# Sudoku from http://www.sudokuwiki.org/Print_Weekly_Sudoku.asp?unsolvable=91

level: medium

position: ....38....4.2....52.....1....73...6..5..9..7..9...53....9.....64....6.2....91.... initial

# Autosolve

position: ....38....4.2....52.....1....73..569.5..9..7..9...53....9.....64....692....91.... autosolve

# Pair Reduction Variants

## Pair Reduction Analysis

The following important HDP chains were detected:

```* DIS # D1: 6,7 => CTR => D1: 1,4,5
* DIS # E3: 6,7 => CTR => E3: 4,5
* DIS # A2: 6,7 => CTR => A2: 1,3,8,9
* DIS # C5: 1,2 => CTR => C5: 3,4,6,8
* DIS # I5: 1,2 => CTR => I5: 4,8
* CNT   5 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: 6,7 => CTR => D1: 1,4,5
* DIS D1: 1,4,5 # E3: 6,7 => CTR => E3: 4,5
* DIS D1: 1,4,5 + E3: 4,5 # A2: 6,7 => CTR => A2: 1,3,8,9
* DIS D1: 1,4,5 + E3: 4,5 + A2: 1,3,8,9 # B4: 1,8 => CTR => B4: 2
* PRF D1: 1,4,5 + E3: 4,5 + A2: 1,3,8,9 + B4: 2 # A5: 1,8 => SOL
* STA D1: 1,4,5 + E3: 4,5 + A2: 1,3,8,9 + B4: 2 + A5: 1,8
* CNT   5 HDP CHAINS /  21 HYP OPENED
```

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

# Details

## Positions

 ....38....4.2....52.....1....73...6..5..9..7..9...53....9.....64....6.2....91.... initial ....38....4.2....52.....1....73..569.5..9..7..9...53....9.....64....692....91.... autosolve 975138642341267895286459137127384569853691274694725381539842716418576923762913458 solved

level: medium

## Pairing Analysis

```--------------------------------------------------
* PAIRS (4)
E2: 6,7
H1: 4,9
A4: 1,8
F5: 1,2

--------------------------------------------------
* CONSTRAINT PAIRS (AUTO SOLVE)
D1,F2: 1.. / D1 = 1  =>  8 pairs (_) / F2 = 1  =>  0 pairs (X)
H7,I8: 1.. / H7 = 1  =>  5 pairs (_) / I8 = 1  =>  5 pairs (_)
H6,H7: 1.. / H6 = 1  =>  5 pairs (_) / H7 = 1  =>  5 pairs (_)
G1,I1: 2.. / G1 = 2  =>  0 pairs (X) / I1 = 2  => 10 pairs (_)
G1,G5: 2.. / G1 = 2  =>  0 pairs (X) / G5 = 2  => 10 pairs (_)
A5,C5: 3.. / A5 = 3  =>  4 pairs (_) / C5 = 3  =>  9 pairs (_)
F7,F9: 3.. / F7 = 3  =>  4 pairs (_) / F9 = 3  =>  7 pairs (_)
C5,C6: 4.. / C5 = 4  =>  6 pairs (_) / C6 = 4  =>  5 pairs (_)
E4,F4: 4.. / E4 = 4  => 10 pairs (_) / F4 = 4  =>  8 pairs (_)
H7,H9: 5.. / H7 = 5  =>  6 pairs (_) / H9 = 5  =>  4 pairs (_)
G1,G2: 6.. / G1 = 6  => 11 pairs (_) / G2 = 6  => 14 pairs (_)
D6,E6: 7.. / D6 = 7  =>  5 pairs (_) / E6 = 7  =>  0 pairs (X)
A1,A2: 9.. / A1 = 9  =>  5 pairs (_) / A2 = 9  =>  5 pairs (_)
F2,F3: 9.. / F2 = 9  =>  9 pairs (_) / F3 = 9  =>  5 pairs (_)
A1,H1: 9.. / A1 = 9  =>  5 pairs (_) / H1 = 9  =>  5 pairs (_)
F3,H3: 9.. / F3 = 9  =>  5 pairs (_) / H3 = 9  =>  9 pairs (_)
* DURATION: 0:00:18.487500  START: 05:06:41.974055  END: 05:07:00.461555 2017-05-01
* CP COUNT: (16)
* CLUE FOUND

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (A4,E2,F5,H1)
* ....38....4.2....52.....1....73..569.5..9..7..9...53....9.....64....692....91....
* PAIR E2: 6,7 BLK 2
D1: 6,7,1,4,5                            # reduction candidate for 6,7
D1: 6,7 => CTR
* ..16382...4.2716.52.5.491..127384569.5..92.7..9.765312.1942...64..856921..291....
D1: 1,4,5                                #  4 pairs
D3: 6,7,4,5                              # reduction candidate for 6,7
D3: 6,7                                  # 17 pairs
E3: 6,7,4,5                              # reduction candidate for 6,7
E3: 6,7 => CTR
* 965138.4.741269..52..5.419...734.569.5.69..7.69.78531...942...6473856921...91....
E3: 4,5                                  #  5 pairs
* PAIR E2: 6,7 ROW 2
A2: 6,7,1,3,8,9                          # reduction candidate for 6,7
A2: 6,7 => CTR
* 9..138642.412798352....4197..7342569.5..9127..927.53.4..942.7.647...692....91....
A2: 1,3,8,9                              #  4 pairs
G2: 6,7,8                                # reduction candidate for 6,7
G2: 6,7                                  #  6 pairs
* PAIR E2: 6,7 COL E
E6: 6,7,2,8                              # reduction candidate for 6,7
E6: 2,8                                  # 11 pairs
E6: 6,7                                  #  7 pairs
* PAIR H1: 4,9 BLK 3
H3: 4,9,3,8                              # reduction candidate for 4,9
H3: 4,9                                  #  8 pairs
H3: 3,8                                  #  6 pairs
* PAIR A4: 1,8 BLK 4
B4: 1,8,2                                # reduction candidate for 1,8
B4: 1,8                                  # 14 pairs
A5: 1,8,3,6                              # reduction candidate for 1,8
A5: 1,8                                  # 10 pairs
C5: 1,8,2,3,4,6                          # reduction candidate for 1,8
C5: 1,8                                  #  9 pairs
A6: 1,8,6                                # reduction candidate for 1,8
A6: 1,8                                  #  9 pairs
C6: 1,8,2,4,6                            # reduction candidate for 1,8
C6: 1,8                                  # 10 pairs
* PAIR A4: 1,8 COL A
A2: 1,8,3,6,7,9                          # reduction candidate for 1,8
A2: 1,8                                  #  8 pairs
A7: 1,8,3,5,7                            # reduction candidate for 1,8
A7: 1,8                                  #  7 pairs
* PAIR F5: 1,2 BLK 5
F4: 1,2,4                                # reduction candidate for 1,2
F4: 1,2                                  # 10 pairs
F4: 4                                    #  8 pairs
* PAIR F5: 1,2 ROW 5
C5: 1,2,3,4,6,8                          # reduction candidate for 1,2
C5: 1,2 => CTR
* ...1382...4.2796.52..56419...734.56935.69..7.694785312..942...6473856921...91....
C5: 3,4,6,8                              #  4 pairs
I5: 1,2,4,8                              # reduction candidate for 1,2
I5: 1,2 => CTR
* 9.5138247.4.2796.527...419...734.569.5..9..7..9.7.53....94....64....692....91....
I5: 4,8                                  #  6 pairs
* INCONCLUSIVE
* SAVE PR GRAPH zz-www.sudokuwiki.org-0091-base-pr-000.dot
* REASONING
* DIS # D1: 6,7 => CTR => D1: 1,4,5
* DIS # E3: 6,7 => CTR => E3: 4,5
* DIS # A2: 6,7 => CTR => A2: 1,3,8,9
* DIS # C5: 1,2 => CTR => C5: 3,4,6,8
* DIS # I5: 1,2 => CTR => I5: 4,8
* CNT   5 HDP CHAINS /  25 HYP OPENED

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (A4,E2,F5,H1)
* ....38....4.2....52.....1....73..569.5..9..7..9...53....9.....64....692....91....
* PAIR E2: 6,7 BLK 2
D1: 6,7,1,4,5                            # reduction candidate for 6,7
D1: 6,7 => CTR
* ..16382...4.2716.52.5.491..127384569.5..92.7..9.765312.1942...64..856921..291....
D3: 6,7,4,5                              # reduction candidate for 6,7
D3: 6,7                                  # 17 pairs
E3: 6,7,4,5                              # reduction candidate for 6,7
E3: 6,7 => CTR
* 965138.4.741269..52..5.419...734.569.5.69..7.69.78531...942...6473856921...91....
* RESTART
* PAIR E2: 6,7 BLK 2
D3: 6,7,4,5                              # reduction candidate for 6,7
D3: 6,7                                  # 17 pairs
D3: 4,5                                  # 13 pairs
* PAIR E2: 6,7 ROW 2
A2: 6,7,1,3,8,9                          # reduction candidate for 6,7
A2: 6,7 => CTR
* 97..3864264.27.8.52..6..1.7..73..569.56.91274.947.53....9.....64....692....91....
G2: 6,7,8                                # reduction candidate for 6,7
G2: 6,7                                  #  7 pairs
* RESTART
* PAIR E2: 6,7 ROW 2
G2: 6,7,8                                # reduction candidate for 6,7
G2: 8                                    # 26 pairs
G2: 6,7                                  #  7 pairs
* PAIR E2: 6,7 COL E
E6: 6,7,2,8                              # reduction candidate for 6,7
E6: 2,8                                  # 16 pairs
E6: 6,7                                  #  7 pairs
* PAIR RESTART
* PAIR E2: 6,7 BLK 2
D3: 6,7,4,5                              # reduction candidate for 6,7
D3: 6,7                                  # 17 pairs
D3: 4,5                                  # 13 pairs
* PAIR E2: 6,7 ROW 2
G2: 6,7,8                                # reduction candidate for 6,7
G2: 8                                    # 26 pairs
G2: 6,7                                  #  7 pairs
* PAIR E2: 6,7 COL E
E6: 6,7,2,8                              # reduction candidate for 6,7
E6: 2,8                                  # 16 pairs
E6: 6,7                                  #  7 pairs
* PAIR H1: 4,9 BLK 3
H3: 4,9,3,8                              # reduction candidate for 4,9
H3: 4,9                                  #  9 pairs
H3: 3,8                                  #  7 pairs
* PAIR A4: 1,8 BLK 4
B4: 1,8,2                                # reduction candidate for 1,8
B4: 1,8 => CTR
* 975138642.4126..352.....197..73..569354691278692..5314..9.....64....6921...91...3
A5: 1,8,3,6                              # reduction candidate for 1,8
A5: 1,8 => SOLVED
* 975138642341267895286459137127384569853691274694725381539842716418576923762913458
* DURATION: 0:00:17.292781  START: 05:07:16.610382  END: 05:07:33.903163 2017-05-01
* SOLUTION FOUND
* SAVE PR GRAPH zz-www.sudokuwiki.org-0091-base-pr-001.dot
* REASONING
* DIS # D1: 6,7 => CTR => D1: 1,4,5
* DIS D1: 1,4,5 # E3: 6,7 => CTR => E3: 4,5
* DIS D1: 1,4,5 + E3: 4,5 # A2: 6,7 => CTR => A2: 1,3,8,9
* DIS D1: 1,4,5 + E3: 4,5 + A2: 1,3,8,9 # B4: 1,8 => CTR => B4: 2
* PRF D1: 1,4,5 + E3: 4,5 + A2: 1,3,8,9 + B4: 2 # A5: 1,8 => SOL
* STA D1: 1,4,5 + E3: 4,5 + A2: 1,3,8,9 + B4: 2 + A5: 1,8
* CNT   5 HDP CHAINS /  21 HYP OPENED
```

```http://www.sudokuwiki.org/Print_Weekly_Sudoku.asp?unsolvable=91
```

# Solution

position: 975138642341267895286459137127384569853691274694725381539842716418576923762913458 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: 6,7 => CTR => D1: 1,4,5
* INC # D1: 1,4,5 => UNS
* INC # D3: 6,7 => UNS
* DIS # E3: 6,7 => CTR => E3: 4,5
* INC # E3: 4,5 => UNS
* DIS # A2: 6,7 => CTR => A2: 1,3,8,9
* INC # A2: 1,3,8,9 => UNS
* INC # G2: 6,7 => UNS
* INC # E6: 6,7 => UNS
* INC # E6: 2,8 => UNS
* INC # H3: 4,9 => UNS
* INC # H3: 3,8 => UNS
* INC # B4: 1,8 => UNS
* INC # A5: 1,8 => UNS
* INC # C5: 1,8 => UNS
* INC # A6: 1,8 => UNS
* INC # C6: 1,8 => UNS
* INC # A2: 1,8 => UNS
* INC # A7: 1,8 => UNS
* INC # F4: 1,2 => UNS
* INC # F4: 4 => UNS
* DIS # C5: 1,2 => CTR => C5: 3,4,6,8
* INC # C5: 3,4,6,8 => UNS
* DIS # I5: 1,2 => CTR => I5: 4,8
* INC # I5: 4,8 => UNS
* CNT  25 HDP CHAINS /  25 HYP OPENED
```

## A2. Pair Reduction

Full list of HDP chains traversed:

```* DIS # D1: 6,7 => CTR => D1: 1,4,5
* INC D1: 1,4,5 # D3: 6,7 => UNS
* DIS D1: 1,4,5 # E3: 6,7 => CTR => E3: 4,5
* INC D1: 1,4,5 + E3: 4,5 # D3: 6,7 => UNS
* INC D1: 1,4,5 + E3: 4,5 # D3: 4,5 => UNS
* DIS D1: 1,4,5 + E3: 4,5 # A2: 6,7 => CTR => A2: 1,3,8,9
* INC D1: 1,4,5 + E3: 4,5 + A2: 1,3,8,9 # G2: 6,7 => UNS
* INC D1: 1,4,5 + E3: 4,5 + A2: 1,3,8,9 # G2: 6,7 => UNS
* INC D1: 1,4,5 + E3: 4,5 + A2: 1,3,8,9 # G2: 8 => UNS
* INC D1: 1,4,5 + E3: 4,5 + A2: 1,3,8,9 # E6: 6,7 => UNS
* INC D1: 1,4,5 + E3: 4,5 + A2: 1,3,8,9 # E6: 2,8 => UNS
* INC D1: 1,4,5 + E3: 4,5 + A2: 1,3,8,9 # D3: 6,7 => UNS
* INC D1: 1,4,5 + E3: 4,5 + A2: 1,3,8,9 # D3: 4,5 => UNS
* INC D1: 1,4,5 + E3: 4,5 + A2: 1,3,8,9 # G2: 6,7 => UNS
* INC D1: 1,4,5 + E3: 4,5 + A2: 1,3,8,9 # G2: 8 => UNS
* INC D1: 1,4,5 + E3: 4,5 + A2: 1,3,8,9 # E6: 6,7 => UNS
* INC D1: 1,4,5 + E3: 4,5 + A2: 1,3,8,9 # E6: 2,8 => UNS
* INC D1: 1,4,5 + E3: 4,5 + A2: 1,3,8,9 # H3: 4,9 => UNS
* INC D1: 1,4,5 + E3: 4,5 + A2: 1,3,8,9 # H3: 3,8 => UNS
* DIS D1: 1,4,5 + E3: 4,5 + A2: 1,3,8,9 # B4: 1,8 => CTR => B4: 2
* PRF D1: 1,4,5 + E3: 4,5 + A2: 1,3,8,9 + B4: 2 # A5: 1,8 => SOL
* STA D1: 1,4,5 + E3: 4,5 + A2: 1,3,8,9 + B4: 2 + A5: 1,8
* CNT  21 HDP CHAINS /  21 HYP OPENED
```