The following puzzle (Puzzle #77 from top1465) is currently considered the toughest known Sudoku puzzle:

- Code: Select all
`*-----------------------*`

| 7 . . | . . . | 4 . . |

| . 2 . | . 7 . | . 8 . |

| . . 3 | . . 8 | . . 9 |

|-------+-------+-------|

| . . . | 5 . . | 3 . . |

| . 6 . | . 2 . | . 9 . |

| . . 1 | . . 7 | . . 6 |

|-------+-------+-------|

| . . . | 3 . . | 9 . . |

| . 3 . | . 4 . | . 6 . |

| . . 9 | . . 1 | . . 5 |

*-----------------------*

Due to that fact, this puzzle has a natural interest for all those interested in computer and manual Sudoku solving. So, the purpose of this thread is to collect all logical solutions and other studies regarding Puzzle #77, and I invite here the others users of this forum to do that in this thread.

Having said that, I will begin to make my own contribution to the study of this puzzle by writing in the next paragraphs the solution of it that I have build by my own. The notation that I use in the logical deductions is described here.

After the basic logic is applied to this puzzle we arrive at the following grid:

- Code: Select all
`*---------------------------------------------------------------*`

| 7 1589 568 | 1269 13569 23569 | 4 125 123 |

| 14569 2 456 | 1469 7 34569 | 156 8 13 |

| 1456 145 3 | 1246 156 8 | 12567 1257 9 |

|---------------------+--------------------+--------------------|

| 2489 4789 2478 | 5 1689 469 | 3 1247 12478 |

| 3458 6 4578 | 148 2 34 | 1578 9 1478 |

| 234589 4589 1 | 489 389 7 | 258 245 6 |

|---------------------+--------------------+--------------------|

| 12568 1578 25678 | 3 568 256 | 9 1247 12478 |

| 1258 3 2578 | 2789 4 259 | 1278 6 1278 |

| 2468 478 9 | 2678 68 1 | 278 3 5 |

*---------------------------------------------------------------*

From here, I have solved the puzzle through the following steps:

1. [r3c78](-1-[r3c1245])=6=[r2c7]-6-[r2c3]=6=[r3c1](-6-[r3c4|r7c1])-6-[r3c5](-5-[r79c5]-6,8-[r89c4| r7c6])-5-[r3c2]{(-4-[r6c2])-4-[r9c2]=4=[r9c1]-4-[r5c1]}-4-[r3c4](-2-[r8c4])-2-[r9c4]{-7-[r8c4](-9-[r8c6])-9-[r6c4]}(-7-[r9c7])-7-[r9c2](-8-[r9c7]-2-[r6c7])(-8-[r6c2])-8-[r78c1]=(Almost Unique Rectangle: r78c16)=8|1=[r78c1]-1-[r2c1]=1=[r1c2]-1-[r1c45]=1=[r2c4](-1-[r56c4]-8-[r6c5])=4=[r2c6]-4-[r5c6](-3-[r5c1])-3-[r6c5]-9-[r6c2](-5-[r6c7]-8-[r6c4])-5-[r5c1]-8-[r5c4]-4-[r6c4], => r3c7/r3c8<>1.

2. [r7c6]{-6-[r4c6]=6=[r4c5](-6-[r3c5])=1=[r5c4]-1-[r5c79]}{-6-[r7c13]=6=[r9c1](-6-[r3c125]-4-[r2c3])=4=[r9c2](-4-[r46c2])-4-[r3c2]=4=[r3c1]-4-[r456c1]}(-6-[r7c5])-6-[r9c5](-8-[r9c7]){-8-[r6c5]=8= [r6c4](-8-[r6c7|r6c2])=4=[X-Wing: r45c36]-4-[r45c89]=4=[r6c8]-4-[r7c8]}-8-[r7c5]-5-[r3c5]-1-[r3c2](-5-[r2c3]-6-[r2c7]=6=[r3c7]=7=[r3c8]-7-[r7c8])-5-[r6c2]-9-[r6c5]-3-[r5c6]-4-[r4c6]-9-[r8c6]=9=[r8c4]=7=[r9c4]-7-[r9c7](-2-[r6c7])(-2-[r8c79])-2-[r7c8]-1-[r8c79]=(Almost Unique Rectangle: r58c79)=1|5=[r5c7]-5-[r6c7], => r7c6<>6.

3. [r12c6](-6-[r1c4|r2c4|r3c5])-6-[r4c6]=6=[r4c5](=1=[r5c4]-1-[r1c4|r2c4]){-6-[r9c5](-8-[r7c5|r9c27])-8-[r6c5]=8=[r6c4]-8-[r6c27]}-6-[r7c5](-5-[r3c5]-1-[r2c3|r3c12]-5,6-[r1c3]-8-[r7c3])(-5-[r7c3|r8c6])-5-[r7c6](-2-[r7c3])-2-[r8c6]-9-[r4c6]-4-[r5c6]-3-[r6c5]-9-[r36c2]-4-[r9c2](-7-[r7c3]-6-[r2c3])-7-[r9c7](-2-[r3c7])-2-[r6c7](-5-[r6c8])-5-[r6c2](-4-[r6c8]-2-[r13c8]=2=[r1c9]-2-[r1c4])-4-[r3c2]-5-[r2c3]-4-[r2c4]-9-[r1c4], => r1c6/r2c6<>6.

4. [r5c1]=3=[r5c6]=4=[r2c6]-4-[r3c4]=4=[X-Wing: r39c12]-4-[r5c1], => r5c1<>4.

5. [r6c5](=3=[r1c5]-3-[r1c9])(=3=[r5c6]=4=[r2c6])(-8-[r6c247|r47c5])-8-[r9c5](-6-[r3c5])-6-[r7c5](-5-[r78c6]=5=[r1c6]-5-[r1c89]-1-[r1c2])-5-[r3c5]{-1-[r2c3|r3c12](-5-[r1c2])-5,6-[r1c3]-8-[r1c2]}-1-[r4c5]-9-[r6c4](-4-[r6c2])-4-[r6c78]-5-[r6c2]-9-[r1c2], => r6c5<>8.

6. [r2c1]{(-4-[r46c1])-4-[r9c1]=4=[r9c2]-4-[r46c2]}(-4-[r2c34])-4-[r3c12]=4=[r3c4]{=2=[r3c78](-2-[r1c89|r2c9]-5-[r1c6])=6=[r2c7](-6-[r2c4])-6-[r2c3](-5-[r2c6|r5c3])=6=[r3c1]-6-[r79c1]=6=[r7c3]}-4-[r2c6]=4=[r5c6](-4-[r5c9])(-4-[r5c3]=4=[r4c3]-4-[r4c89]=4=[r6c8]-4-[r7c8]=4=[r7c9])-4-[r256c4]-9-[r8c4]=9=[r8c6](-9-[r1c6])-9-[r2c6](-3-[r1c6]-2-[r7c6])-3-[r2c9]-1-[r2c4]-9-[r6c4](-8-[r6c7])-8-[r5c4](-1-[r5c7])-1-[r5c39]-7,8-[r5c7]-5-[r6c7](-2-[r9c7])-2-[r3c7](=2=[r3c8]-2-[r7c8])-7-[r9c7]-8-[r9c1]-2-[r7c1]=2=[r7c8], => r2c1<>4.

7. [r2c6](-9-[r2c1]=9=[r1c2]-9-[r6c2])(-9-[r8c6]=9=[r8c4]=7=[r9c4]-7-[r9c27])(=3=[r2c9]-3-[r1c9]){=4= [r5c6](-4-[r5c3])=3=[r6c5]}-9-[r1c5]=9=[r4c5]=1=[r5c4](-1-[r123c4])=8=[r6c4]-8-[r6c278]-2-[r4c89]-{Nice Loop: [r6c7]-5-[r6c2]-4-[r9c2]-8-[r9c7]-2-[r6c7]}-{(-8-[r9c1])-8-[r9c5]-6-[r9c1]}-(-5-[r6c8] =5=[r13c8]-5-[r2c7]=5=[r2c13]-5-[r3c2])-4-[r3c2]-1-[r3c5]=1=[r1c5]-1-[r1c9]-2-[r1c46]=2=[r3c4] =4=[r3c1](-4-[r46c1])-4-[r9c1](-2-[r4c1])=4=[r9c2]-4-[r46c2]=4=[r4c3]=2=[r4c1], => r2c6<>9.

8. [r5c4]{=1=[r4c5](-1-[r3c5])-1-[r4c89]}{(-4-[r5c3])-4-[r3c4]=4=[X-Wing: r39c12]-4-[r46c12]=4=[r4c3]-4-[r4c8]}(-4-[r6c4])-4-[r5c6]-3-[r6c5](-9-[r6c2])-9-[r6c4](-8-[r6c7])-8-[r6c2](-5-[r6c78]=5=[r5c7]=1=[r5c9]-1-[r12c9]-2-[r3c8])-5-[r6c7](-2-[r9c7])-2-[r4c8]-7-[r3c8](-5-[r3c5])=7=[r3c7]-7-[r9c7]-8-[r9c5]-6-[r3c5], => r5c4<>4.

9. [r6c1]{-9-[r2c1](=9=[r2c4]-9-[r1c6]=9=[r8c6]=5=[r7c56]-5-[r7c2])=9=[r1c2](-9-[r1c5])=8=[r1c3]-8-[r45c3]}(-9-[r6c24])-9-[r6c5](-3-[r1c5])-3-[r5c6](=3=[r5c1]){-4-[r6c4](-8-[r6c2]=8=[r4c12]-8-[r4c9])-8-[r6c278]-2-[r4c9]}-4-[r2c6]=4=[r2c3](=6=[r7c3]-6-[r7c5])(=6=[X-Wing: r23c17]-6-[r3c5])-4-[r3c12]=4=[r3c4](=2=[X-Wing: r36c78]-2-[r9c7])=2=[r3c78]-2-[r1c89|r2c9](-1-[r4c9])-5-[r1c5]-{Nice Loop: [r6c2]=5=[r3c2]-5-[r3c5]-1-[r1c5]-6-[r9c5](-8-[r9c2])-8-[r9c7]-7-[r9c2]-4-[r6c2]}-4-[r6c2]{-5-[r3c2](-1-[r7c2])-1-[r3c5]-5-[r7c5]-8-[r7c2]}(-5-[r6c8])-5-[r6c7](-2-[r3c7]-7-[r89c7])-2-[r6c8]-4-[r4c9]-7-[r78c9]=7=[r7c8]-7-[r7c2], => r6c1<>9.

10. [r2c1]{(=9=[r4c1]-9-[r4c5])=9=[r2c4](-9-[r1c6])-9-[r6c4]=9=[r6c5]=3=[r1c5]-3-[r1c69]}(-6-[r2c3|r3c1])(-6-[r2c7]=6=[r3c7]-6-[r3c5])-6-[r1c89|r2c7]-2-[r1c6]-5-[r3c5]-1-[r3c12]-4,5-[r2c3], => r2c1<>6.

11. [r2c1]{(=9=[r4c1]-9-[r4c5])=9=[r2c4](-9-[r1c6])-9-[r6c4]=9=[r6c5]=3=[r5c6](-3-[r1c6])=4=[r2c6](-4-[r2c3])=3=[r2c9]-3-[r1c9]}(-5-[r2c3|r3c12]-1-[r3c5])(-5-[r2c7])-5-[r2c3]-6-[r2c7]=6=[r3c7]-6-[r3c5]-5-[r1c6]-2-[r1c9]-1-[r2c7], => r2c1<>5.

12. [r1c5](-6-[r2c4|r3c5]){=3=[r6c5](-3-[r6c1])(-3-[r5c6]-4-[r5c3])=9=[r4c5](=1=[r5c4]=8= [r6c4]-8-[r6c1])-9-[r4c12]=9=[r6c2]-9-[r1c2]=9=[r2c1]-9-[r2c4]}-6-[r79c5]=6=[r9c4](-6-[r9c1])=7=[r8c4]=9=[r8c6]=2=[r7c6](-2-[r7c3])=5=[r7c5](-5-[r7c3])(=8=[r9c5]-8-[r9c1])-5-[r3c5](-1-[r2c4]-4-[r2c3]=4=[X-Wing: r39c12]-4-[r6c1])-1-[r2c3|r3c12](-5,6-[r1c3]-8-[r57c3])-5-[r1c2]=5=[r3c2](=4=[r3c1]-4-[r9c1]-2-[r6c1])-5-[r2c3]-6-[r7c3]-7-[r5c3]-5-[r6c1], => r1c5<>6.

13. [r1c3](-5-[r1c8|r5c3])(=8=[r1c2]-8-[r6c2])(-5-[r2c3|r3c12]-1-[r3c5])=6=[r1c4]-6-[r3c5]-5-[r3c78]=5=[r2c7]-5-[r56c7]=5=[r6c8]-5-[r6c2|r6c1]=5=[r5c1]=3=[r5c6](-3-[r6c5]-9-[r6c2])=4=[r6c4]-4-[r6c2], => r1c3<>5.

14. [r5c1](-8-[r789c1|r46c2|r5c379])(-8-[r5c4]-1-[r5c79])=3=[r6c1]-3-[r6c5](-9-[r6c2|r1c5])=3=[r5c6](-3-[r1c6])=4=[r6c4](-4-[r6c2]-5-[r5c3])(-4-[r3c4]=4=[X-Wing: r39c12]-4-[r4c2])=8=[r4c5](-8-[r4c9]=8=[r6c7]=2=[r6c8]-2-[r1c8])-8-[r79c5](-5-[r78c6]-2,9-[r1c6]-5-[r1c8]-1-[r4c8]=1=[r4c9])-5,6-[r3c5]-1-[r2c3|r3c12]-6-[r1c3](-8-[r78c3])=6=[r1c4]=9=[r2c4]-9-[r2c1](-1-[r78c1]=1=[r7c2] =8=[r7c9])=9=[r1c2]-9-[r4c2]-7-[r5c3]-4-[r5c9]=4=[r4c8]-4-[r7c8]=4=[r7c9], => r5c1<>8.

15. [r1c5](-5-[r1c89|r2c9]-2-[r1c46]=2=[r3c4]=4=[X-Wing: r39c12]-4-[r46c12]){(-5-[r3c5])-5-[r79c5]-6-[r3c5]-1-[r2c3|r3c12](-5-[r1c2])-6-[r1c3](=6=[r1c4])-8-[r5c3]}=3=[r6c5](=9=[r4c5]-9-[r4c12]=9=[r6c2]-9-[r1c2]=9=[r1c6]-9-[r8c6]=9=[r8c4]=7=[r9c4]-7-[r9c2])-3-[r5c6](-4-[r5c9])-4-[r5c3]=4=[r4c3]-4-[r4c89]=4=[r6c8]=5=[r3c8](-5-[r3c12]=5=[r2c3]-5-[r5c3])-5-[r3c2]-4-[r9c2]-8-[r4c2]-7-[r5c3], => r1c5<>5.

16. [r3c7](-5-[r3c12])(-5-[r56c7]=5=[r6c8]-5-[r6c12])(=6=[r2c7]-6-[r2c3])-5-[r3c5]=5=[r7c5]-5-[r7c123]=5=[X-Wing: r58c13]-5-[r2c3]=5=[r1c2]=8=[r1c3]=6=[r1c4]-6-[r3c5]-1-[r3c2]-4-[r2c3], => r3c7<>5.

17. [r1c6]{(-2-[r1c8])-2-[r12c9]-1-[r1c8]-5-[r1c2]}-2-[r7c6](-5-[r79c5]-8-[r4c5])(-5-[r7c123])-5-[r7c5]=5=[r3c5]-5-[r3c12]=5=[r2c3](-5-[r5c3])-5-[r8c3]=5=[r8c1]-5-[r56c1]=5=[r6c2]=9=[r6c45]-9-[r4c5]-1-[r4c89]=1=[X-Wing: r58c79]-1-[r2c7|r12c9]=1=[r1c8], => r1c6<>2.

18. [r3c4]=2=[r1c4]=6=[r1c3]-6-[r2c3|r3c12]-1-[r3c4], => r3c4<>1.

19. [r46c12](-4-[r3c1/2])-4-[r9c1/2]=4=[r9c2/1]-4-[r3c2/1]=4=[r3c4]=2=[r1c4]{(-2-[r1c8])-2-[r12c9](-1-[r458c9])(-1-[r2c7])-1-[r1c8]-5-[r1c6]}=6=[r1c3](=8=[r1c2]=9=[r2c1])-6-[r3c1|r3c2]-5-[r3c5](=5=[r7c5]-5-[r7c123])=5=[r2c6]=3=[r2c9]=1=[r2c4]-1-[r5c4]=1=[r4c5]-1-[r4c8]=1=[r5c7](=5= [r6c78]-5-[r6c12]=5=[X-Wing: r58c13]-5-[r3c1])-1-[r8c7]=1=[r8c1]-1-[r3c1], => r4c1/r4c2/r6c1/r6c2<>4.

20. [r3c1]{(-5-[r5678c1])-5-[r3c5]=5=[r7c5]-5-[r7c23]=5=[r8c3]-5-[r5c3]=5=[r6c2]-5-[r6c7]}(=4=[r9c1] =2=[r9c7]-2-[r6c7]-8-[r6c124])-5-[r2c3]-6-[r1c3](-8-[r5c3])=6=[r1c4]-6-[r3c5]-1-[r4c5]=1=[r5c4]=8=[r4c5]-8-[r4c123]=8=[r5c3], => r3c1<>5.

21. [r3c8]{=7=[r3c7](-7-[r89c7])=6=[r2c7]-6-[r2c3]-5-[r578c3]}-5-[r3c5]=5=[r7c5]-5-[r7c12]=5=[r8c1] (-5-[r6c1])-5-[r5c1](-3-[r6c1])(=5=[r6c2]-5-[r69c7]-2,8-[r8c7]-1-[r5c7])-3-[r5c6]{(-4-[r5c3])-4-[r6c4]=4=[r6c8]}-4-[r2c6]=4=[r2c4]=9=[r2c1](=1=[r2c9])-9-[r46c1](-8-[r5c3]-7-[r5c79])-2-[r9c1]=2=[r9c7]-2-[r6c7]-(Hidden Pair: r4c89 – “2,7”)-1-[r4c89]=1=[r5c9]-1-[r2c9], => r3c8<>5.

22. [r2c6]{=4=[r2c4]=9=[r2c1](-9-[r1c2])-9-[r4c1]}{=4=[r5c6](=3=[r1c6])-4-[r6c4]=4=[r6c8] =2=[X-Wing: r69c17]-2-[r4c1]}(-5-[r3c5])-5-[r2c3]-6-[r1c3]=6=[r1c4](-6-[r3c5]-1-[r4c5])=9=[r1c5]-9-[r4c5]-8-[r4c1], => r2c6<>5.

23. [r5c3]-5-[r2c3]=5=[r2c7]-5-[r56c7]=5=[r6c8]=4=[r6c4]-4-[r5c6]-3-[r5c1]-5-[r5c3], => r5c3<>5.

24. [r6c4]=4=[r6c8]=5=[r1c8]-5-[r1c6]-9-[r8c6]=9=[r8c4]-9-[r6c4], => r6c4<>9.

25. [r6c2]=9=[r6c5]=3=[r5c6]-3-[r5c1]-5-[r6c2], => r6c2<>5.

26. [r1c4]=6=[r1c3]=8=[r1c2]-8-[r6c2]-9-[r6c5]-3-[r1c5](-1-[r1c4])-1-[r3c5]-5-[r1c6]-9-[r1c4], => r1c4<>1,9.

27. [r5c1]=5=[r5c7](=1=[r8c7]-1-[r8c1])(-5-[r6c7])-5-[r6c8]=5=[r1c8]-5-[r1c6]-9-[r1c2]=9=[r2c1]-9-[r48c1]-2-[r9c1]=2=[r9c7]-2-[r6c7]-8-[r6c4]-4-[r5c6]-3-[r5c1],

which implies r5c1<>3 and that solve the puzzle.

Carcul