The New Sudoku Players' Forum

[wychwood]

Beginner wychwood here again, stuck on this one (and I thought I had got going so well).

Starting position:

Code: Select all

 *-----------*
 |..1|..6|..8|
 |2..|...|...|
 |...|3.7|.9.|
 |---+---+---|
 |..6|...|7..|
 |5..|.9.|..2|
 |..4|...|1..|
 |---+---+---|
 |.3.|1.8|...|
 |...|...|..4|
 |9..|2..|5..|
 *-----------*

Position I got to with techniques I undertand and therefore can use, and then got stuck:

Code: Select all

 *-----------------------------------------------------------------------------*
 | 3       457     1       | 9       245     6       | 24      2457    8       |
 | 2       45678   9       | 458     1458    145     | 346     134567  13567   |
 | 468     4568    58      | 3       12458   7       | 246     9       156     |
 |-------------------------+-------------------------+-------------------------|
 | 18      29      6       | 458     13458   12345   | 7       3458    359     |
 | 5       18      3       | 7       9       14      | 468     468     2       |
 | 7       29      4       | 568     3568    235     | 1       358     359     |
 |-------------------------+-------------------------+-------------------------|
 | 46      3       257     | 1       4567    8       | 9       267     67      |
 | 16      1568    2578    | 56      3567    9       | 2368    123678  4       |
 | 9       1468    78      | 2       3467    34      | 5       13678   1367    |
 *-----------------------------------------------------------------------------*

All tips and help gratefully received.
Thanks
Wychwood

[re'born]

The following two moves don't solve the puzzle, put will definitely get you going in the right direction:

Code: Select all

 *-----------------------------------------------------------------------------*
 | 3       457     1       | 9       245     6       | 24      2457    8       |
 | 2       45678   9       | 458     1458    145     | 346     134567  13567   |
 | 468     4568    58      | 3       12458   7       | 246     9       156     |
 |-------------------------+-------------------------+-------------------------|
 | 18      29      6       | 458     13458   12345   | 7       3458    359     |
 | 5       18B     3       | 7       9       14      | 468     468     2       |
 | 7       29      4       | 568     3568    235     | 1       358     359     |
 |-------------------------+-------------------------+-------------------------|
 | 46A     3       257     | 1       4567    8       | 9       267     67      |
 | 16A     1568-   2578    | 56      3567    9       | 2368    123678  4       |
 | 9       1468A   78      | 2       3467    34      | 5       13678   1367    |
 *-----------------------------------------------------------------------------*


The ALS xz-rule with A={1468} on r78c1, r9c2 and B={18} on r5c2, x=8, z=1 gives r8c2<>1. After this, one can shift A around a little to yield:

Code: Select all

 *-----------------------------------------------------------------------------*
 | 3       457     1       | 9       245     6       | 24      2457    8       |
 | 2       45678   9       | 458     1458    145     | 346     134567  13567   |
 | 468     4568    58      | 3       12458   7       | 246     9       156     |
 |-------------------------+-------------------------+-------------------------|
 | 18-     29      6       | 458     13458   12345   | 7       3458    359     |
 | 5       18B     3       | 7       9       14      | 468     468     2       |
 | 7       29      4       | 568     3568    235     | 1       358     359     |
 |-------------------------+-------------------------+-------------------------|
 | 46      3       257     | 1       4567    8       | 9       267     67      |
 | 16A     568A    2578    | 56A     3567    9       | 2368    123678  4       |
 | 9       1468-   78      | 2       3467    34      | 5       13678   1367    |
 *-----------------------------------------------------------------------------*


A={1568} on r8c124, B={18} on r5c2, x=8, z=1 giving r4c1, r9c2<>1.

[udosuk]

I can't find an elegant ALS that will crack the puzzle into singles, but r3c3 is a critical cell (or "magic cell" as some likes to call it). Assuming r3c3=8 will allow you to reaching contradiction through a long chain of singles, where as setting r3c3=5 would enable you to reach the solution via another long chain of singles.

Now hopefully someone can find an elegant logical move to eliminate 8 from r3c3.

[wychwood]

Hi Rep'nA
Thanks but can you just remind me what the ALS-XZ rule is, and where I fidn out more about it?
Cheers

[re'born]

Hi Rep'nA
Thanks but can you just remind me what the ALS-XZ rule is, and where I fidn out more about it?
Cheers

Sure, the ALS xz-rule was written up by Bennys here. Here is a quick explanation, in the context of your example. First, a little terminology.

If you have n cells in the same row, column, or block (or briefly, in the same group) which contain a total of n possible candidates, you call the n cells a locked set. In this case, we know that each of the n cells contains a different candidate and that each of the n possible candidates shows up in exactly one of the cells. Evidently, you can not have n cells in the same group with only n-1 possible candidates.

Now, if you have n+1 candidates in those same n cells, we call the cells an almost locked set. If we are able to remove one candidate from all of the cells, then of course we reduce ourselves to a locked set.

In your situation, look at my first move. The 3 cells I marked A contain the 5 candidates {1468}, while the cell I marked B contains the 2 candidates {18}. Therefore, A and B are both almost locked sets. Now, notice that if an 8 occurs in A, it cannot occur in B. Conversely, if an 8 occurs in B, it cannot occur in A. We therefore call x=8 a restricted common candidate (common since it is in both almost locked sets, but restricted since it cannot be in both). Note also that z=1 is a common candidate for both A and B.

Here comes the ALS xz-rule: Any cell that sees every instance of 1 in both A and B cannot be a 1.

Why? Well, assume r8c2=1. Then every 1 in both A and B is killed. Thus, both become locked sets. We therefore know that the remaining candidates must all appear. For A this means that {468} must all appear, while for B this means that {8} must appear. But we already established that 8 cannot appear in both A and B. This gives a contradiction.

I apologize for my exposition of the rule. If you need any clarification, please feel free to ask.

[udosuk]

After the 2 ALS-xz moves demonstrated by rep'nA earlier (which can in fact be grouped as one single ALS-XY-wing move), and a series of singles, you reach the following state:

Code: Select all

 *-----------------------------------------------------------*
 | 3     57    1     | 9     2     6     | 4    A57    8     |
 | 2     678   9     | 4     18    5     | 36    1367 -1367  |
 | 46    4568  58    | 3     18    7     | 2     9    B156   |
 |-------------------+-------------------+-------------------|
 | 8     2     6     | 5     3     1     | 7     4     9     |
 | 5     1     3     | 7     9     4     | 68    68    2     |
 | 7     9     4     | 8     6     2     | 1     35    35    |
 |-------------------+-------------------+-------------------|
 | 46    3     257   | 1     457   8     | 9    -267  B67    |
 | 1     58    2578  | 6     57    9     | 38   -2378  4     |
 | 9     468   78    | 2     47    3     | 5    -1678 B167   |
 *-----------------------------------------------------------*

ALS A: r1c8={57}
ALS B: r379c9={1567}
restricted common: x=5
common: z=7

So r2c9, r789c8 cannot be 7.

Code: Select all

 *-----------------------------------------------------------*
 | 3     57    1     | 9     2     6     | 4     57    8     |
 | 2     678   9     | 4     18    5     | 36    1367  136   |
 | 46    4568  58    | 3     18    7     | 2     9     156   |
 |-------------------+-------------------+-------------------|
 | 8     2     6     | 5     3     1     | 7     4     9     |
 | 5     1     3     | 7     9     4     | 68   A68    2     |
 | 7     9     4     | 8     6     2     | 1     35    35    |
 |-------------------+-------------------+-------------------|
 | 46    3     257   | 1     457   8     | 9    -26   B67    |
 | 1     58    2578  | 6     57    9     | 38    238   4     |
 | 9     468   78    | 2     47    3     | 5    B168  B167   |
 *-----------------------------------------------------------*

ALS A: r5c8={68}
ALS B: r7c9+r9c89={1678}
restricted common: x=8
common: z=6

So r7c8 cannot be 6.

... and the rest can be solved with singles.

So all in all, it takes 4 ALS-xz moves plus singles to solve this puzzle. A pretty decent challenge!

[wychwood]

Thanks guys, brilliant (to me anyway).

Looks like I shall have to get to grips with these ALS's and their various types, especially the xz rule.

Cheers :D

[udosuk]

After the 2 ALS-xz moves demonstrated by rep'nA earlier (which can in fact be grouped as one single ALS-XY-wing move)...

I'm afraid I've made a mistake there. The 2 moves can in fact be grouped as one 2 ALS 2 restricted common move:
(cf bennys)

Code: Select all

 *-----------------------------------------------------------------------------*
 | 3       457     1       | 9       245     6       | 24      2457    8       |
 | 2       45678   9       | 458     1458    145     | 346     134567  13567   |
 | 468     4568    58      | 3       12458   7       | 246     9       156     |
 |-------------------------+-------------------------+-------------------------|
 | 18      29      6       | 458     13458   12345   | 7       3458    359     |
 | 5      A18      3       | 7       9       14      | 468     468     2       |
 | 7       29      4       | 568     3568    235     | 1       358     359     |
 |-------------------------+-------------------------+-------------------------|
 |B46      3       257     | 1       4567    8       | 9       267     67      |
 |-16     A1568    2578    |C56      3567    9       | 2368    123678  4       |
 | 9      A1468    78      | 2       3467    34      | 5       13678   1367    |
 *-----------------------------------------------------------------------------*

A=r589c2={14568}
B=r7c1={46}
C=r8c4={56}
x=4 (restricted common to A,B)
y=5 (restricted common to A,C)
z=6 (common to A,B,C)

So r8c1 cannot be 6, must be 1.

And yes, good idea about picking up the ALS techniques, at least the (simplest) xz-rule.