The New Sudoku Players' Forum

[ronk]

In puzzle #1325 (the first one) I could use the AUR to show that r9c4<>8.

I hadn't found an elimination yet but, after that clue, it was pretty easy. I assume you used digit 1 of the AUR and went thru r1c1.

In the second puzzle, well, I leave the use of the AUR with you.

Yes, that one was too easy.

[ronk]

Consider the following grid:

Code: Select all

  468  5    468  | 346  2    1    | 7    9    348   
  2    7    69   | 69   348  348  | 34   5    1     
  1    3    489  | 49   5    7    | 48   2    6     
 ----------------+----------------+----------------
  349  6    7    | 1    34   349  | 5    8    2     
  5    49   2    | 8    7    349  | 1    6    34   
  34   8    1    | 2    6    5    | 9    34   7     
 ----------------+----------------+----------------
  469  49   5    | 34   1    2    | 3468 7    348   
  7    2    346  | 5    348  348  | 346  1    9     
  48   1    348  | 7    9    6    | 2    34   5     


We have a type E3 AUR in cells r2c56/r8c56. The AUR (48) can be used in the following way:

[r9c3]=3=[r8c3]-3-[r8c5|r8c6]=3=[r2c5|r2c6]-3-[r2c7]=3=[r7c7|r8c7]-3-[r9c8]=3=[r9c3], => r9c3=3 and that solve the puzzle.

That quote is from your opening post, and I've used that AUR type 3 technique without fail a few times since. But upon re-reading it, I either missed something or the technique is invalid. In brief and using the above example, this is the way I see the type 3 AUR today.

There are three different digits (3,4,8) that must ultimately occupy four cells (r2c5, r2c6, r8c5, r8c6). Thusly, one of those digits must occur twice, implying that each of the following strong inferences cannot [edit: but they can, they can] be true simultaneously:

1. [r8c5|r8c6]=3=[r2c5|r2c6]
2. [r8c5|r8c6]=4=[r2c5|r2c6]
3. [r8c5|r8c6]=8=[r2c5|r2c6]

Therefore, unless we have beforehand knowledge of the repeated digit, I don't see how we can validly use either of the above inferences. [edit: But we can, we can.]

Can you shed some light on this? [edit: No longer any need to.)

Ron

[vidarino]

Carcul's usage of the AUR above is valid, as far as I can see.

If R8C3 contains a 3 then the AUR becomes a type 2 UR, which in turn means that R2C7 cannot be 3. And vice versa. And exactly the same is valid for the other digits in the AUR.

[ronk]

Carcul's usage of the AUR above is valid, as far as I can see.

If R8C3 contains a 3 then the AUR becomes a type 2 UR, which in turn means that R2C7 cannot be 3. And vice versa. And exactly the same is valid for the other digits in the AUR.

But I think it depends ... somehow ... on the fact that the AUR contains the only 8s in both rows 2 and 8. IOW there is "beforehand knowledge" that 8s occur twice in the AUR. Which also means "the same" is NOT true for the 8, doesn't it?

[edit: Oh, never mind. Without realizing it, I was trying to streeeeeeetch a strong inference into a strong link. Inference ... link ... inference ... link ... inference ... link ......]

Ron

[vidarino]

Carcul's usage of the AUR above is valid, as far as I can see.

If R8C3 contains a 3 then the AUR becomes a type 2 UR, which in turn means that R2C7 cannot be 3. And vice versa. And exactly the same is valid for the other digits in the AUR.

But I think it depends ... somehow ... on the fact that the AUR contains the only 8s in both rows 2 and 8. IOW there is "beforehand knowledge" that 8s occur twice in the AUR. Which also means "the same" is NOT true for the 8, doesn't it?

Hmm, I don't think it would matter. Not in this loop, anyway... Since the 8s are locked in both rows you wouldn't find a candidate to link them to. The rest of the rows are obviously empty. Same for the rest of the boxes, since they would be the victims of Locked Candidates, and none in the same columns in the other rows since they would be X-Winged away. I might be wrong, though, but I really don't see any problems with the usage in this case...

Speaking of locked AUR-candidates, it is also possible to use the fact that the 8s are locked to our advantage. If we - as before - have a 3 in R8C3, the AUR also turns into a type 4 UR, which allows us to eliminate the 4s from the row 2 corners.

In this particular grid that just leaves a lonely 4 in the same cell as the one we eliminated 3 from earlier - not much gained. But in a hypothetical example it might be useful to be able to form a link like this; ...-R8C3-3|4-R2C56=4=R2C7=...

Anyway, I think I'm ranting now. Sleep deprivation doing its stuff, no doubt. Wear protective gear at all times. Good night. ;)

Vidar

[vidarino]

I found another sneaky one in Carcul's puzzle, using the same AUR rule:

Code: Select all

  468  5    468  | 346  2    1    | 7    9    348   
  2    7    69   | 69   348  348  | 34   5    1     
  1    3    489  | 49   5    7    | 48   2    6     
 ----------------+----------------+----------------
  349  6    7    | 1    34   349  | 5    8    2     
  5    49   2    | 8    7    349  | 1    6    34   
  34   8    1    | 2    6    5    | 9    34   7     
 ----------------+----------------+----------------
  469  49   5    | 34   1    2    | 3468 7    348   
  7    2    346  | 5    348  348  | 346  1    9     
  48   1    348  | 7    9    6    | 2    34   5     

R7C7-3-R7C4-4-R8C56=4=R2C56-4-R2C7-3-R7C7 => R7C7 <> 3

Then an X-Wing with 3 in R17C49 => R5C9 <> 3

And that solves the puzzle.

[ronk]

I found another sneaky one in Carcul's puzzle, using the same AUR rule:
............
R7C7-3-R7C4-4-R8C56=4=R2C56-4-R2C7-3-R7C7 => R7C7 <> 3

I see you've dropped the square brackets. I've been tempted to do the same keeping them only for grouped cells, e.g., for [r1c1,r2c2] but not for r12c1.

BTW I posted that same deduction here.

Ron

[ronk]

Here is a library of 55 puzzles with AUR type E3 ... after being advanced to the degree possible by "basic techniques" including multi-coloring. Depending on your own set of basic techniques, mileage may vary.

Code: Select all

.5..2179.27.....5113..57.26.671..5825.287.16..812659.7..5.12.7.72.5...19.1.7962.5
1.......3.6.3..7...7...5..121.7...9...7........8.1..2....8.64....9.2..6....4.....
...4...26.7...9...28.5.....8..3....79......8.....9..64...7..1....6.....2..3.15...
15.3......7..4.2....4.72.....8.........9..1.8.1..8.79......38...........6....7423
......5.88..651...9..2..3...4.7..9.5....2....5.1.4.....1.56.8.9..3............61.
...9.2.....8....753..4..9....5.83....7.......61.2.......1..7.64...8..3.....6...9.
5..3...8.........9.79.28...1......7.....3..2..8..69.5.8....2....46.......9....4.7
.9...71....8...26........5...6..3..27.1.95......6......738.........4...66.2...7..
.3..8...4......36..6...1..2..3..2.......3815...8....2.2...64..9.862......7.......
....8..5.93....7.....6.1..2698.4.5.........4.5.....2.84...2..7..69.......52...9..
.......31.8..4.....7.......1.63...7.3............8....54....8.....6..2.....1.....
.......72.8.6......1.......4...97.........8..3........7.3....4....18.6.....5.....
.....3.1..5..7.......4.....1.38.........6.7..2...........1.23...6....5.7.........
....2.7...41.......3..5....2.....5.....4...8....1.....3...7..6....8...1.......2..
....31....8.....6..........54.2....83...7....1..6...........51..2.8...........3..
....5.1..4.7...........8...6..3...4........9..5.......2..7..5........8.13..4.....
....61....84......1........6.......2...5...3..5.4.....3...2..........4.8......57.
....92.6...4....8..1.......9.....72.....5.......4.....3....71........4.5...6.....
...1..5.2.7...46...........2.5.3..........79.1........68.....3.....2.......5.....
...2...3.4.5.........6.........5..67.3.1.......8...4...2....1..7....8.......4....
...2..3..9......1...........7..5..4..28...........1...1.5.6.......3..2.7......8..
...2..5..1..7......3....8...5..3.4.....6....2.8.......2......1.6.7..........5....
...286...7......45..........36...2...5.4.........1.........36..4..5............1.
...3..1.58..4..6..9........2...8.......7....3.............9..8..5...2....3.....4.
...42...6.1..........6.........6.31.4..7.5................31.5.2.6............8..
...6.5...2.....4............76....4....1..2.....83...........57.....4..61...2....
...6.5...2.....9............76....4....1..2.....83...........57.....4..61...2....
...7..1..62.........3....5.7.......9....8..3.1..4........15....4.....7......3....
...7..34.28................5...21.....7....3.....8......34..1...1......26........
...8..4...67...................17.6.....6.5..4.........9.....718..4.5......2.....
..2....54.1.6..............4..3..6..7.5.2..........1.....1.2....8....3......5....
..4....23.5...6................756..2......8...31........34.....6....5..1........
..6.31....2.....9.............25..8.1...........9.....4.....1.3...82....7.....4..
..71..........85..6...............41...2...7.3........58....3.....72....1...4....
..71..........89..6...............41...2...7.3........58....3.....72....1...4....
..8.15...6.....2......4.....5....74.3....1......8.....72.6............51.........
.1...46......9...3..........2.8..7.....35....4..............21...35.....9.....4..
.23..4.........17..........94....8.....23........1....7.8....9....5....21........
.28.......7..9..........1..6...3.45......2..71...........7.8.........63....2.....
.283...........71..........1..5..6..4...7............2...2....8.6..3....7......4.
.683...........7.2.........5...4..6.2...7.........61..71..........6...8.........3
.69..........4..3..8..........8..61.4.....9..5...2....2..9....5...6.1............
.725.........1..6..........61......3...2.85..4........1...6..........82....3.....
.763...........8.2.........5...4..6.2...8.........71..81..........6...7.........3
1......3..2..7........2...8.4....7.....6...5....3.....6..14.2..........43........
1......727..5........3........6..8.34...1...............3...5...2...4.......7..4.
38.....7....5..1............6..73....4....5..............42..3.9.1......5....8...
4..8............1.........717.3........6..25..9.........2...4..3.....6......19...
5.....72.....46...............25..8...6....9..1.......3..7..1........6.4.....8...
5..4............9.........86....25...3..1........8.4.....6..75..8.5.......1......
5.46........1...7........38.3..71.....2...5......8....1..5..2...7................
6...7.3..1...9.....2.......7.3.........5...2........8.4.....7.....2..4.....8.5...
6..2.87.....3............4.1.......3...9..2..54........32..........5..1.....4....
7.....4...6.5................16.3...4.....7.8......2.....27........4..5..9.....3.
71..2...8.......6.4..........65...1..35............4..2.....9.....6.3......1.....

The first is Carcul's example opening this thread. The next nine are from the top1465 including #1325 and #1391 that I posted recently. The last 45 are from Gordon Royle's library of 32930 puzzles with 17 clues.

Ron

P.S. Carcul, would you please supply the source and the original "minimal" clues for your example? (Sorry if I missed it.)
P.P.S If anyone thinks it's worthwhile, I am willing to add the puzzle numbers to this post.

[vidarino]

I see you've dropped the square brackets. I've been tempted to do the same keeping them only for grouped cells, e.g., for [r1c1,r2c2] but not for r12c1.

Ah, yes, that's the very notation I'm using, actually. I rarely need to group cells in brackets, though, as the RyyCxx usually covers my needs.

BTW I posted that same deduction here.
[/quote]

Oops, indeed you did. Good work.

Also, again, good work, Carcul. (And not so good work, Vidar, for not discovering this interesting pattern earlier.) ;)

Vidar

[Carcul]

Carcul, would you please supply the source and the original "minimal" clues for your example?

The puzzle that I have used as first example was posted by Pep in the thread "Nice loops for elementary level players - the x-cycle", where he also didn't provide the original clues.

Here is a library of 55 puzzles with AUR type E3 ...

This is a good work. If possible, could you also provide the starting grids for those puzzles? Thanks.

If anyone thinks it's worthwhile, I am willing to add the puzzle numbers to this post.

What do you mean by "puzzle numbers"?

Also, again, good work, Carcul.

Thanks.

[vidarino]

Here's another puzzle which has an AUR E3 which can be used for several loops;

Code: Select all

+-------+-------+-------+
| . . . | . 6 5 | 7 . . |
| . . 6 | . . 7 | . . . |
| . . 8 | 3 . . | . 5 . |
+-------+-------+-------+
| 9 . . | . . 2 | . . . |
| . . 5 | . 4 . | . . . |
| . . . | 6 . . | 8 2 . |
+-------+-------+-------+
| 6 9 . | . 5 . | . 3 . |
| . . 3 | . 1 . | . 9 . |
| . 5 2 | . . . | 4 . 7 |
+-------+-------+-------+


But it has another interesting step, even before the AUR E3 emerges fully, too. Basic techniques take you here;

Code: Select all

   1234   1234      9 |     148      6      5 |       7    148  12348
      5   1234      6 |    1489    289      7 |    1239    148 123489
   1247   1247      8 |       3     29     19 |    1269      5  12469
----------------------+-----------------------+----------------------
      9     68     17 |       5    378      2 |     136    146   1346
     28    268      5 |      18      4     38 |    1369      7   1369
   1347   1347    147 |       6     79     19 |       8      2      5
----------------------+-----------------------+----------------------
      6      9    147 |      27      5     48 |      12      3    128
    478    478      3 |      27      1    468 |       5      9    268
     18      5      2 |      89    389    368 |       4    168      7


Now, note the 148-AUR in R12C48. It has an extra 9 in R2C4. But there also exists a pair 89 in R9C4. This indirectly creates a strong link between the 8 in R5C4 to the other half of the AUR, R12C8. Not too useful for this particular puzzle, as far as I can see (haven't looked to hard, admittedly), but it might be worth noting that even AUR E3s can have "satellite" tuples similar to a UR type 3.

Vidar

[ronk]

This is a good work. If possible, could you also provide the starting grids for those puzzles?

Thanks. I'm sure you understand that each line in the "Code:" section is a starting grid, so I'm not sure what you're asking. If you mean present the starting grid in the *form* of a grid, that would be a post with 11 (or so) times as many text lines, which is impractical IMO.

If you mean the grid at the stage with the type E3 AUR, that would be a LOT of work since I have no way to automate that step. However, I'm perfectly willing to do either for a few of the puzzles -- once I'm sure what the "either" is.

If anyone thinks it's worthwhile, I am willing to add the puzzle numbers to this post.

What do you mean by "puzzle numbers"?

Like the 9th puzzle in the list is #1325 from the top1465.

Ron

[ronk]

Now, note the 148-AUR in R12C48. It has an extra 9 in R2C4. But there also exists a pair 89 in R9C4. This indirectly creates a strong link between the 8 in R5C4 to the other half of the AUR, R12C8. Not too useful for this particular puzzle, as far as I can see (haven't looked to hard, admittedly), but it might be worth noting that even AUR E3s can have "satellite" tuples similar to a UR type 3.

You certainly lost me. If you're suggesting r12c4 can be treated as a 'phantom cell' and combined with r5c4 and r9c4 to eliminate candidates elsewhere in col 4 (were there any), would you please explain further?

TIA, Ron

[vidarino]

You certainly lost me. If you're suggesting r12c4 can be treated as a 'phantom cell' and combined with r5c4 and r9c4 to eliminate candidates elsewhere in col 4 (were there any), would you please explain further?

I can try. There are no immediate eliminations, though, just a weak inference.

Imagine that R5C4=8. Then, R9C4=9, and our AUR E3 would have been reduced to a type 2 UR (14 14 / 148 148), which eliminates 8 from R9C8.

Approaching it from the other end, imagine that R9C8 was 8. This would make the AUR E3 into a Unique Rect type 3, since the "extras" 8 and 9 in R12C4 form a pair with R9C4. Hence, R5C4<>8.

Vidar

[Carcul]

I'm sure you understand that each line in the "Code:" section is a starting grid, so I'm not sure what you're asking.

Ahh, ok. My confusion originated because you have stated "Here is a library of 55 puzzles with AUR type E3 ... after being advanced to the degree possible by" and the word "after" lead me to conclude that the grids have already been advanced from the start, and I was only interested in the starting grids. Ok, so we have 54 starting grids, good job.

Regards, Carcul