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Look at columns 1 and 6 which only have two unsolved cells each, two of them connected by row 4 and the other two sharing the bottom band. If both columns are missing at least one common digit, you have a Skyscraper pattern allowing eliminations in r8c4 and r9c12. Are they?

This is an easy way to find Skyscrapers (one of the most common and useful patterns in puzzles of this level) when solving without pencil marks. Note that if both columns miss the same two digits, then you have a double-Skyscraper (with eliminations on both digits) which is also a Remote Pair. We don't have that here, but it's quite common too.

Another relatively simple pattern that solves this is W-Wing, in pretty much the same cells as the Skyscraper. It can also be seen without pencil marks, but at least for me Skyscraper is clearly easiest here.

There's also an X-Wing in columns 5 and 9, but it's both harder to see (in this case -- in general it's easier) and doesn't solve the puzzle. It's still good practice to see if you can spot it, and it does simplify the puzzle.

29...1...4...2....5....923....9...63..12.49..95...8....391....8....3...2...8...46

SpAce wrote:Look at columns 1 and 6 which only have two unsolved cells each, two of them connected by row 4 and the other two sharing the bottom band. If both columns are missing at least one common digit, you have a Skyscraper pattern allowing eliminations in r8c4 and r9c12. Are they?

This is an easy way to find Skyscrapers (one of the most common and useful patterns in puzzles of this level) when solving without pencil marks. Note that if both columns miss the same two digits, then you have a double-Skyscraper (with eliminations on both digits) which is also a Remote Pair. We don't have that here, but it's quite common too.

Another relatively simple pattern that solves this is W-Wing, in pretty much the same cells as the Skyscraper. It can also be seen without pencil marks, but at least for me Skyscraper is clearly easiest here.

There's also an X-Wing in columns 5 and 9, but it's both harder to see (in this case -- in general it's easier) and doesn't solve the puzzle. It's still good practice to see if you can spot it, and it does simplify the puzzle.

29...1...4...2....5....923....9...63..12.49..95...8....391....8....3...2...8...46

Full answer: Show

Code: Select all

.-------------------.------------------.----------------.
|   2    9     3    | 4567  578   1    | 5678  578  457 |
|   4    678   678  | 567   2     3    | 5678  1    9   |
|   5    1     678  | 467   78    9    | 2     3    47  |
:-------------------+------------------+----------------:
|  b78   2478  2478 | 9     1    c57   | 4578  6    3   |
|   3    678   1    | 2     567   4    | 9     578  57  |
|   9    5     467  | 3     67    8    | 47    2    1   |
:-------------------+------------------+----------------:
|   6    3     9    | 1     4     2    | 57    57   8   |
| a(7)8  478   4578 | 5-7   3     6    | 1     9    2   |
|   1    2-7   25-7 | 8     9    d5(7) | 3     4    6   |
'-------------------'------------------'----------------'


Skyscraper: (7)r8c1 = r4c1 - r4c6 = (7)r9c6 => -7 r8c4,r9c23; stte

Code: Select all

.-----------------.-------------------.----------------.
|  2   9     3    |   4567  578   1   | 5678  578  457 |
|  4   678   678  |   567   2     3   | 5678  1    9   |
|  5   1     678  |   467   78    9   | 2     3    47  |
:-----------------+-------------------+----------------:
| b78  2478  2478 |   9     1   a(5)7 | 4578  6    3   |
|  3   678   1    |   2     567   4   | 9     578  57  |
|  9   5     467  |   3     67    8   | 47    2    1   |
:-----------------+-------------------+----------------:
|  6   3     9    |   1     4     2   | 57    57   8   |
| c78  478   4578 | d(5)7   3     6   | 1     9    2   |
|  1   27    257  |   8     9     7-5 | 3     4    6   |
'-----------------'-------------------'----------------'

W-Wing: (5=7)r4c6 - r4c1 = r8c1 - (7=5)r8c4 => -5 r9c6; stte

Both patterns prove that at least one of the bracketed candidates (start and end nodes of the chains) must contain that digit. Thus any other cells that see both of them can't hold those digits, so they can be eliminated.

Here's also the X-Wing:

Code: Select all

.----------------.-------------------.---------------------.
| 2   9     3    | 467-5  *(5)78  1  | 678-5  78-5  *(5)47 |
| 4   678   678  | 567      2     3  | 5678   1       9    |
| 5   1     678  | 467      78    9  | 2      3       47   |
:----------------+-------------------+---------------------:
| 78  2478  2478 | 9        1     57 | 4578   6       3    |
| 3   678   1    | 2      *(5)67  4  | 9      78-5  *(5)7  |
| 9   5     467  | 3        67    8  | 47     2       1    |
:----------------+-------------------+---------------------:
| 6   3     9    | 1        4     2  | 57     57      8    |
| 78  478   4578 | 57       3     6  | 1      9       2    |
| 1   27    257  | 8        9     57 | 3      4       6    |
'----------------'-------------------'---------------------'

X-Wing: 5:C59\r15 => -5 r1c478,r5c8

Many thanks. Being relatively new to Sudoku puzzles I get the concept of skyscrapers but where you say...

Look at columns 1 and 6 which only have two unsolved cells each, two of them connected by row 4 and the other two sharing the bottom band. If both columns are missing at least one common digit, you have a Skyscraper pattern allowing eliminations in r8c4 and r9c12. Are they?

I’m confused by the last line r8c4 I presume is row 8 column 4 and is r9c12 row9 columns 1 and 2?

So can you tell me.....what would be the next number you would put in and where would you put it.

Sorry for my ignorance. Still learning!

Shpongle wrote:Many thanks. Being relatively new to Sudoku puzzles I get the concept of skyscrapers but where you say...

SpAce wrote:Look at columns 1 and 6 which only have two unsolved cells each, two of them connected by row 4 and the other two sharing the bottom band. If both columns are missing at least one common digit, you have a Skyscraper pattern allowing eliminations in r8c4 and r9c12. Are they?

I’m confused by the last line r8c4 I presume is row 8 column 4 and is r9c12 row9 columns 1 and 2?

Yes, you got that right! They're standard cell references used on this forum.

So can you tell me.....what would be the next number you would put in and where would you put it.

Based on the Skyscraper, the easiest first placement would be either 5 at r8c4 or 7 at r9c6 (or both at once), since the Skyscraper eliminates 7 from r8c4 -- leaving the 5 there as a naked single and the 7 at r9c6 as a hidden single (the only 7 left in the box). Even without pencil marks it's easy to see that box 8 has only two digits left, so if you can eliminate one of them from one or the other cell, they both get solved immediately. The Skyscraper also eliminates the 7 from r9c12 which leaves the 7 at r9c6 as a row-based hidden single too, so that's another way to see that placement.

The Skyscraper also gives you another direct placement, 2 at r9c2, because it only has two candidates 2 and 7 (and the 7 gets eliminated). It's a bit harder to see without pencil marks, though, and unlike the others it doesn't solve the puzzle alone. With pencil marks all of these immediate placements (marked with [+]) are easy to see if the Skyscraper eliminations (marked with '-') are performed:

Code: Select all

.-----------------.---------------------.----------------.
| 2    9     3    |   4567  578   1     | 5678  578  457 |
| 4    678   678  |   567   2     3     | 5678  1    9   |
| 5    1     678  |   467   78    9     | 2     3    47  |
:-----------------+---------------------+----------------:
| 78   2478  2478 |   9     1     57    | 4578  6    3   |
| 3    678   1    |   2     567   4     | 9     578  57  |
| 9    5     467  |   3     67    8     | 47    2    1   |
:-----------------+---------------------+----------------:
| 6    3     9    |   1     4     2     | 57    57   8   |
| 78   478   4578 | [+5]-7  3     6     | 1     9    2   |
| 1  [+2]-7  25-7 |   8     9     5[+7] | 3     4    6   |
'-----------------'---------------------'----------------'


Once you place either 5r8c4 or 7r9c6 there's only singles left, so either of them solves the bottleneck. If you use the W-Wing instead, it eliminates 5 from r9c6 leaving 7 there as a naked single and 5 in r8c4 as a hidden single, so you get the same result with that, just a bit differently.

Sorry for my ignorance. Still learning!

As we all (I hope)!