DIS A1: 9 + B1: 2 + C5: 3,7 + D1: 1,4,7 # F3: 1,9 => CTR => F3: 8
* PRF A1: 9 + B1: 2 + C5: 3,7 + D1: 1,4,7 + F3: 8 # H2: 7 => SOL
* STA A1: 9 + B1: 2 + C5: 3,7 + D1: 1,4,7 + F3: 8 + H2: 7
* CNT   6 HDP CHAINS /  20 HYP OPENED
```

```http://www.sudokuwiki.org/Print_Weekly_Sudoku.asp?unsolvable=57
```

# Solution

position: 926475138851369472374128965769841253513296784482537619245613897137984526698752341 solved

See section Pair Reduction for the HDP chains leading to this result.

# Appendix: Full HDP Chains

## A1. Pair Reduction Analysis

Full list of HDP chains traversed:

```* DIS # A1: 5,6 => CTR => A1: 9
* INC # A1: 9 => UNS
* DIS # B1: 5,6 => CTR => B1: 2,9
* INC # B1: 2,9 => UNS
* INC # B2: 5,6 => UNS
* INC # D1: 5,6 => UNS
* INC # E1: 5,6 => UNS
* INC # F1: 5,6 => UNS
* DIS # C5: 5,6 => CTR => C5: 3,7
* INC # C5: 3,7 => UNS
* INC # C7: 5,6 => UNS
* INC # C9: 5,6 => UNS
* DIS # D1: 1,9 => CTR => D1: 4,5,6,7
* INC # D1: 4,5,6,7 => UNS
* INC # F1: 1,9 => UNS
* INC # F3: 1,9 => UNS
* INC # G3: 1,9 => UNS
* INC # G3: 8 => UNS
* INC # H1: 2,4 => UNS
* DIS # I1: 2,4 => CTR => I1: 1,3,8
* INC # I1: 1,3,8 => UNS
* INC # H2: 2,4 => UNS
* INC # I9: 2,4 => UNS
* INC # I9: 1,3,8 => UNS
* DIS # A4: 4,5 => CTR => A4: 6,7
* INC # A4: 6,7 => UNS
* INC # A5: 4,5 => UNS
* INC # D6: 4,5 => UNS
* INC # E6: 4,5 => UNS
* INC # H6: 4,5 => UNS
* DIS # H4: 3,4 => CTR => H4: 1,5,7
* INC # H4: 1,5,7 => UNS
* INC # I4: 3,4 => UNS
* INC # G5: 3,4 => UNS
* INC # H6: 3,4 => UNS
* INC # I1: 3,4 => UNS
* INC # I9: 3,4 => UNS
* INC # B9: 3,9 => UNS
* INC # B9: 5,6 => UNS
* INC # F7: 5,6 => UNS
* DIS # D9: 5,6 => CTR => D9: 2,7,9
* INC # D9: 2,7,9 => UNS
* INC # E9: 5,6 => UNS
* DIS # F9: 5,6 => CTR => F9: 2,3,8,9
* INC # F9: 2,3,8,9 => UNS
* INC # C7: 5,6 => UNS
* INC # C7: 3,8 => UNS
* INC # D1: 5,6 => UNS
* INC # D5: 5,6 => UNS
* INC # G9: 3,8 => UNS
* INC # I9: 3,8 => UNS
* INC # C7: 3,8 => UNS
* INC # F7: 3,8 => UNS
* INC # G1: 3,8 => UNS
* INC # G1: 1,4,7,9 => UNS
* CNT  55 HDP CHAINS /  55 HYP OPENED
```

## A2. Pair Reduction

Full list of HDP chains traversed:

```* DIS # A1: 5,6 => CTR => A1: 9
* DIS A1: 9 # B1: 5,6 => CTR => B1: 2
* INC A1: 9 + B1: 2 # D1: 5,6 => UNS
* INC A1: 9 + B1: 2 # E1: 5,6 => UNS
* INC A1: 9 + B1: 2 # F1: 5,6 => UNS
* DIS A1: 9 + B1: 2 # C5: 5,6 => CTR => C5: 3,7
* INC A1: 9 + B1: 2 + C5: 3,7 # C7: 5,6 => UNS
* INC A1: 9 + B1: 2 + C5: 3,7 # C9: 5,6 => UNS
* DIS A1: 9 + B1: 2 + C5: 3,7 # D1: 5,6 => CTR => D1: 1,4,7
* INC A1: 9 + B1: 2 + C5: 3,7 + D1: 1,4,7 # E1: 5,6 => UNS
* INC A1: 9 + B1: 2 + C5: 3,7 + D1: 1,4,7 # F1: 5,6 => UNS
* INC A1: 9 + B1: 2 + C5: 3,7 + D1: 1,4,7 # C7: 5,6 => UNS
* INC A1: 9 + B1: 2 + C5: 3,7 + D1: 1,4,7 # C9: 5,6 => UNS
* INC A1: 9 + B1: 2 + C5: 3,7 + D1: 1,4,7 # E1: 5,6 => UNS
* INC A1: 9 + B1: 2 + C5: 3,7 + D1: 1,4,7 # F1: 5,6 => UNS
* INC A1: 9 + B1: 2 + C5: 3,7 + D1: 1,4,7 # C7: 5,6 => UNS
* INC A1: 9 + B1: 2 + C5: 3,7 + D1: 1,4,7 # C9: 5,6 => UNS
* DIS A1: 9 + B1: 2 + C5: 3,7 + D1: 1,4,7 # F3: 1,9 => CTR => F3: 8
* INC A1: 9 + B1: 2 + C5: 3,7 + D1: 1,4,7 + F3: 8 # H2: 2,4 => UNS
* PRF A1: 9 + B1: 2 + C5: 3,7 + D1: 1,4,7 + F3: 8 # H2: 7 => SOL
* STA A1: 9 + B1: 2 + C5: 3,7 + D1: 1,4,7 + F3: 8 + H2: 7
* CNT  20 HDP CHAINS /  20 HYP OPENED
```