Sudoku from http://www.sudokuwiki.org/Print_Weekly_Sudoku.asp?unsolvable=78

level: medium

position: .9...8.....2....98...3...547..5....6..3.4.9..2....1..316...9...92....4.....4...1. initial

Autosolve

position: 49...8.....2..4.98...39..547.95....6..3.4.9..2..9.1..3164..9...92....4.7...4...19 autosolve

Pair Reduction Variants

Pair Reduction Analysis

The following important HDP chains were detected:

```* DIS # C3: 6,8 => CTR => C3: 1,7
* DIS # C3: 7,8 => CTR => C3: 1,6
* DIS # G1: 1,2 => CTR => G1: 3,6,7
* DIS # E1: 1,2 => CTR => E1: 5,6,7
* DIS # I5: 1 => CTR => I5: 2,5
* CNT   5 HDP CHAINS /  27 HYP OPENED
```

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

Pair Reduction

The following important HDP chains were detected:

```* DIS # C3: 6,8 => CTR => C3: 1,7
* DIS C3: 1,7 # A5: 5 => CTR => A5: 6,8
* DIS C3: 1,7 + A5: 6,8 # G1: 1,2 => CTR => G1: 3,6,7
* DIS C3: 1,7 + A5: 6,8 + G1: 3,6,7 # G3: 6,7 => CTR => G3: 1,2
* DIS C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 # B9: 5,8 => CTR => B9: 3,7
* DIS C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 # G7: 3,8 => CTR => G7: 2,5
* DIS C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 + G7: 2,5 # B2: 7 => CTR => B2: 3,5
* DIS C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 + G7: 2,5 + B2: 3,5 # C1: 6 => CTR => C1: 1,7
* DIS C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 + G7: 2,5 + B2: 3,5 + C1: 1,7 # D1: 6,7 => CTR => D1: 2
* PRF C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 + G7: 2,5 + B2: 3,5 + C1: 1,7 + D1: 2 => SOL
* STA C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 + G7: 2,5 + B2: 3,5 + C1: 1,7 + D1: 2
* CNT  10 HDP CHAINS /  22 HYP OPENED
```

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

Details

Positions

 .9...8.....2....98...3...547..5....6..3.4.9..2....1..316...9...92....4.....4...1. initial 49...8.....2..4.98...39..547.95....6..3.4.9..2..9.1..3164..9...92....4.7...4...19 autosolve 497258361532164798681397254749583126813642975256971843164739582925816437378425619 solved

level: medium

Pairing Analysis

```--------------------------------------------------
* PAIRS (6)
A3: 6,8
B3: 7,8
I1: 1,2
F4: 2,3
C8: 5,8
I7: 2,5

--------------------------------------------------
* CONSTRAINT PAIRS (AUTO SOLVE)
C1,C3: 1.. / C1 = 1  =>  0 pairs (X) / C3 = 1  =>  6 pairs (_)
B4,B5: 1.. / B4 = 1  =>  0 pairs (X) / B5 = 1  =>  8 pairs (_)
G4,I5: 1.. / G4 = 1  =>  8 pairs (_) / I5 = 1  =>  0 pairs (X)
D8,E8: 1.. / D8 = 1  =>  9 pairs (_) / E8 = 1  =>  7 pairs (_)
C3,G3: 1.. / C3 = 1  =>  6 pairs (_) / G3 = 1  =>  0 pairs (X)
B4,G4: 1.. / B4 = 1  =>  0 pairs (X) / G4 = 1  =>  8 pairs (_)
B5,I5: 1.. / B5 = 1  =>  8 pairs (_) / I5 = 1  =>  0 pairs (X)
I1,I5: 1.. / I1 = 1  =>  8 pairs (_) / I5 = 1  =>  0 pairs (X)
F3,G3: 2.. / F3 = 2  => 22 pairs (_) / G3 = 2  => 11 pairs (_)
A2,B2: 3.. / A2 = 3  =>  8 pairs (_) / B2 = 3  =>  7 pairs (_)
G1,H1: 3.. / G1 = 3  =>  6 pairs (_) / H1 = 3  => 12 pairs (_)
E4,F4: 3.. / E4 = 3  =>  0 pairs (X) / F4 = 3  =>  7 pairs (_)
A9,B9: 3.. / A9 = 3  =>  7 pairs (_) / B9 = 3  =>  8 pairs (_)
A2,A9: 3.. / A2 = 3  =>  8 pairs (_) / A9 = 3  =>  7 pairs (_)
B2,B9: 3.. / B2 = 3  =>  7 pairs (_) / B9 = 3  =>  8 pairs (_)
F4,F8: 3.. / F4 = 3  =>  7 pairs (_) / F8 = 3  =>  0 pairs (X)
G1,G7: 3.. / G1 = 3  =>  6 pairs (_) / G7 = 3  => 12 pairs (_)
B4,B6: 4.. / B4 = 4  =>  9 pairs (_) / B6 = 4  =>  8 pairs (_)
H4,H6: 4.. / H4 = 4  =>  8 pairs (_) / H6 = 4  =>  9 pairs (_)
B4,H4: 4.. / B4 = 4  =>  9 pairs (_) / H4 = 4  =>  8 pairs (_)
B6,H6: 4.. / B6 = 4  =>  8 pairs (_) / H6 = 4  =>  9 pairs (_)
E1,E2: 5.. / E1 = 5  =>  8 pairs (_) / E2 = 5  => 14 pairs (_)
I5,G6: 5.. / I5 = 5  => 15 pairs (_) / G6 = 5  =>  8 pairs (_)
F8,F9: 5.. / F8 = 5  => 11 pairs (_) / F9 = 5  => 13 pairs (_)
G7,I7: 5.. / G7 = 5  => 15 pairs (_) / I7 = 5  =>  8 pairs (_)
C1,E1: 5.. / C1 = 5  => 14 pairs (_) / E1 = 5  =>  8 pairs (_)
C8,F8: 5.. / C8 = 5  => 13 pairs (_) / F8 = 5  => 11 pairs (_)
G6,G7: 5.. / G6 = 5  =>  8 pairs (_) / G7 = 5  => 15 pairs (_)
I5,I7: 5.. / I5 = 5  => 15 pairs (_) / I7 = 5  =>  8 pairs (_)
A5,C6: 6.. / A5 = 6  => 17 pairs (_) / C6 = 6  =>  8 pairs (_)
H8,G9: 6.. / H8 = 6  =>  9 pairs (_) / G9 = 6  => 12 pairs (_)
C6,E6: 6.. / C6 = 6  =>  8 pairs (_) / E6 = 6  => 17 pairs (_)
H1,H8: 6.. / H1 = 6  => 12 pairs (_) / H8 = 6  =>  9 pairs (_)
B9,C9: 7.. / B9 = 7  => 15 pairs (_) / C9 = 7  =>  6 pairs (_)
D7,E7: 7.. / D7 = 7  =>  7 pairs (_) / E7 = 7  => 10 pairs (_)
F3,F5: 7.. / F3 = 7  => 21 pairs (_) / F5 = 7  =>  9 pairs (_)
* DURATION: 0:00:52.628146  START: 04:48:48.451775  END: 04:49:41.079921 2017-05-01
* CP COUNT: (36)
* CLUE FOUND

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (A3,B3,C8,F4,I1,I7)
* 49...8.....2..4.98...39..547.95....6..3.4.9..2..9.1..3164..9...92....4.7...4...19
* PAIR A3: 6,8 BLK 1
C3: 6,8,1,7                              # reduction candidate for 6,8
C3: 6,8 => CTR
* 491.58.62..2..4798.7.3921547195.3.46..3.479.124.9.1573164.79..5925.36487..7425619
C3: 1,7                                  #  7 pairs
* PAIR A3: 6,8 COL A
A5: 6,8,5                                # reduction candidate for 6,8
A5: 5                                    # 20 pairs
A5: 6,8                                  # 11 pairs
* PAIR B3: 7,8 BLK 1
C3: 7,8,1,6                              # reduction candidate for 7,8
C3: 7,8 => CTR
* 491.58..2..2..4.986..392154719523846..364792124698157316427938592.8..4.7...4...19
C3: 1,6                                  #  7 pairs
* PAIR B3: 7,8 COL B
B9: 7,8,3,5                              # reduction candidate for 7,8
B9: 7,8                                  # 13 pairs
B9: 3,5                                  #  7 pairs
* PAIR I1: 1,2 BLK 3
G1: 1,2,3,6,7                            # reduction candidate for 1,2
G1: 1,2 => CTR
* 49..58.3...2.14.98..1392.547.95.3.46..3.479..2469.1573164..93.5928135467...4.6.19
G1: 3,6,7                                #  6 pairs
G3: 1,2,6,7                              # reduction candidate for 1,2
G3: 1,2                                  #  8 pairs
* PAIR I1: 1,2 ROW 1
D1: 1,2,6,7                              # reduction candidate for 1,2
D1: 1,2                                  #  7 pairs
E1: 1,2,5,6,7                            # reduction candidate for 1,2
E1: 1,2 => CTR
* 495.18..2..2.54198..139..54719523846..3.4.921246981573164879235928135467..74.2.19
E1: 5,6,7                                #  6 pairs
* PAIR I1: 1,2 COL I
I5: 1,2,5                                # reduction candidate for 1,2
I5: 5                                    # 15 pairs
I5: 1,2                                  #  8 pairs
* PAIR F4: 2,3 BLK 5
E4: 2,3,8                                # reduction candidate for 2,3
E4: 8                                    # 18 pairs
E4: 2,3                                  # 10 pairs
* PAIR C8: 5,8 BLK 7
A9: 5,8,3                                # reduction candidate for 5,8
A9: 5,8                                  #  8 pairs
B9: 5,8,3,7                              # reduction candidate for 5,8
B9: 5,8                                  # 10 pairs
C9: 5,8,7                                # reduction candidate for 5,8
C9: 5,8                                  # 15 pairs
* PAIR C8: 5,8 COL C
C6: 5,8,6                                # reduction candidate for 5,8
C6: 5,8                                  # 17 pairs
C6: 6                                    #  8 pairs
* PAIR I7: 2,5 BLK 9
G7: 2,5,3,8                              # reduction candidate for 2,5
G7: 2,5                                  # 10 pairs
G7: 3,8                                  #  9 pairs
* PAIR I7: 2,5 COL I
I5: 2,5,1                                # reduction candidate for 2,5
I5: 1 => CTR
* 4917583625326147988763921547195832466532479812489615731648.9..5925..64.7..7425619
I5: 2,5                                  #  8 pairs
* INCONCLUSIVE
* SAVE PR GRAPH zz-www.sudokuwiki.org-0078-base-pr-000.dot
* REASONING
* DIS # C3: 6,8 => CTR => C3: 1,7
* DIS # C3: 7,8 => CTR => C3: 1,6
* DIS # G1: 1,2 => CTR => G1: 3,6,7
* DIS # E1: 1,2 => CTR => E1: 5,6,7
* DIS # I5: 1 => CTR => I5: 2,5
* CNT   5 HDP CHAINS /  27 HYP OPENED

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (A3,B3,C8,F4,I1,I7)
* 49...8.....2..4.98...39..547.95....6..3.4.9..2..9.1..3164..9...92....4.7...4...19
* PAIR A3: 6,8 BLK 1
C3: 6,8,1,7                              # reduction candidate for 6,8
C3: 6,8 => CTR
* 491.58.62..2..4798.7.3921547195.3.46..3.479.124.9.1573164.79..5925.36487..7425619
* PAIR A3: 6,8 COL A
A5: 6,8,5                                # reduction candidate for 6,8
A5: 5 => CTR
* 495218...632.54.9887139..547.95...465.3647982246981573164879..59281.54673574..819
* PAIR RESTART
* PAIR B3: 7,8 COL B
B9: 7,8,3,5                              # reduction candidate for 7,8
B9: 3,5                                  # 16 pairs
B9: 7,8                                  # 14 pairs
* PAIR I1: 1,2 BLK 3
G1: 1,2,3,6,7                            # reduction candidate for 1,2
G1: 1,2 => CTR
* 49..58132532.14.98.813927547195.3.468536479.124.9.1573164..93.5928135467...4.6.19
G3: 1,2,6,7                              # reduction candidate for 1,2
G3: 1,2                                  # 14 pairs
* RESTART
* PAIR I1: 1,2 BLK 3
G3: 1,2,6,7                              # reduction candidate for 1,2
G3: 6,7 => CTR
* 496.58.....26147988713926547.95.31.6613.479.52.5961873164.795..92..35467..74.6.19
* PAIR F4: 2,3 BLK 5
E4: 2,3,8                                # reduction candidate for 2,3
E4: 8                                    # 25 pairs
E4: 2,3                                  # 20 pairs
* PAIR C8: 5,8 BLK 7
B9: 5,8,3,7                              # reduction candidate for 5,8
B9: 5,8 => CTR
* 49...83.1532..4.988713962547.95..1.6613.479.52..961..3164..95.292.61.4.73.74..619
C9: 5,8,7                                # reduction candidate for 5,8
C9: 5,8                                  # 17 pairs
* RESTART
* PAIR C8: 5,8 BLK 7
C9: 5,8,7                                # reduction candidate for 5,8
C9: 7                                    # 18 pairs
C9: 5,8                                  # 17 pairs
* PAIR I7: 2,5 BLK 9
G7: 2,5,3,8                              # reduction candidate for 2,5
G7: 3,8 => CTR
* 491.58..2532..4.9868739215471952384685364792124698157316427938592.8..4.7...4...19

* LEVEL 0 PASS 1 ROUND 2 (AUTO SOLVE) (A2,A3,A5,A9,B3,B5,B9,C3,C6,C8,D7,F4,G2,G3,G4,G7,G9,H1,H7,I1,I5,I7)
* 49...83....2..4.98...39..547.95....6..3.4.9..2..9.1..3164..9...92....4.7...4...19
* PAIR A2: 3,5 BLK 1
B2: 3,5,7                                # reduction candidate for 3,5
B2: 7 => CTR
* 49...83..372.54698.813972547.95..1.6813.4.9.52569.1743164.7953292..134675374..819
* PAIR C3: 1,7 BLK 1
C1: 1,7,6                                # reduction candidate for 1,7
C1: 6 => CTR
* 496.58371..2..46988713962547.95..1.6613.479.52.59617.3164..95.29286154375374.2819
* PAIR H1: 6,7 ROW 1
D1: 6,7,2                                # reduction candidate for 6,7
D1: 6,7 => CTR
* 49..583.2532..4.9868.3921547195.384685324.9..2.69.1..3164..9...92....4.737.4...19
D1: 2 => SOLVED
* 497258361532164798681397254749583126813642975256971843164739582925816437378425619
* DURATION: 0:00:26.212413  START: 04:50:02.822021  END: 04:50:29.034434 2017-05-01
* SOLUTION FOUND
* SAVE PR GRAPH zz-www.sudokuwiki.org-0078-base-pr-001.dot
* REASONING
* DIS # C3: 6,8 => CTR => C3: 1,7
* DIS C3: 1,7 # A5: 5 => CTR => A5: 6,8
* DIS C3: 1,7 + A5: 6,8 # G1: 1,2 => CTR => G1: 3,6,7
* DIS C3: 1,7 + A5: 6,8 + G1: 3,6,7 # G3: 6,7 => CTR => G3: 1,2
* DIS C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 # B9: 5,8 => CTR => B9: 3,7
* DIS C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 # G7: 3,8 => CTR => G7: 2,5
* DIS C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 + G7: 2,5 # B2: 7 => CTR => B2: 3,5
* DIS C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 + G7: 2,5 + B2: 3,5 # C1: 6 => CTR => C1: 1,7
* DIS C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 + G7: 2,5 + B2: 3,5 + C1: 1,7 # D1: 6,7 => CTR => D1: 2
* PRF C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 + G7: 2,5 + B2: 3,5 + C1: 1,7 + D1: 2 => SOL
* STA C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 + G7: 2,5 + B2: 3,5 + C1: 1,7 + D1: 2
* CNT  10 HDP CHAINS /  22 HYP OPENED
```

```http://www.sudokuwiki.org/Print_Weekly_Sudoku.asp?unsolvable=78
```

Solution

position: 497258361532164798681397254749583126813642975256971843164739582925816437378425619 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 # C3: 6,8 => CTR => C3: 1,7
* INC # C3: 1,7 => UNS
* INC # A5: 6,8 => UNS
* INC # A5: 5 => UNS
* DIS # C3: 7,8 => CTR => C3: 1,6
* INC # C3: 1,6 => UNS
* INC # B9: 7,8 => UNS
* INC # B9: 3,5 => UNS
* DIS # G1: 1,2 => CTR => G1: 3,6,7
* INC # G1: 3,6,7 => UNS
* INC # G3: 1,2 => UNS
* INC # D1: 1,2 => UNS
* DIS # E1: 1,2 => CTR => E1: 5,6,7
* INC # E1: 5,6,7 => UNS
* INC # I5: 1,2 => UNS
* INC # I5: 5 => UNS
* INC # E4: 2,3 => UNS
* INC # E4: 8 => UNS
* INC # A9: 5,8 => UNS
* INC # B9: 5,8 => UNS
* INC # C9: 5,8 => UNS
* INC # C6: 5,8 => UNS
* INC # C6: 6 => UNS
* INC # G7: 2,5 => UNS
* INC # G7: 3,8 => UNS
* INC # I5: 2,5 => UNS
* DIS # I5: 1 => CTR => I5: 2,5
* CNT  27 HDP CHAINS /  27 HYP OPENED
```

A2. Pair Reduction

Full list of HDP chains traversed:

```* DIS # C3: 6,8 => CTR => C3: 1,7
* INC C3: 1,7 # A5: 6,8 => UNS
* DIS C3: 1,7 # A5: 5 => CTR => A5: 6,8
* INC C3: 1,7 + A5: 6,8 # B9: 7,8 => UNS
* INC C3: 1,7 + A5: 6,8 # B9: 3,5 => UNS
* DIS C3: 1,7 + A5: 6,8 # G1: 1,2 => CTR => G1: 3,6,7
* INC C3: 1,7 + A5: 6,8 + G1: 3,6,7 # G3: 1,2 => UNS
* INC C3: 1,7 + A5: 6,8 + G1: 3,6,7 # G3: 1,2 => UNS
* DIS C3: 1,7 + A5: 6,8 + G1: 3,6,7 # G3: 6,7 => CTR => G3: 1,2
* INC C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 # E4: 2,3 => UNS
* INC C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 # E4: 8 => UNS
* DIS C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 # B9: 5,8 => CTR => B9: 3,7
* INC C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 # C9: 5,8 => UNS
* INC C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 # C9: 5,8 => UNS
* INC C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 # C9: 7 => UNS
* INC C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 # G7: 2,5 => UNS
* DIS C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 # G7: 3,8 => CTR => G7: 2,5
* INC C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 + G7: 2,5 # B2: 3,5 => UNS
* DIS C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 + G7: 2,5 # B2: 7 => CTR => B2: 3,5
* INC C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 + G7: 2,5 + B2: 3,5 # C1: 1,7 => UNS
* DIS C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 + G7: 2,5 + B2: 3,5 # C1: 6 => CTR => C1: 1,7
* DIS C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 + G7: 2,5 + B2: 3,5 + C1: 1,7 # D1: 6,7 => CTR => D1: 2
* PRF C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 + G7: 2,5 + B2: 3,5 + C1: 1,7 + D1: 2 => SOL
* STA C3: 1,7 + A5: 6,8 + G1: 3,6,7 + G3: 1,2 + B9: 3,7 + G7: 2,5 + B2: 3,5 + C1: 1,7 + D1: 2
* CNT  23 HDP CHAINS /  22 HYP OPENED
```