# Sudoku from http://www.sudokuwiki.org/Print_Weekly_Sudoku.asp?unsolvable=67

level: medium

position: .....6....9.15...6..7...18..5..1...2...9.5...6...3..4..84...6..5...81.9....2..... initial

# Autosolve

position: ..5..6....9.15...6.67...185.5..1...2...9.5...6...3.54..845..6..5...81.9....2...5. autosolve

# Pair Reduction Variants

## Pair Reduction Analysis

The following important HDP chains were detected:

```* DIS # D1: 3,4 => CTR => D1: 7,8
* DIS # F2: 3,4 => CTR => F2: 2,7,8
* DIS # F3: 3,4 => CTR => F3: 2,9
* DIS # A3: 2 => CTR => A3: 3,4
* DIS # D4: 7,8 => CTR => D4: 4,6
* PRF # F6: 7,8 => SOL
* DIS # I6: 7,8 => CTR => I6: 1,9
* DIS # D1: 3,4 => CTR => D1: 7,8
* DIS # E9: 7,9 => CTR => E9: 4,6
* DIS # A7: 7,9 => CTR => A7: 1,2,3
* CNT  10 HDP CHAINS /  25 HYP OPENED
```

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

## Pair Reduction

The following important HDP chains were detected:

```* DIS # D1: 3,4 => CTR => D1: 7,8
* DIS D1: 7,8 # F2: 3,4 => CTR => F2: 2,7,8
* DIS D1: 7,8 + F2: 2,7,8 # F3: 3,4 => CTR => F3: 2,9
* DIS D1: 7,8 + F2: 2,7,8 + F3: 2,9 # F4: 7,8 => CTR => F4: 4
* PRF D1: 7,8 + F2: 2,7,8 + F3: 2,9 + F4: 4 => SOL
* STA D1: 7,8 + F2: 2,7,8 + F3: 2,9 + F4: 4
* CNT   5 HDP CHAINS /   4 HYP OPENED
```

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

# Details

## Positions

 .....6....9.15...6..7...18..5..1...2...9.5...6...3..4..84...6..5...81.9....2..... initial ..5..6....9.15...6.67...185.5..1...2...9.5...6...3.54..845..6..5...81.9....2...5. autosolve 815746329392158476467392185758614932143925768629837541284579613536481297971263854 solved

level: medium

## Pairing Analysis

```--------------------------------------------------
* PAIRS (3)
D3: 3,4
D6: 7,8
E7: 7,9

--------------------------------------------------
* CONSTRAINT PAIRS (AUTO SOLVE)
A1,B1: 1.. / A1 = 1  =>  6 pairs (_) / B1 = 1  =>  9 pairs (_)
H5,H7: 1.. / H5 = 1  =>  0 pairs (X) / H7 = 1  => 10 pairs (_)
E5,F6: 2.. / E5 = 2  =>  0 pairs (*) / F6 = 2  =>  0 pairs (X)
H7,G8: 2.. / H7 = 2  =>  0 pairs (X) / G8 = 2  =>  9 pairs (_)
A7,H7: 2.. / A7 = 2  =>  9 pairs (_) / H7 = 2  =>  0 pairs (X)
B1,B5: 4.. / B1 = 4  =>  9 pairs (_) / B5 = 4  =>  3 pairs (_)
D4,E5: 6.. / D4 = 6  => 10 pairs (_) / E5 = 6  =>  0 pairs (X)
H4,H5: 6.. / H4 = 6  =>  0 pairs (X) / H5 = 6  => 10 pairs (_)
C8,C9: 6.. / C8 = 6  => 10 pairs (_) / C9 = 6  =>  0 pairs (X)
D8,E9: 6.. / D8 = 6  =>  0 pairs (X) / E9 = 6  => 10 pairs (_)
D4,H4: 6.. / D4 = 6  => 10 pairs (_) / H4 = 6  =>  0 pairs (X)
E5,H5: 6.. / E5 = 6  =>  0 pairs (X) / H5 = 6  => 10 pairs (_)
C8,D8: 6.. / C8 = 6  => 10 pairs (_) / D8 = 6  =>  0 pairs (X)
C9,E9: 6.. / C9 = 6  =>  0 pairs (X) / E9 = 6  => 10 pairs (_)
D4,D8: 6.. / D4 = 6  => 10 pairs (_) / D8 = 6  =>  0 pairs (X)
E5,E9: 6.. / E5 = 6  =>  0 pairs (X) / E9 = 6  => 10 pairs (_)
D1,F2: 8.. / D1 = 8  =>  6 pairs (_) / F2 = 8  => 11 pairs (_)
G9,I9: 8.. / G9 = 8  =>  4 pairs (_) / I9 = 8  =>  3 pairs (_)
A1,D1: 8.. / A1 = 8  => 11 pairs (_) / D1 = 8  =>  6 pairs (_)
E3,F3: 9.. / E3 = 9  => 10 pairs (_) / F3 = 9  =>  5 pairs (_)
G1,I1: 9.. / G1 = 9  =>  3 pairs (_) / I1 = 9  =>  4 pairs (_)
G4,I6: 9.. / G4 = 9  =>  4 pairs (_) / I6 = 9  =>  3 pairs (_)
C6,I6: 9.. / C6 = 9  =>  4 pairs (_) / I6 = 9  =>  3 pairs (_)
G1,G4: 9.. / G1 = 9  =>  3 pairs (_) / G4 = 9  =>  4 pairs (_)
I1,I6: 9.. / I1 = 9  =>  4 pairs (_) / I6 = 9  =>  3 pairs (_)
* DURATION: 0:00:29.996824  START: 04:36:22.632212  END: 04:36:52.629036 2017-05-01
* CP COUNT: (25)
* SOLUTION FOUND

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (D3,D6,E7)
* ..5..6....9.15...6.67...185.5..1...2...9.5...6...3.54..845..6..5...81.9....2...5.
* PAIR D3: 3,4 BLK 2
D1: 3,4,7,8                              # reduction candidate for 3,4
D1: 3,4 => CTR
* 815.76....9.158..6.67.29185.5..1...2.4.965...67.83254..8459362.526.81.9..3.247.5.
D1: 7,8                                  #  5 pairs
F2: 3,4,2,7,8                            # reduction candidate for 3,4
F2: 3,4 => CTR
* 145876.2..9.15.47636742918545.618732.2394586161873254928459.61.5.638129493126..5.
F2: 2,7,8                                #  3 pairs
F3: 3,4,2,9                              # reduction candidate for 3,4
F3: 3,4 => CTR
* ..5.269.449.15.2.6267.9.185.59.1..62.42965.1.671832549.8457.62.523681497716249.5.
F3: 2,9                                  #  4 pairs
* PAIR D3: 3,4 ROW 3
A3: 3,4,2                                # reduction candidate for 3,4
A3: 2 => CTR
* ..5.269.449.15.2.6267...185.59.1..62.42965.1.671832549.845..62.523681497.162...5.
A3: 3,4                                  #  7 pairs
* PAIR D3: 3,4 COL D
D8: 3,4,6,7                              # reduction candidate for 3,4
D8: 3,4                                  # 12 pairs
D8: 6,7                                  #  4 pairs
* PAIR D6: 7,8 BLK 5
D4: 7,8,4,6                              # reduction candidate for 7,8
D4: 7,8 => CTR
* 815..6....9.158..6.67...185.5..14.62.4.965...67.83254..845..6..523681.9...624..5.
D4: 4,6                                  #  4 pairs
F4: 7,8,4                                # reduction candidate for 7,8
F4: 7,8                                  #  7 pairs
F6: 7,8,2                                # reduction candidate for 7,8
F6: 7,8 => SOLVED
* 815746329392158476467392185758614932143925768629837541284579613536481297971263854
F6: 2                                    #  4 pairs
* PAIR D6: 7,8 ROW 6
I6: 7,8,1,9                              # reduction candidate for 7,8
I6: 7,8 => CTR
* 1458.673989215.4.6.67...185453618972.289453616197325482845..61.5.6.8129.9.126.85.
I6: 1,9                                  #  4 pairs
* PAIR D6: 7,8 COL D
D1: 7,8,3,4                              # reduction candidate for 7,8
D1: 3,4 => CTR
* 815.76....9.158..6.67.29185.5..1...2.4.965...67.83254..8459362.526.81.9..3.247.5.
D1: 7,8                                  #  5 pairs
* PAIR E7: 7,9 BLK 8
F7: 7,9,3                                # reduction candidate for 7,9
F7: 7,9                                  #  7 pairs
E9: 7,9,4,6                              # reduction candidate for 7,9
E9: 7,9 => CTR
* 8157.6....9.158..6.673.918535.417.62.4.965...67.83254..845.36..523681.9...62.4.5.
E9: 4,6                                  #  4 pairs
F9: 7,9,3,4                              # reduction candidate for 7,9
F9: 7,9                                  #  7 pairs
* PAIR E7: 7,9 ROW 7
A7: 7,9,1,2,3                            # reduction candidate for 7,9
A7: 7,9 => CTR
* 815.269.4.931582762673..185.59417862.4.965.1.671832549.845.36215..681.9...62...5.
A7: 1,2,3                                #  3 pairs
* INCONCLUSIVE
* SAVE PR GRAPH zz-www.sudokuwiki.org-0067-base-pr-000.dot
* REASONING
* DIS # D1: 3,4 => CTR => D1: 7,8
* DIS # F2: 3,4 => CTR => F2: 2,7,8
* DIS # F3: 3,4 => CTR => F3: 2,9
* DIS # A3: 2 => CTR => A3: 3,4
* DIS # D4: 7,8 => CTR => D4: 4,6
* PRF # F6: 7,8 => SOL
* DIS # I6: 7,8 => CTR => I6: 1,9
* DIS # D1: 3,4 => CTR => D1: 7,8
* DIS # E9: 7,9 => CTR => E9: 4,6
* DIS # A7: 7,9 => CTR => A7: 1,2,3
* CNT  10 HDP CHAINS /  25 HYP OPENED

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (D3,D6,E7)
* ..5..6....9.15...6.67...185.5..1...2...9.5...6...3.54..845..6..5...81.9....2...5.
* PAIR D3: 3,4 BLK 2
D1: 3,4,7,8                              # reduction candidate for 3,4
D1: 3,4 => CTR
* 815.76....9.158..6.67.29185.5..1...2.4.965...67.83254..8459362.526.81.9..3.247.5.
F2: 3,4,2,7,8                            # reduction candidate for 3,4
F2: 3,4 => CTR
* 145876.2..9.15.47636742918545.618732.2394586161873254928459.61.5.638129493126..5.
F3: 3,4,2,9                              # reduction candidate for 3,4
F3: 3,4 => CTR
* ..5.269.449.15.2.6267394185.5941..62.42965.1.671832549.8457.62.523681497716249.5.
* PAIR D6: 7,8 BLK 5
F4: 7,8,4                                # reduction candidate for 7,8
F4: 7,8 => CTR
* 8157.6....9.158..6.673.918535.417.62.4.965...67.83254..845.36..523681.9...62.4.5.
F4: 4 => SOLVED
* 815746329392158476467392185758614932143925768629837541284579613536481297971263854
* DURATION: 0:00:05.613828  START: 04:37:10.370849  END: 04:37:15.984677 2017-05-01
* SOLUTION FOUND
* SAVE PR GRAPH zz-www.sudokuwiki.org-0067-base-pr-001.dot
* REASONING
* DIS # D1: 3,4 => CTR => D1: 7,8
* DIS D1: 7,8 # F2: 3,4 => CTR => F2: 2,7,8
* DIS D1: 7,8 + F2: 2,7,8 # F3: 3,4 => CTR => F3: 2,9
* DIS D1: 7,8 + F2: 2,7,8 + F3: 2,9 # F4: 7,8 => CTR => F4: 4
* PRF D1: 7,8 + F2: 2,7,8 + F3: 2,9 + F4: 4 => SOL
* STA D1: 7,8 + F2: 2,7,8 + F3: 2,9 + F4: 4
* CNT   5 HDP CHAINS /   4 HYP OPENED
```

```http://www.sudokuwiki.org/Print_Weekly_Sudoku.asp?unsolvable=67
```

# Solution

position: 815746329392158476467392185758614932143925768629837541284579613536481297971263854 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 # D1: 3,4 => CTR => D1: 7,8
* INC # D1: 7,8 => UNS
* DIS # F2: 3,4 => CTR => F2: 2,7,8
* INC # F2: 2,7,8 => UNS
* DIS # F3: 3,4 => CTR => F3: 2,9
* INC # F3: 2,9 => UNS
* INC # A3: 3,4 => UNS
* DIS # A3: 2 => CTR => A3: 3,4
* INC # D8: 3,4 => UNS
* INC # D8: 6,7 => UNS
* DIS # D4: 7,8 => CTR => D4: 4,6
* INC # D4: 4,6 => UNS
* INC # F4: 7,8 => UNS
* PRF # F6: 7,8 => SOL
* INC # F6: 2 => UNS
* DIS # I6: 7,8 => CTR => I6: 1,9
* INC # I6: 1,9 => UNS
* INC # D1: 7,8 => UNS
* DIS # D1: 3,4 => CTR => D1: 7,8
* INC # F7: 7,9 => UNS
* DIS # E9: 7,9 => CTR => E9: 4,6
* INC # E9: 4,6 => UNS
* INC # F9: 7,9 => UNS
* DIS # A7: 7,9 => CTR => A7: 1,2,3
* INC # A7: 1,2,3 => UNS
* CNT  25 HDP CHAINS /  25 HYP OPENED
```

## A2. Pair Reduction

Full list of HDP chains traversed:

```* DIS # D1: 3,4 => CTR => D1: 7,8
* DIS D1: 7,8 # F2: 3,4 => CTR => F2: 2,7,8
* DIS D1: 7,8 + F2: 2,7,8 # F3: 3,4 => CTR => F3: 2,9
* DIS D1: 7,8 + F2: 2,7,8 + F3: 2,9 # F4: 7,8 => CTR => F4: 4
* PRF D1: 7,8 + F2: 2,7,8 + F3: 2,9 + F4: 4 => SOL
* STA D1: 7,8 + F2: 2,7,8 + F3: 2,9 + F4: 4
* CNT   5 HDP CHAINS /   4 HYP OPENED
```