# Sudoku from http://www.sudokuwiki.org/Print_Weekly_Sudoku.asp?unsolvable=140

level: medium

position: 2....47....8..1.9.9...2...14..8...3...1...9...3...5..73...6...9.2.4..3....47....8 initial

# Autosolve

position: 21...47....8..1.9.9...2...14..8...3...1...9...3...5..73...6...9.2.4..3....47....8 autosolve

# Pair Reduction Variants

## Pair Reduction Analysis

The following important HDP chains were detected:

```* DIS # B7: 5,7 => CTR => B7: 8
* DIS # C3: 5,7 => CTR => C3: 3,6
* DIS # C4: 5,7 => CTR => C4: 2,6,9
* DIS # H8: 5,6 => CTR => H8: 1,7
* DIS # I1: 5,6 => CTR => I1: 3
* DIS # I2: 5,6 => CTR => I2: 2,3,4
* CNT   6 HDP CHAINS /  30 HYP OPENED
```

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

## Pair Reduction

The following important HDP chains were detected:

```* DIS # B7: 5,7 => CTR => B7: 8
* DIS B7: 8 # C3: 5,7 => CTR => C3: 3,6
* DIS B7: 8 + C3: 3,6 # C4: 5,7 => CTR => C4: 2,6,9
* DIS B7: 8 + C3: 3,6 + C4: 2,6,9 # E8: 1,5 => CTR => E8: 8,9
* DIS B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 # H8: 5,6 => CTR => H8: 1,7
* DIS B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 + H8: 1,7 # A8: 5,6 => CTR => A8: 1
* DIS B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 + H8: 1,7 + A8: 1 # I1: 5,6 => CTR => I1: 3
* DIS B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 + H8: 1,7 + A8: 1 + I1: 3 # I2: 5,6 => CTR => I2: 2,4
* DIS B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 + H8: 1,7 + A8: 1 + I1: 3 + I2: 2,4 # I5: 5,6 => CTR => I5: 2,4
* PRF B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 + H8: 1,7 + A8: 1 + I1: 3 + I2: 2,4 + I5: 2,4 # G9: 5,6 => SOL
* STA B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 + H8: 1,7 + A8: 1 + I1: 3 + I2: 2,4 + I5: 2,4 + G9: 5,6
* CNT  10 HDP CHAINS /  28 HYP OPENED
```

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

# Details

## Positions

 2....47....8..1.9.9...2...14..8...3...1...9...3...5..73...6...9.2.4..3....47....8 initial 21...47....8..1.9.9...2...14..8...3...1...9...3...5..73...6...9.2.4..3....47....8 autosolve 216984753748351692953627841462879135571236984839145267387562419125498376694713528 solved

level: medium

## Pairing Analysis

```--------------------------------------------------
* PAIRS (5)
A6: 6,8
C7: 5,7
F7: 2,8
F8: 8,9
I8: 5,6

--------------------------------------------------
* CONSTRAINT PAIRS (AUTO SOLVE)
A8,A9: 1.. / A8 = 1  =>  7 pairs (_) / A9 = 1  =>  8 pairs (_)
E4,G4: 1.. / E4 = 1  =>  7 pairs (_) / G4 = 1  =>  6 pairs (_)
D6,D7: 1.. / D6 = 1  =>  9 pairs (_) / D7 = 1  =>  6 pairs (_)
G2,I2: 2.. / G2 = 2  =>  5 pairs (_) / I2 = 2  =>  9 pairs (_)
C4,C6: 2.. / C4 = 2  => 12 pairs (_) / C6 = 2  =>  6 pairs (_)
C1,C3: 3.. / C1 = 3  =>  0 pairs (X) / C3 = 3  =>  7 pairs (_)
I1,I2: 3.. / I1 = 3  =>  7 pairs (_) / I2 = 3  =>  0 pairs (X)
E9,F9: 3.. / E9 = 3  => 12 pairs (_) / F9 = 3  =>  5 pairs (_)
B2,B3: 4.. / B2 = 4  =>  6 pairs (_) / B3 = 4  => 13 pairs (_)
E5,E6: 4.. / E5 = 4  => 18 pairs (_) / E6 = 4  =>  6 pairs (_)
G7,H7: 4.. / G7 = 4  =>  5 pairs (_) / H7 = 4  =>  5 pairs (_)
I2,I5: 4.. / I2 = 4  => 11 pairs (_) / I5 = 4  =>  6 pairs (_)
E2,F3: 7.. / E2 = 7  =>  8 pairs (_) / F3 = 7  =>  9 pairs (_)
H7,H8: 7.. / H7 = 7  =>  8 pairs (_) / H8 = 7  =>  5 pairs (_)
E1,F3: 8.. / E1 = 8  =>  6 pairs (_) / F3 = 8  =>  0 pairs (X)
B7,A8: 8.. / B7 = 8  =>  6 pairs (_) / A8 = 8  =>  0 pairs (X)
E1,H1: 8.. / E1 = 8  =>  6 pairs (_) / H1 = 8  =>  0 pairs (X)
B7,F7: 8.. / B7 = 8  =>  6 pairs (_) / F7 = 8  =>  0 pairs (X)
B5,B7: 8.. / B5 = 8  =>  0 pairs (X) / B7 = 8  =>  6 pairs (_)
E1,E8: 8.. / E1 = 8  =>  6 pairs (_) / E8 = 8  =>  0 pairs (X)
G3,G6: 8.. / G3 = 8  =>  6 pairs (_) / G6 = 8  =>  9 pairs (_)
D1,E1: 9.. / D1 = 9  =>  5 pairs (_) / E1 = 9  =>  0 pairs (X)
C8,B9: 9.. / C8 = 9  =>  0 pairs (X) / B9 = 9  =>  8 pairs (_)
B4,B9: 9.. / B4 = 9  =>  0 pairs (X) / B9 = 9  =>  8 pairs (_)
D1,D6: 9.. / D1 = 9  =>  5 pairs (_) / D6 = 9  =>  0 pairs (X)
* DURATION: 0:00:27.730144  START: 06:06:25.293699  END: 06:06:53.023843 2017-05-01
* CP COUNT: (25)
* CLUE FOUND

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (A6,C7,F7,F8,I8)
* 21...47....8..1.9.9...2...14..8...3...1...9...3...5..73...6...9.2.4..3....47....8
* PAIR A6: 6,8 BLK 4
A5: 6,8,5,7                              # reduction candidate for 6,8
A5: 6,8                                  # 13 pairs
B5: 6,8,5,7                              # reduction candidate for 6,8
B5: 6,8                                  #  8 pairs
* PAIR A6: 6,8 ROW 6
G6: 6,8,1,2,4                            # reduction candidate for 6,8
G6: 6,8                                  #  7 pairs
H6: 6,8,1,2,4                            # reduction candidate for 6,8
H6: 6,8                                  #  9 pairs
* PAIR A6: 6,8 COL A
A8: 6,8,1,5,7                            # reduction candidate for 6,8
A8: 6,8                                  # 11 pairs
A8: 1,5,7                                #  6 pairs
* PAIR C7: 5,7 BLK 7
B7: 5,7,8                                # reduction candidate for 5,7
B7: 5,7 => CTR
* 2159847637.8..1.9.9...2...14..8..13..81...95.63.1.5..73..268419826419375194753..8
B7: 8                                    #  6 pairs
A8: 5,7,1,6,8                            # reduction candidate for 5,7
A8: 5,7                                  # 11 pairs
C8: 5,7,6,9                              # reduction candidate for 5,7
C8: 5,7                                  # 21 pairs
* PAIR C7: 5,7 ROW 7
H7: 5,7,1,2,4                            # reduction candidate for 5,7
H7: 5,7                                  #  7 pairs
H7: 1,2,4                                #  5 pairs
* PAIR C7: 5,7 COL C
C3: 5,7,3,6                              # reduction candidate for 5,7
C3: 5,7 => CTR
* 213.9478..48..12939...28..14..81..32..12..9.4.3294581738.162..912.4893...94753128
C3: 3,6                                  #  6 pairs
C4: 5,7,2,6,9                            # reduction candidate for 5,7
C4: 5,7 => CTR
* 216984753..83512949436278.149.876132..12439...32195..7387562419.2941837.1.4739.28
C4: 2,6,9                                #  5 pairs
* PAIR F8: 8,9 BLK 8
E8: 8,9,1,5                              # reduction candidate for 8,9
E8: 8,9                                  # 10 pairs
E8: 1,5                                  #  7 pairs
* PAIR I8: 5,6 BLK 9
H8: 5,6,1,7                              # reduction candidate for 5,6
H8: 5,6 => CTR
* 2169847537.8.316929.352.8.14.28.9536.61..798.8396.5..7385162479127498365694753..8
H8: 1,7                                  #  6 pairs
G9: 5,6,1,2                              # reduction candidate for 5,6
G9: 5,6                                  #  8 pairs
H9: 5,6,1,2                              # reduction candidate for 5,6
H9: 5,6                                  #  8 pairs
* PAIR I8: 5,6 ROW 8
A8: 5,6,1,7,8                            # reduction candidate for 5,6
A8: 5,6                                  # 10 pairs
C8: 5,6,7,9                              # reduction candidate for 5,6
C8: 5,6                                  # 10 pairs
* PAIR I8: 5,6 COL I
I1: 5,6,3                                # reduction candidate for 5,6
I1: 5,6 => CTR
* 21..9478..48..12939...28..14..81..32..12..9.4.3294581738.162..912.4893...94753128
I1: 3                                    #  7 pairs
I2: 5,6,2,3,4                            # reduction candidate for 5,6
I2: 5,6 => CTR
* 21..94783.48..129.9.3.28..14..81..32..12..9.4.3294581738.162..912.4893...94753128
I2: 2,3,4                                #  5 pairs
I4: 5,6,2                                # reduction candidate for 5,6
I4: 5,6                                  # 10 pairs
I5: 5,6,2,4                              # reduction candidate for 5,6
I5: 5,6                                  # 19 pairs
* INCONCLUSIVE
* SAVE PR GRAPH zz-www.sudokuwiki.org-0140-base-pr-000.dot
* REASONING
* DIS # B7: 5,7 => CTR => B7: 8
* DIS # C3: 5,7 => CTR => C3: 3,6
* DIS # C4: 5,7 => CTR => C4: 2,6,9
* DIS # H8: 5,6 => CTR => H8: 1,7
* DIS # I1: 5,6 => CTR => I1: 3
* DIS # I2: 5,6 => CTR => I2: 2,3,4
* CNT   6 HDP CHAINS /  30 HYP OPENED

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (A6,C7,F7,F8,I8)
* 21...47....8..1.9.9...2...14..8...3...1...9...3...5..73...6...9.2.4..3....47....8
* PAIR A6: 6,8 BLK 4
A5: 6,8,5,7                              # reduction candidate for 6,8
A5: 6,8                                  # 13 pairs
B5: 6,8,5,7                              # reduction candidate for 6,8
B5: 6,8                                  #  8 pairs
* PAIR A6: 6,8 ROW 6
G6: 6,8,1,2,4                            # reduction candidate for 6,8
G6: 6,8                                  #  7 pairs
H6: 6,8,1,2,4                            # reduction candidate for 6,8
H6: 6,8                                  #  9 pairs
* PAIR A6: 6,8 COL A
A8: 6,8,1,5,7                            # reduction candidate for 6,8
A8: 6,8                                  # 11 pairs
A8: 1,5,7                                #  6 pairs
* PAIR C7: 5,7 BLK 7
B7: 5,7,8                                # reduction candidate for 5,7
B7: 5,7 => CTR
* 2159847637.8..1.9.9...2...14..8..13..81...95.63.1.5..73..268419826419375194753..8
A8: 5,7,1,6                              # reduction candidate for 5,7
A8: 5,7                                  # 11 pairs
C8: 5,7,6,9                              # reduction candidate for 5,7
C8: 5,7                                  # 21 pairs
* PAIR C7: 5,7 ROW 7
H7: 5,7,1,4                              # reduction candidate for 5,7
H7: 5,7                                  #  7 pairs
H7: 1,4                                  #  6 pairs
* PAIR C7: 5,7 COL C
C3: 5,7,3,6                              # reduction candidate for 5,7
C3: 5,7 => CTR
* 213.9478..48..12939..328..14..81..328.12..9.463294581738.162..912648937...47...28
C4: 5,7,2,6,9                            # reduction candidate for 5,7
C4: 5,7 => CTR
* 21.9847.3..8351294943627..1495876132..12439...32195..7387562419.2941837.1.4739.28
* RESTART
* PAIR RESTART
* PAIR C7: 5,7 BLK 7
C8: 5,7,6,9                              # reduction candidate for 5,7
C8: 5,7                                  # 21 pairs
C8: 6,9                                  # 10 pairs
* PAIR C7: 5,7 ROW 7
H7: 5,7,1,4                              # reduction candidate for 5,7
H7: 5,7                                  #  8 pairs
H7: 1,4                                  #  7 pairs
* PAIR F8: 8,9 BLK 8
E8: 8,9,1,5                              # reduction candidate for 8,9
E8: 1,5 => CTR
* 215984763..83.1.94943.27..149.876135..1243986.3..95..7387.62..9.294.837...4739..8
* PAIR I8: 5,6 BLK 9
H8: 5,6,1,7                              # reduction candidate for 5,6
H8: 5,6 => CTR
* 2169847537.8.316929.352.8.14.28.9536.61..798.8396.5..7385162479127498365694753..8
G9: 5,6,1,2                              # reduction candidate for 5,6
G9: 5,6                                  # 14 pairs
H9: 5,6,1,2                              # reduction candidate for 5,6
H9: 5,6                                  # 16 pairs
* PAIR I8: 5,6 ROW 8
A8: 5,6,1                                # reduction candidate for 5,6
A8: 5,6 => CTR
* 2169847537.8.31.9.9...2.8.14.2819536.61..798.839645127385162479627498315194753268
* RESTART
* PAIR I8: 5,6 COL I
I1: 5,6,3                                # reduction candidate for 5,6
I1: 5,6 => CTR
* 21..9478..48.7129397..28..14.98...32..123.9.4.32945..7387162..912.4.937..94753.28
I2: 5,6,2,4                              # reduction candidate for 5,6
I2: 5,6 => CTR
* 21..94783.4837129.973.28..14.98...32..123.9.4.32945..7387162..912.4.937..94753128
I4: 5,6,2                                # reduction candidate for 5,6
I4: 5,6                                  # 16 pairs
I5: 5,6,2,4                              # reduction candidate for 5,6
I5: 5,6 => CTR
* 21...47.3..8..1.94943.27..14..8...32..1236945.32.45..7387.624.91254.83766947.3.28
* RESTART
* PAIR RESTART
* PAIR I8: 5,6 BLK 9
G9: 5,6,1,2                              # reduction candidate for 5,6
G9: 5,6 => SOLVED
* 216984753748351692953627841462879135571236984839145267387562419125498376694713528
* DURATION: 0:00:24.367126  START: 06:07:12.208112  END: 06:07:36.575238 2017-05-01
* SOLUTION FOUND
* SAVE PR GRAPH zz-www.sudokuwiki.org-0140-base-pr-001.dot
* REASONING
* DIS # B7: 5,7 => CTR => B7: 8
* DIS B7: 8 # C3: 5,7 => CTR => C3: 3,6
* DIS B7: 8 + C3: 3,6 # C4: 5,7 => CTR => C4: 2,6,9
* DIS B7: 8 + C3: 3,6 + C4: 2,6,9 # E8: 1,5 => CTR => E8: 8,9
* DIS B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 # H8: 5,6 => CTR => H8: 1,7
* DIS B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 + H8: 1,7 # A8: 5,6 => CTR => A8: 1
* DIS B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 + H8: 1,7 + A8: 1 # I1: 5,6 => CTR => I1: 3
* DIS B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 + H8: 1,7 + A8: 1 + I1: 3 # I2: 5,6 => CTR => I2: 2,4
* DIS B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 + H8: 1,7 + A8: 1 + I1: 3 + I2: 2,4 # I5: 5,6 => CTR => I5: 2,4
* PRF B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 + H8: 1,7 + A8: 1 + I1: 3 + I2: 2,4 + I5: 2,4 # G9: 5,6 => SOL
* STA B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 + H8: 1,7 + A8: 1 + I1: 3 + I2: 2,4 + I5: 2,4 + G9: 5,6
* CNT  10 HDP CHAINS /  28 HYP OPENED
```

```http://www.sudokuwiki.org/Print_Weekly_Sudoku.asp?unsolvable=140
```

# Solution

position: 216984753748351692953627841462879135571236984839145267387562419125498376694713528 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:

```* INC # A5: 6,8 => UNS
* INC # B5: 6,8 => UNS
* INC # G6: 6,8 => UNS
* INC # H6: 6,8 => UNS
* INC # A8: 6,8 => UNS
* INC # A8: 1,5,7 => UNS
* DIS # B7: 5,7 => CTR => B7: 8
* INC # B7: 8 => UNS
* INC # A8: 5,7 => UNS
* INC # C8: 5,7 => UNS
* INC # H7: 5,7 => UNS
* INC # H7: 1,2,4 => UNS
* DIS # C3: 5,7 => CTR => C3: 3,6
* INC # C3: 3,6 => UNS
* DIS # C4: 5,7 => CTR => C4: 2,6,9
* INC # C4: 2,6,9 => UNS
* INC # E8: 8,9 => UNS
* INC # E8: 1,5 => UNS
* DIS # H8: 5,6 => CTR => H8: 1,7
* INC # H8: 1,7 => UNS
* INC # G9: 5,6 => UNS
* INC # H9: 5,6 => UNS
* INC # A8: 5,6 => UNS
* INC # C8: 5,6 => UNS
* DIS # I1: 5,6 => CTR => I1: 3
* INC # I1: 3 => UNS
* DIS # I2: 5,6 => CTR => I2: 2,3,4
* INC # I2: 2,3,4 => UNS
* INC # I4: 5,6 => UNS
* INC # I5: 5,6 => UNS
* CNT  30 HDP CHAINS /  30 HYP OPENED
```

## A2. Pair Reduction

Full list of HDP chains traversed:

```* INC # A5: 6,8 => UNS
* INC # B5: 6,8 => UNS
* INC # G6: 6,8 => UNS
* INC # H6: 6,8 => UNS
* INC # A8: 6,8 => UNS
* INC # A8: 1,5,7 => UNS
* DIS # B7: 5,7 => CTR => B7: 8
* INC B7: 8 # A8: 5,7 => UNS
* INC B7: 8 # C8: 5,7 => UNS
* INC B7: 8 # H7: 5,7 => UNS
* INC B7: 8 # H7: 1,4 => UNS
* DIS B7: 8 # C3: 5,7 => CTR => C3: 3,6
* DIS B7: 8 + C3: 3,6 # C4: 5,7 => CTR => C4: 2,6,9
* INC B7: 8 + C3: 3,6 + C4: 2,6,9 # C8: 5,7 => UNS
* INC B7: 8 + C3: 3,6 + C4: 2,6,9 # C8: 6,9 => UNS
* INC B7: 8 + C3: 3,6 + C4: 2,6,9 # H7: 5,7 => UNS
* INC B7: 8 + C3: 3,6 + C4: 2,6,9 # H7: 1,4 => UNS
* INC B7: 8 + C3: 3,6 + C4: 2,6,9 # E8: 8,9 => UNS
* DIS B7: 8 + C3: 3,6 + C4: 2,6,9 # E8: 1,5 => CTR => E8: 8,9
* DIS B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 # H8: 5,6 => CTR => H8: 1,7
* INC B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 + H8: 1,7 # G9: 5,6 => UNS
* INC B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 + H8: 1,7 # H9: 5,6 => UNS
* DIS B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 + H8: 1,7 # A8: 5,6 => CTR => A8: 1
* DIS B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 + H8: 1,7 + A8: 1 # I1: 5,6 => CTR => I1: 3
* DIS B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 + H8: 1,7 + A8: 1 + I1: 3 # I2: 5,6 => CTR => I2: 2,4
* INC B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 + H8: 1,7 + A8: 1 + I1: 3 + I2: 2,4 # I4: 5,6 => UNS
* DIS B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 + H8: 1,7 + A8: 1 + I1: 3 + I2: 2,4 # I5: 5,6 => CTR => I5: 2,4
* PRF B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 + H8: 1,7 + A8: 1 + I1: 3 + I2: 2,4 + I5: 2,4 # G9: 5,6 => SOL
* STA B7: 8 + C3: 3,6 + C4: 2,6,9 + E8: 8,9 + H8: 1,7 + A8: 1 + I1: 3 + I2: 2,4 + I5: 2,4 + G9: 5,6
* CNT  28 HDP CHAINS /  28 HYP OPENED
```