The New Sudoku Players' Forum

[swaatacba]

The 7Sudoku site offers some "very difficult" puzzles. I'd be grateful help with this one. https://drive.google.com/file/d/1ucbMUxAnk_52Whgjr4jzzk7JCpZwUiXC/view?usp=sharing

The link should open for any one to view.

Notes Cells with a green background are given or solved; Bold red values are from the original puzzle; bold black values I have solved using standard methods.

[jco]

Hello swaatacba,

Looking at your grid, you can use an X-Wing on the digit 5 in rows 3,5, columns 6,9
(each row has only two candidates 5 in the same columns).
That will eliminate fives from (r2c6,r2c9,r6c6).
After that eliminations, you need to find a skyscraper in the same digit 5.

Regards,

[swaatacba]

Thank you.

I was not certain about the X-Wing in 5. I'm pleased that works, but it does not leave the puzzle in an easily solvable state.

Skyscraper is new to me. I'll have to look it up.

[jco]

Thank you.

I was not certain about the X-Wing in 5. I'm pleased that works, but it does not leave the puzzle in an easily solvable state.

Skyscraper is new to me. I'll have to look it up.

You can find about it here.

[swaatacba]

Thank you for the link. It will take a little while to understand the techique fully.

However, after applying the X-Wing you suggested earlier, we have two strongly paired Id 5s in C1 and C8 with a weakly paired baseline at row 6. Consequently, the values of Id 5 can be eliminated from Cells 1_13 and 8_74 respectively.

The connections look more like half a swastika than a skyscraper. But the principle is the same, and they lead to the satisfactory solution to the puzzle.

[jco]

Hello swaatacba,

The eliminations you mentioned are correct, but the
pattern you mentioned does not justify them.

Code: Select all

.----------------------------------------------.
| 4    1      25   | 9   3   8  | 6    25   7  |
| 8    25     3    | 6   57  47 | 9    1    24 |
| 79   69     67   | 2   1   45 | 8    3    45 |
|------------------+------------+--------------|
| 1    45     457  | 57  2   6  | 3    8    9  |
| 6    3      9    | 8   4   15 | 2    7    15 |
| 257  8      257  | 3   9   17 | 14   45   6  |
|------------------+------------+--------------|
| 259  2459   8    | 57  6   3  | 147  249  12 |
| 29   7      1    | 4   8   29 | 5    6    3  |
| 3    24569  2456 | 1   57  29 | 47   249  8  |
'----------------------------------------------'

Before mentioning directly the skyscraper, I take the opportunity
to show how to make eliminations with colouring.
I have coloured the 5s using two pairs of opposite colors (',")
and (!,!!). For instance, choosing the color (') for (5)r1c3, we
can color (5")r1c8 because one of these two 5s must be true (strong link).
We have two clusters (',") and (!,!!), using the two pairs of opposite
colors, as shown below.
In each pair of opposite colors only one color must be true.

Code: Select all

.-------------------------------.
|  .  .  '5 | .  .  . | . "5  . |
|  . "5   . | . '5  . | .  .  . |
|  .  .   . | .  . "5 | .  . '5 |
|-----------+---------+---------|
|  . -5  -5 |"5  .  . | .  .  . |
|  .  .   . | .  . '5 | .  . "5 |
| !5  .   5 | .  .  . | . '5  . |
|-----------+---------+---------|
|!!5  5   . |'5  .  . | .  .  . |
|  .  .   . | .  .  . | .  .  . |
|  .  5  -5 | . "5  . | .  .  . |
'-------------------------------'

Now, (5)r4c3 sees ('5)r1c3 and ("5)r4c4.
Since one of the opposite colors must be true, one of these
colored 5s must be true, so (5)r4c3 can be eliminated (marked
with a minus sign at the grid). Something similar happens to (5)r9c3,
that sees ('5) r1c3 and ("5)r9c5.
A key observation is that since (!!5)r7c1 and ('5)r7c4 are in
the same row, both colors cannot be true, so one of the colors
(!,") must be true.
This is useful, since (5)r4c2 and (5)r4c3 both see (!5)r6c1 and ("5) r4c4.
This implies that (5)r4c2,(5)r4c3 can both be eliminated. One of
these eliminations was already justified, but I mention this
second way to get both elimination because the pattern abcd below
is the skyscraper.

Code: Select all

.-------------------------------.
|  .  .   5 | .  .  . | .  5  . |
|  .  5   . | .  5  . | .  .  . |
|  .  .   . | .  .  5 | .  .  5 |
|-----------+---------+---------|
|  . -5  -5 |d5  .  . | .  .  . |
|  .  .   . | .  .  5 | .  .  5 |
| a5  .   5 | .  .  . | .  5  . |
|-----------+---------+---------|
| b5  5   . |c5  .  . | .  .  . |
|  .  .   . | .  .  . | .  .  . |
|  .  5   5 | .  5  . | .  .  . |
'-------------------------------'

Here abcd is not a notation for colors. I am using labels a,b,c,d just to show the cells that are part of the
skyscraper pattern: two walls "ab" and "cd", floor "bc" and tilted roof "da".

The logic to justify the skyscraper elimination is simple:
If (5)r6c1 (marked a) is true, it eliminates (5)r4c23.
If (5)r6c1 is false, then (5)r7c1 (marked b) must be true
(column 1 has only two 5s), so (5)r7c4 must be false, but this implies
that (5)r4c4 must be true (column 4 has only two 5s), which eliminates
the same (5)r4c23. This reasoning show that one of (5)r6c1,(5)r4c4 must
be true. Any cell that sees these two cells and has a 5 in it, must have
a false 5 that can be eliminate. So,(5)r4c23 can eliminated.

When you get used to skyscrapers, you will see these eliminations instantly.

Did you noticed something neat and powerful in that colouring argument
shown in the second grid? ('5)r6c8 sees (!5)r6c1 and (5")r5c9, so it must
be false. But that implies that the color (') is false. So, all digits marked (') are
false and all digits marked (") are true!

Regards,
JCO

Edit: improved text at the beginning.

[swaatacba]

Thank you. That makes a coherent logic that I was missing before.

[swaatacba]

So, in the original query, I eliminated a 5 in cells 1_13 and 8_74 each of which had only 2 possible solutions. Since this was not a logical elimination (in SkyScraper logic), I had only a 1 : 4 chance of delivering the correct solution and took it.

I still need a method (other than by guessing) of determining the correct solution.

[jco]

Hello swaatacba,

I still need a method (other than by guessing) of determining the correct solution.

I will repeat in other terms what is implied by your previous comment. When solving a given puzzle from a known source, one assumes that the puzzle has a unique solution.
In this case, if in each step one applies eliminations justified by logic, one will find the only solution of the puzzle. In case one makes a wrong step, two scenarios can occur: one ends up getting a contradiction (like two equal digits in the same row, column or block, or like an empty cell), or if the mistake was in making a correct elimination without correct justification, one will get the solution with a step that can be considered as guessing. How to avoid the second situation ? You can see from the previous post that colouring can help. Also, checking the logic for each elimination (also shown in the previous post) is a way to reduce that possibility.
If you really need to make sure no mistake was made, my suggestion would be to check each step with a solver (like HoDoKu or YZF_Sudoku) in a post-mortem analysis like chess players do after the game.

Regarding applying techniques to make progress towards the solution, I am assuming that your aim is to solve puzzles manually. I recommend the following explanation given by Keith link and the following explanation by Havard. These references explain how one can approach the problem of solving a puzzle manually (first link) and the basics for finding basic single digit patterns (like skyscrapers) (both links explain this) and more. So, a unique method for a puzzle is hard to conceive because the puzzle maker makes puzzles with certain patterns in mind and there are very difficult puzzles that require very elaborate techniques.
Many useful techniques can be found here. My suggestion after learning some techniques (single digit basic techniques, basic fishes, xy-wing, xyz-wing and W-wing is a good start) is colouring.
Medusa colouring (well explained in many places that you will find easily) is a method of colouring that extends the colouring I briefly explained in the previous post. For a "pencil and paper" player, Medusa colouring is an almost ideal approach. I hope these comments help.

Regards,

[swaatacba]

Hello swaatacba,

Regarding applying techniques to make progress towards the solution, I am assuming that your aim is to solve puzzles manually.

Your assumption is mostly correct. I use Excel workbooks to record my results.

The solutions are recorded stepwise on separate worksheets. The progress of the solution is managed by hyperlinking the outcome of step 'a' to the starting position of step 'b'. There are some further caveats.

1. The standard elimination of potential solutions to unsolved cells, using the Sudoku rules, are handled automatically via a series of lookup tables. This avoids a lot of tedious dimensional cross checking.
2. Each non-standard process is handled progressively within the linked process. Also each process is self validating. In theory, if you employ the right criteria with the right method, you will always get the right answer.
3. There are a number of the more sophisticated methods, for example x-Wing, that I have not yet modelled. Errors can and do creep in here.

You will appreciate that this generates quite large workbooks, but you can download the workbook that created for this particular puzzle fron the link. https://drive.google.com/file/d/17HvaBujawzkBqfsEcUc6xxaOedojOoc2/view?usp=sharing

Regards,

Stephen