# Original Sudoku

level: hard

position: 98.7..6..7..........5.9..7.5...8..4...4..3..6.....21...5..4..6....1..2.......6..3 initial

# Autosolve

position: 98.7..6..7..........5.9..7.5...8..4...4..3..6...4.21...5..4..6....1..2.......6..3 autosolve

# Pair Reduction Variants

## Pair Reduction Analysis

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

## Pair Reduction

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

## Deep Pair Reduction

Time used: 0:00:11.619173

The following important HDP chains were detected:

```* DIS # B4: 6,9 # E5: 5 => CTR => E5: 1,7
* PRF # B4: 6,9 + E5: 1,7 # C4: 1,7 => SOL
* STA # B4: 6,9 + E5: 1,7 + C4: 1,7
* CNT   2 HDP CHAINS /  14 HYP OPENED
```

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

# Details

## Positions

 98.7..6..7..........5.9..7.5...8..4...4..3..6.....21...5..4..6....1..2.......6..3 initial 98.7..6..7..........5.9..7.5...8..4...4..3..6...4.21...5..4..6....1..2.......6..3 autosolve 981734625746215839325698471567981342214573986893462157152349768638157294479826513 solved

level: hard

## Pairing Analysis

```--------------------------------------------------
* PAIRS (2)
D4: 6,9
D5: 5,9

--------------------------------------------------
* CONSTRAINT PAIRS (AUTO SOLVE)
F4,E5: 1.. / F4 = 1  =>  9 pairs (_) / E5 = 1  =>  4 pairs (_)
I7,H9: 1.. / I7 = 1  =>  2 pairs (_) / H9 = 1  =>  3 pairs (_)
I4,H5: 2.. / I4 = 2  =>  2 pairs (_) / H5 = 2  =>  4 pairs (_)
G4,H6: 3.. / G4 = 3  =>  3 pairs (_) / H6 = 3  =>  4 pairs (_)
D7,E8: 3.. / D7 = 3  =>  3 pairs (_) / E8 = 3  =>  2 pairs (_)
I8,G9: 4.. / I8 = 4  =>  3 pairs (_) / G9 = 4  =>  3 pairs (_)
F1,I1: 4.. / F1 = 4  =>  3 pairs (_) / I1 = 4  =>  4 pairs (_)
D4,E6: 6.. / D4 = 6  =>  2 pairs (_) / E6 = 6  =>  6 pairs (_)
E2,E6: 6.. / E2 = 6  =>  2 pairs (_) / E6 = 6  =>  6 pairs (_)
* DURATION: 0:00:05.314050  START: 17:13:33.917698  END: 17:13:39.231748 2020-12-10
* CP COUNT: (9)
* INCONCLUSIVE

* DEEP PAIR REDUCTION
* DURATION: 0:00:11.537564  START: 17:13:43.967635  END: 17:13:55.505199 2020-12-10
* SOLUTION FOUND
* SAVE PR GRAPH xx-ph-00028970-2011_12-base-pr-002.dot
* REASONING
* DIS # B4: 6,9 # E5: 5 => CTR => E5: 1,7
* PRF # B4: 6,9 + E5: 1,7 # C4: 1,7 => SOL
* STA # B4: 6,9 + E5: 1,7 + C4: 1,7
* CNT   2 HDP CHAINS /  14 HYP OPENED
```

```28970;2011_12;GP;23;11.30;1.20;1.20
```

# Solution

position: 981734625746215839325698471567981342214573986893462157152349768638157294479826513 solved

See section Deep 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 # B4: 6,9 => UNS
* INC # C4: 6,9 => UNS
* INC # G5: 5,9 => UNS
* INC # H5: 5,9 => UNS
* INC # D9: 5,9 => UNS
* INC # D9: 2,8 => UNS
* CNT   6 HDP CHAINS /   6 HYP OPENED
```

## A2. Pair Reduction

Full list of HDP chains traversed:

```* INC # B4: 6,9 => UNS
* INC # C4: 6,9 => UNS
* INC # G5: 5,9 => UNS
* INC # H5: 5,9 => UNS
* INC # D9: 5,9 => UNS
* INC # D9: 2,8 => UNS
* CNT   6 HDP CHAINS /   6 HYP OPENED
```

## A3. Deep Pair Reduction

Full list of HDP chains traversed:

```* INC # B4: 6,9 => UNS
* INC # C4: 6,9 => UNS
* INC # G5: 5,9 => UNS
* INC # H5: 5,9 => UNS
* INC # D9: 5,9 => UNS
* INC # D9: 2,8 => UNS
* INC # B4: 6,9 # B6: 6,9 => UNS
* INC # B4: 6,9 # C6: 6,9 => UNS
* INC # B4: 6,9 # B8: 6,9 => UNS
* INC # B4: 6,9 # B8: 3,4,7 => UNS
* INC # B4: 6,9 # E5: 1,7 => UNS
* DIS # B4: 6,9 # E5: 5 => CTR => E5: 1,7
* PRF # B4: 6,9 + E5: 1,7 # C4: 1,7 => SOL
* STA # B4: 6,9 + E5: 1,7 + C4: 1,7
* CNT  13 HDP CHAINS /  14 HYP OPENED
```