The New Sudoku Players' Forum

[Guest]

Hi,

was around on the forum a couple of weeks ago on the Times "Very Hard" puzzle that evolved into a T&E debate and the hunt for the ledgendary X-wing - liked the way you got to it and so obvious when you see it !

Anyway, decided to have a look at the official PC program over the weekend and am enjoying the challenge of the "Hard" category. Got stuck on the one below and have only come up with one way of getting any further and am not sure if it's "Nishio", T&E or a logical solution (it involves 4s and 6s). Or, more probably, I'm missing something obvious !!!

anyone got any ideas ?

*21 | *** | 564
*** | **5 | *21
*** | 2** | **8
------------------
13* | 592 | ***
64* | *** | **2
**2 | 643 | *1*
------------------
7** | *29 | 1**
2** | 15* | ***
514 | *** | 2**

cheers,

Jim

[Guest]

Not sure if this counts as obvious, but...

In col 7, there are 3 cells (r7c2, r7c3 and r7c5) which between them only have 3 possible digits (3,7 & 9).

This means that 3, 7 & 9 have to go in those cells in some combination (we don't know which way round at this stage) and so cannot appear elsewhere in that row.

This tells you the value of r7c6.

[Guest]

yeah, I had a similar line of thought over the 4s & 6s in col 7 in that they can only go in r4 and r8 leading to the value in r6c7, but wasn't sure if I wasn't using nishio - but looking at it again and re-reading the Times thread, I think this is not nishio and therefore logic !

thanks for your thoughts,

Jim

[shakers]

Anyway, decided to have a look at the official PC program over the weekend and am enjoying the challenge of the "Hard" category. Got stuck on the one below and have only come up with one way of getting any further and am not sure if it's "Nishio", T&E or a logical solution (it involves 4s and 6s). Or, more probably, I'm missing something obvious !!!

I think Wayne said in the X-Wing/Nishio discussion that his program would solve using techniques such as the Nishio, but would not generate puzzles that required their use to be solved. There was a reference to the Help file which discusses this grey area of logic/T&E.

Even if the program did generate them, I would only have expected to see them in the Very Hard puzzles, not the standard Hard ones.

[Guest]

Apology first - just about every cell reference I gave was wrong - hope hope this didn't cause too much confusion! I was talking about column 7, rows 2,3 & 5, leading to the value of r6c7. That might help make sense of what I was saying! It's a straightforward elimination technique.

I see what you mean with the 4 & 6s, I think - r4c7 & r8c7 are the only places that 4 and 6 can live, but does this lead to a definite digit placement? I can't see how - what did you do next?

[shakers]

I see what you mean with the 4 & 6s, I think - r4c7 & r8c7 are the only places that 4 and 6 can live, but does this lead to a definite digit placement? I can't see how - what did you do next?

R6C7 has to be 8 as it's the only possibly place for an 8 in the column now. From there it's plain sailing.

[Animator]

Apology first - just about every cell reference I gave was wrong - hope hope this didn't cause too much confusion!

Then maybe you should register yourself?

This would enable two things:
a) you don't need to type your name with every post
b) you can edit your posts, which gives you the chance to fix your mistakes

[Guest]

I assume that you can get from

Code: Select all

   2   1            5   6   4
               5      2   1
         2               8
1   3      5   9   2         
6   4                     2
      2   6   4   3      1   
7            2   9   1     
2         1   5           
5   1   4            2     


by easy (n=1) reductions to

Code: Select all

389   2   1   3789   378   78   5   6   4
3489   6789   6789   3489   368   5   379   2   1
349   5679   5679   2   136   146   379   379   8
1   3   78   5   9   2   4678   478   67
6   4   59   78   178   178   39   359   2
89   5789   2   6   4   3   789   1   579
7   68   368   348   2   9   1   458   356
2   689   3689   1   5   4678   4678   478   367
5   1   4   378   3678   678   2   789   3679


You now need to open Milo's rules up to n>1 as follows
Rule 2: in column 7 possible 379 exists in exactly 3 rows {2,7}=379 {3,7}=379 {5,7}=39 so eliminate 7 from cell {4,7} old=4678 new=468
Rule 2: in column 7 possible 379 exists in exactly 3 rows {2,7}=379 {3,7}=379 {5,7}=39 so eliminate 7 from cell {6,7} old=789 new=89
Rule 2: in column 7 possible 379 exists in exactly 3 rows {2,7}=379 {3,7}=379 {5,7}=39 so eliminate 9 from cell {6,7} old=89 new=8
Rule 2: in column 7 possible 379 exists in exactly 3 rows {2,7}=379 {3,7}=379 {5,7}=39 so eliminate 7 from cell {8,7} old=4678 new=468
Rule 2: in column 7 possible 3798 exists in exactly 4 rows {2,7}=379 {3,7}=379 {5,7}=39 {6,7}=8 so eliminate 8 from cell {4,7} old=468 new=46
Rule 2: in column 7 possible 3798 exists in exactly 4 rows {2,7}=379 {3,7}=379 {5,7}=39 {6,7}=8 so eliminate 8 from cell {8,7} old=468 new=46
Rule 2: in box 6 possible 4678 exists in exactly 4 cells {4,7}=46 {4,8}=478 {4,9}=67 {6,7}=8 so eliminate 7 from cell {6,9} old=579 new=59
Rule 2: in box 6 possible 8395 exists in exactly 4 cells {5,7}=39 {5,8}=359 {6,7}=8 {6,9}=59 so eliminate 8 from cell {4,8} old=478 new=47
Rule 3: in row 6 possible 57 exists in exactly 2 columns{6,2}=5789 {6,9}=59 so reduce cell {6,2} old=5789 new=57
Rule 3: in row 6 possible 57 exists in exactly 2 columns{6,2}=5789 {6,9}=59 so reduce cell {6,9} old=59 new=5
Rule 3: in box 6 possible 39 exists in exactly 2 cells {5,7}=39 {5,8}=359 so reduce cell {5,8} old=359 new=39
Rule 3: in row 6 possible 897 exists in exactly 3 columns{6,1}=89 {6,2}=57 {6,7}=8 so reduce cell {6,2} old=57 new=7
Rule 3: in row 6 possible 957 exists in exactly 3 columns{6,1}=89 {6,2}=7 {6,9}=5 so reduce cell {6,1} old=89 new=9
Rule 3: in column 1 possible 384 exists in exactly 3 rows {1,1}=389 {2,1}=3489 {3,1}=349 so reduce cell {1,1} old=389 new=38
Rule 3: in column 1 possible 384 exists in exactly 3 rows {1,1}=389 {2,1}=3489 {3,1}=349 so reduce cell {2,1} old=3489 new=348
Rule 3: in column 1 possible 384 exists in exactly 3 rows {1,1}=389 {2,1}=3489 {3,1}=349 so reduce cell {3,1} old=349 new=34
Rule 3: in box 4 possible 785 exists in exactly 3 cells {4,3}=78 {5,3}=59 {6,2}=7 so reduce cell {5,3} old=59 new=5
Rule 3: in box 4 possible 859 exists in exactly 3 cells {4,3}=78 {5,3}=5 {6,1}=9 so reduce cell {4,3} old=78 new=8
Rule 3: in column 2 possible 6895 exists in exactly 4 rows {2,2}=6789 {3,2}=5679 {7,2}=68 {8,2}=689 so reduce cell {2,2} old=6789 new=689
Rule 3: in column 2 possible 6895 exists in exactly 4 rows {2,2}=6789 {3,2}=5679 {7,2}=68 {8,2}=689 so reduce cell {3,2} old=5679 new=569
Rule 3: in column 3 possible 6793 exists in exactly 4 rows {2,3}=6789 {3,3}=5679 {7,3}=368 {8,3}=3689 so reduce cell {2,3} old=6789 new=679
Rule 3: in column 3 possible 6793 exists in exactly 4 rows {2,3}=6789 {3,3}=5679 {7,3}=368 {8,3}=3689 so reduce cell {3,3} old=5679 new=679
Rule 3: in column 3 possible 6793 exists in exactly 4 rows {2,3}=6789 {3,3}=5679 {7,3}=368 {8,3}=3689 so reduce cell {7,3} old=368 new=36
Rule 3: in column 3 possible 6793 exists in exactly 4 rows {2,3}=6789 {3,3}=5679 {7,3}=368 {8,3}=3689 so reduce cell {8,3} old=3689 new=369
Rule 3: in column 9 possible 6739 exists in exactly 4 rows {4,9}=67 {7,9}=356 {8,9}=367 {9,9}=3679 so reduce cell {7,9} old=356 new=36
Rule 3: in box 1 possible 6975 exists in exactly 4 cells {2,2}=689 {2,3}=679 {3,2}=569 {3,3}=679 so reduce cell {2,2} old=689 new=69
Rule 1: in column 7 possible 7 exists only in box 3 so eliminate it from cell {3,8} old=379 new=39
Rule 1: in row 1 possible 9 exists only in box 2 so eliminate it from cell {2,4} old=3489 new=348
Rule 1: in column 9 possible 9 exists only in box 9 so eliminate it from cell {9,8} old=789 new=78
Rule 2: in row 7 possible 63 exists in exactly 2 columns {7,3}=36 {7,9}=36 so eliminate 6 from cell {7,2} old=68 new=8
Rule 2: in row 7 possible 63 exists in exactly 2 columns {7,3}=36 {7,9}=36 so eliminate 3 from cell {7,4} old=348 new=48
Rule 2: in row 7 possible 84 exists in exactly 2 columns {7,2}=8 {7,4}=48 so eliminate 8 from cell {7,8} old=458 new=45
Rule 2: in row 7 possible 84 exists in exactly 2 columns {7,2}=8 {7,4}=48 so eliminate 4 from cell {7,8} old=45 new=5
Rule 2: in row 7 possible 85 exists in exactly 2 columns {7,2}=8 {7,8}=5 so eliminate 8 from cell {7,4} old=48 new=4
Rule 2: in row 1 possible 387 exists in exactly 3 columns {1,1}=38 {1,5}=378 {1,6}=78 so eliminate 3 from cell {1,4} old=3789 new=789
Rule 2: in row 1 possible 387 exists in exactly 3 columns {1,1}=38 {1,5}=378 {1,6}=78 so eliminate 8 from cell {1,4} old=789 new=79
Rule 2: in row 1 possible 387 exists in exactly 3 columns {1,1}=38 {1,5}=378 {1,6}=78 so eliminate 7 from cell {1,4} old=79 new=9
Rule 2: in column 2 possible 698 exists in exactly 3 rows {2,2}=69 {7,2}=8 {8,2}=689 so eliminate 6 from cell {3,2} old=569 new=59
Rule 2: in column 2 possible 698 exists in exactly 3 rows {2,2}=69 {7,2}=8 {8,2}=689 so eliminate 9 from cell {3,2} old=59 new=5
Rule 2: in column 9 possible 673 exists in exactly 3 rows {4,9}=67 {7,9}=36 {8,9}=367 so eliminate 6 from cell {9,9} old=3679 new=379
Rule 2: in column 9 possible 673 exists in exactly 3 rows {4,9}=67 {7,9}=36 {8,9}=367 so eliminate 7 from cell {9,9} old=379 new=39
Rule 2: in column 9 possible 673 exists in exactly 3 rows {4,9}=67 {7,9}=36 {8,9}=367 so eliminate 3 from cell {9,9} old=39 new=9
Rule 3: in column 2 possible 69 exists in exactly 2 rows {2,2}=69 {8,2}=689 so reduce cell {8,2} old=689 new=69
Rule 3: in column 4 possible 387 exists in exactly 3 rows {2,4}=348 {5,4}=78 {9,4}=378 so reduce cell {2,4} old=348 new=38
Rule 3: in box 2 possible 614 exists in exactly 3 cells {2,5}=368 {3,5}=136 {3,6}=146 so reduce cell {2,5} old=368 new=6
Rule 3: in box 2 possible 614 exists in exactly 3 cells {2,5}=368 {3,5}=136 {3,6}=146 so reduce cell {3,5} old=136 new=16
Rule 3: in row 2 possible 3487 exists in exactly 4 columns{2,1}=348 {2,3}=679 {2,4}=38 {2,7}=379 so reduce cell {2,3} old=679 new=7
Rule 3: in row 2 possible 3487 exists in exactly 4 columns{2,1}=348 {2,3}=679 {2,4}=38 {2,7}=379 so reduce cell {2,7} old=379 new=37
Rule 3: in row 2 possible 4869 exists in exactly 4 columns{2,1}=348 {2,2}=69 {2,4}=38 {2,5}=6 so reduce cell {2,1} old=348 new=48
Rule 3: in row 2 possible 4869 exists in exactly 4 columns{2,1}=348 {2,2}=69 {2,4}=38 {2,5}=6 so reduce cell {2,4} old=38 new=8
Rule 3: in column 5 possible 3781 exists in exactly 4 rows {1,5}=378 {3,5}=16 {5,5}=178 {9,5}=3678 so reduce cell {3,5} old=16 new=1
Rule 3: in column 5 possible 3781 exists in exactly 4 rows {1,5}=378 {3,5}=16 {5,5}=178 {9,5}=3678 so reduce cell {9,5} old=3678 new=378
Rule 3: in column 5 possible 3786 exists in exactly 4 rows {1,5}=378 {2,5}=6 {5,5}=178 {9,5}=378 so reduce cell {5,5} old=178 new=78
Rule 3: in box 2 possible 3784 exists in exactly 4 cells {1,5}=378 {1,6}=78 {2,4}=8 {3,6}=146 so reduce cell {3,6} old=146 new=4
Rule 3: in box 2 possible 3714 exists in exactly 4 cells {1,5}=378 {1,6}=78 {3,5}=1 {3,6}=4 so reduce cell {1,5} old=378 new=37
Rule 3: in box 2 possible 3714 exists in exactly 4 cells {1,5}=378 {1,6}=78 {3,5}=1 {3,6}=4 so reduce cell {1,6} old=78 new=7
Rule 3: in box 2 possible 3814 exists in exactly 4 cells {1,5}=37 {2,4}=8 {3,5}=1 {3,6}=4 so reduce cell {1,5} old=37 new=3
Rule 3: in box 8 possible 6783 exists in exactly 4 cells {8,6}=4678 {9,4}=378 {9,5}=378 {9,6}=678 so reduce cell {8,6} old=4678 new=678
Rule 3: in row 2 possible 48693 exists in exactly 5 columns{2,1}=48 {2,2}=69 {2,4}=8 {2,5}=6 {2,7}=37 so reduce cell {2,7} old=37 new=3
Rule 3: in row 2 possible 48937 exists in exactly 5 columns{2,1}=48 {2,2}=69 {2,3}=7 {2,4}=8 {2,7}=3 so reduce cell {2,2} old=69 new=9
Rule 3: in row 2 possible 46937 exists in exactly 5 columns{2,1}=48 {2,2}=9 {2,3}=7 {2,5}=6 {2,7}=3 so reduce cell {2,1} old=48 new=4
Rule 3: in box 1 possible 38467 exists in exactly 5 cells {1,1}=38 {2,1}=4 {2,3}=7 {3,1}=34 {3,3}=679 so reduce cell {3,3} old=679 new=67
Rule 3: in box 1 possible 38469 exists in exactly 5 cells {1,1}=38 {2,1}=4 {2,2}=9 {3,1}=34 {3,3}=67 so reduce cell {3,3} old=67 new=6
Rule 3: in box 1 possible 38679 exists in exactly 5 cells {1,1}=38 {2,2}=9 {2,3}=7 {3,1}=34 {3,3}=6 so reduce cell {3,1} old=34 new=3
Rule 3: in box 1 possible 84679 exists in exactly 5 cells {1,1}=38 {2,1}=4 {2,2}=9 {2,3}=7 {3,3}=6 so reduce cell {1,1} old=38 new=8
Rule 1: in row 2 possible 3 exists only in box 3 so eliminate it from cell {3,7} old=379 new=79
Rule 1: in row 2 possible 3 exists only in box 3 so eliminate it from cell {3,8} old=39 new=9
Rule 1: in column 4 possible 3 exists only in box 8 so eliminate it from cell {9,5} old=378 new=78
Rule 1: in column 8 possible 3 exists only in box 6 so eliminate it from cell {5,7} old=39 new=9
Rule 1: in row 9 possible 6 exists only in box 8 so eliminate it from cell {8,6} old=678 new=78
Rule 1: in column 2 possible 6 exists only in box 7 so eliminate it from cell {7,3} old=36 new=3
Rule 1: in column 2 possible 6 exists only in box 7 so eliminate it from cell {8,3} old=369 new=39
Rule 1: in box 7 possible 6 exists only in row 8 so eliminate it from cell {8,7} old=46 new=4
Rule 1: in box 7 possible 6 exists only in row 8 so eliminate it from cell {8,9} old=367 new=37
Rule 1: in box 9 possible 6 exists only in column 9 so eliminate it from cell {4,9} old=67 new=7
Rule 1: in box 2 possible 7 exists only in column 6 so eliminate it from cell {5,6} old=178 new=18
Rule 1: in box 2 possible 7 exists only in column 6 so eliminate it from cell {8,6} old=78 new=8
Rule 1: in box 2 possible 7 exists only in column 6 so eliminate it from cell {9,6} old=678 new=68
Rule 1: in box 8 possible 7 exists only in row 9 so eliminate it from cell {9,8} old=78 new=8
Rule 1: in box 2 possible 8 exists only in column 4 so eliminate it from cell {5,4} old=78 new=7
Rule 1: in box 2 possible 8 exists only in column 4 so eliminate it from cell {9,4} old=378 new=37
Rule 1: in column 3 possible 9 exists only in box 7 so eliminate it from cell {8,2} old=69 new=6

The final few reductions are simple and obvious
The solution emerges as

Code: Select all

8   2   1   9   3   7   5   6   4
4   9   7   8   6   5   3   2   1
3   5   6   2   1   4   7   9   8
1   3   8   5   9   2   6   4   7
6   4   5   7   8   1   9   3   2
9   7   2   6   4   3   8   1   5
7   8   3   4   2   9   1   5   6
2   6   9   1   5   8   4   7   3
5   1   4   3   7   6   2   8   9


[/code]

[Tony Williams]

anyone got any ideas ?

*21 | *** | 564
*** | **5 | *21
*** | 2** | **8
------------------
13* | 592 | ***
64* | *** | **2
**2 | 643 | *1*
------------------
7** | *29 | 1**
2** | 15* | ***
514 | *** | 2**

cheers,

Jim

Hi Jim,
You have a solution in a later post using Milo's rule for N-Cells contain N-Digits - haven't completed my coding of this yet, but can be solved as below.

The puzzle is hard in the sense that the first 3 iterations only exercised rules to exclude options (as Milo's rule does) and only in Iteartion 4 do we actually start placing digits into the Puzzle.

Code: Select all

 - - - - - - - - - - - - - - -
Iteration Number  1
 - - - - - - - - - - - - - - -
Testing for Unique Placements
Test all Digits for unique to Row/Col/Sq
Testing only Row/col in Square
Digit 7 occurs only in Row 5 - Square 5
Digit  7 not allowed Cell( 5, 3)
Digit  7 not allowed Cell( 5, 7)
Digit  7 not allowed Cell( 5, 8)
Digit 8 occurs only in Row 5 - Square 5
Digit  8 not allowed Cell( 5, 3)
Digit  8 not allowed Cell( 5, 7)
Digit  8 not allowed Cell( 5, 8)
Digit 3 occurs only in Column 3 - Square 7
Digit  3 not allowed Cell( 2, 3)
Digit  3 not allowed Cell( 3, 3)
Digit 9 occurs only in Row 8 - Square 7
Digit  9 not allowed Cell( 8, 7)
Digit  9 not allowed Cell( 8, 8)
Digit  9 not allowed Cell( 8, 9)
 - - - - - - - - - - - - - - -
Iteration Number  2
 - - - - - - - - - - - - - - -
Testing for Unique Placements
Test all Digits for unique to Row/Col/Sq
Testing only Row/col in Square
Testing for Pairs/Triplets [Rows]
Row  1 - Digit 7 is a Pair in the same Square ~ Columns =  4, 5
Digit  7 not allowed Cell( 2, 4)
Digit  7 not allowed Cell( 2, 5)
Digit  7 not allowed Cell( 3, 5)
Digit  7 not allowed Cell( 3, 6)
Testing for Pairs/Triplets [Columns]
Col  9 - Digit 3 is a Triplet in the same Square ~ Rows =  7, 8, 9
Digit  3 not allowed Cell( 8, 7)
Digit  3 not allowed Cell( 7, 8)
Digit  3 not allowed Cell( 8, 8)
Digit  3 not allowed Cell( 9, 8)
 - - - - - - - - - - - - - - -
Iteration Number  3
 - - - - - - - - - - - - - - -
Testing for Unique Placements
Test all Digits for unique to Row/Col/Sq
Testing only Row/col in Square
Testing for Pairs/Triplets [Rows]
Testing for Pairs/Triplets [Columns]
Testing for x,y Pairs of Digits [Rows]
Testing for x,y Pairs of Digits [Columns]
Col  7 - Digits   4 & 6 are a Pair into Rows =  48
Digit  7 not allowed Cell( 4, 7)
Digit  8 not allowed Cell( 4, 7)
Digit  7 not allowed Cell( 8, 7)
Digit  8 not allowed Cell( 8, 7)
 - - - - - - - - - - - - - - -
Iteration Number  4
 - - - - - - - - - - - - - - -
Testing for Unique Placements
Test all Digits for unique to Row/Col/Sq
 35 - Digit  8 Only position in Column( 7) is Cell( 6, 7)
 - - - - - - - - - - - - - - -
Iteration Number  5
 - - - - - - - - - - - - - - -
Testing for Unique Placements
 36 - Digit  9 only candidate for Cell( 6, 1)
 37 - Digit  5 only candidate for Cell( 5, 3)
 38 - Digit  7 only candidate for Cell( 6, 2)
 39 - Digit  5 only candidate for Cell( 6, 9)
 40 - Digit  8 only candidate for Cell( 4, 3)
Test all Digits for unique to Row/Col/Sq
 41 - Cell( 3, 2) = 5
 42 - Cell( 7, 8) = 5
All Digit =  5 allocated
 43 - Digit  9 Only position in Row( 1) is Cell( 1, 4)
 44 - Digit  9 Only position in Column( 9) is Cell( 9, 9)
 - - - - - - - - - - - - - - -
Iteration Number  6
 - - - - - - - - - - - - - - -
Testing for Unique Placements
Test all Digits for unique to Row/Col/Sq
 45 - Digit  4 Only position in Row( 7) is Cell( 7, 4)
 46 - Digit  8 Only position in Row( 7) is Cell( 7, 2)
 - - - - - - - - - - - - - - -
Iteration Number  7
 - - - - - - - - - - - - - - -
Testing for Unique Placements
Test all Digits for unique to Row/Col/Sq
 47 - Digit  4 Only position in Row( 2) is Cell( 2, 1)
 48 - Cell( 3, 6) = 4
 49 - Cell( 1, 1) = 8
 - - - - - - - - - - - - - - -
Iteration Number  8
 - - - - - - - - - - - - - - -
Testing for Unique Placements
 50 - Digit  7 only candidate for Cell( 1, 6)
 51 - Digit  3 only candidate for Cell( 3, 1)
 52 - Digit  3 only candidate for Cell( 1, 5)
 53 - Digit  8 only candidate for Cell( 2, 4)
 54 - Digit  6 only candidate for Cell( 2, 5)
 55 - Digit  1 only candidate for Cell( 3, 5)
 56 - Digit  7 only candidate for Cell( 5, 4)
 57 - Digit  8 only candidate for Cell( 5, 5)
 58 - Digit  1 only candidate for Cell( 5, 6)
All Digit =  1 allocated
 59 - Digit  3 only candidate for Cell( 9, 4)
 60 - Digit  7 only candidate for Cell( 9, 5)
 61 - Digit  8 only candidate for Cell( 9, 8)
 62 - Digit  9 only candidate for Cell( 2, 2)
 63 - Digit  7 only candidate for Cell( 2, 3)
 64 - Digit  3 only candidate for Cell( 2, 7)
 65 - Digit  6 only candidate for Cell( 3, 3)
 66 - Digit  9 only candidate for Cell( 5, 7)
 67 - Digit  3 only candidate for Cell( 5, 8)
 68 - Digit  3 only candidate for Cell( 7, 3)
 69 - Digit  6 only candidate for Cell( 7, 9)
 70 - Digit  6 only candidate for Cell( 8, 2)
 71 - Digit  9 only candidate for Cell( 8, 3)
 72 - Digit  8 only candidate for Cell( 8, 6)
All Digit =  8 allocated
 73 - Digit  4 only candidate for Cell( 8, 7)
 74 - Digit  7 only candidate for Cell( 8, 8)
 75 - Digit  3 only candidate for Cell( 8, 9)
All Digit =  3 allocated
 76 - Digit  6 only candidate for Cell( 9, 6)
 77 - Digit  7 only candidate for Cell( 3, 7)
 78 - Digit  9 only candidate for Cell( 3, 8)
All Digit =  9 allocated
 79 - Digit  6 only candidate for Cell( 4, 7)
All Digit =  6 allocated
 80 - Digit  4 only candidate for Cell( 4, 8)
All Digit =  4 allocated
 81 - Digit  7 only candidate for Cell( 4, 9)
All Digit =  7 allocated
 ====================
  Puzzle Solved
 ====================



8 Iterations also indicates a fairly HARD Puzzle, so don't be too surprised you had some trouble with this manually !!