The New Sudoku Players' Forum

[stumble]

I want to exercise my new-found ‘expertise’ in using AIC (Alternating Reference Chain) in solving this 11-26-07 BrainBusters.com VeryHard Sudoku. After eliminating the ‘usual suspects’ I arrived at this mark-up grid.

Code: Select all

.---------------------.---------------------.---------------------.
| 9      3      8     | 126    46     124   | 12     7      5     |
| 27     6      17    | 9      3      5     | 8      12     4     |
| 25     125    4     | 7      8      12    | 6      3      9     |
:---------------------+---------------------+---------------------:
| 1      4      69    | 3      2      8     | 79     5      67    |
| 23568  258    356   | 4      7      9     | 123    12     126   |
| 23     7      39    | 5      1      6     | 239    4      8     |
:---------------------+---------------------+---------------------:
| 4678   18     2     | 16     46     3     | 5      9      17    |
| 3457   159    13579 | 8      459    1247  | 127    6      127   |
| 567    159    15679 | 126    569    127   | 4      8      3     |
'---------------------'---------------------'---------------------'


I have not yet any skill in CHOOSING the targets for my AIC, but I proceeded thusly:
1: r1c7 = r8c7 – r7c9 = r7c4 => r1c4 <> 1
....(12)....(127)...(17).... (16).....(126)
Not ENTIRELY sure I did that right, but it didn’t contradict the solution.

But then I tried this AIC

Code: Select all

.---------------------.---------------------.---------------------.
| 9      3      8     | 26     46     124   | 12     7      5     |
| 27     6      17    | 9      3      5     | 8      12     4     |
| 25     125    4     | 7      8      12    | 6      3      9     |
:---------------------+---------------------+---------------------:
| 1      4      69    | 3      2      8     | 79     5      67    |
| 23568  258    356   | 4      7      9     | 123    12     126   |
| 23     7      39    | 5      1      6     | 239    4      8     |
:---------------------+---------------------+---------------------:
| 4678   18     2     | 16     46     3     | 5      9      17    |
| 3457   159    13579 | 8      459    247   | 127    6      127   |
| 567    159    15679 | 126    569    27    | 4      8      3     |
'---------------------'---------------------'---------------------'

2: r1c4 = r9c4 – r9c6 = r3c6 => r1c6<>2
....(26)....(126)...(27).....(12)
Except that r1c6 DOES EQUAL EXACTLY THAT, equal ‘2’ in the solution. Apparently my AIC usage leaves something to be desired.
Could someone tell what I did wrong? In either of my AIC’s. I figure the first one may have come out right by luck.


I decided to try an ALS XZ, which I may know how to work.

Code: Select all

.---------------------.---------------------.---------------------.
| 9      3      8     | 26     46     124   | 12     7      5     |
| 27     6      17    | 9      3      5     | 8      12     4     |
| 25     125    4     | 7      8      12    | 6      3      9     |
:---------------------+---------------------+---------------------:
| 1      4      69    | 3      2      8     | 79     5      67    |
| 23568  258    356   | 4      7      9     | 123    12     126   |
| 23     7      39    | 5      1      6     | 239    4      8     |
:---------------------+---------------------+---------------------:
| 4678   18     2     | 16     46     3     | 5      9      17    |
| 3457   159    13579 | 8      459    247   | 127    6      127   |
| 567    159    15679 | 126    569    27    | 4      8      3     |
'---------------------'---------------------'---------------------'


Set A r7c2, r7c4, r7c5 {1468}
.......(18)...(16)..(46)
Set B r1c5 {46}
X=6
Z=4
r1c6<>4

After this operation I was able to complete the puzzle using my usual simple weapons, leading me to believe my ALS was OK. Unless I was just very lucky. Anyway, does my ALS usage appear correct?

Edit: the reason for the 3 grids is that I was trying to color the markup numbers to highlight my AIC and ALS parameter choices. Forum software doesn't allow it.

[Para]

Your ALS is just a naked pair {46} in R17C5 locked for C5.

There's a Empty Rectangle(sorry wrong name) on digit 2 that eliminates 2 from R2C1. After that it is just singles.
R6C1(2) = R6C7(2) - R1C7(2) = R2C8(2) -> R2C1 <> 2

For your AIC's, i am not the best in them.
For the first one: There is no strong link on 1 between R1C7 and R8C7(weak link).
For the second one: There is no strong link on 2 between R9C6 and R3C6(weak link).
Others might explain it better

greetings

Para

[Carcul]

The UR at r89c25 solves the puzzle.

[stumble]

For your AIC's, i am not the best in them.
For the first one: There is no strong link on 1 between R1C7 and R8C7(weak link).
For the second one: There is no strong link on 2 between R9C6 and R3C6(weak link).
Para

Thank you for finally making me understand strong and weak links. (to have a strong link on '1' between r1c7(12) and r8c7(127) there would have had to be only two 1s in that column, and there are three, right?)

Balancing the definitions (filled with mysterious unfamiliar words) and the examples I've studied, I came up with a complete misconception.

I'm still studying your first two statements and will have still more questions later, I'm pretty sure.

[stumble]

Your ALS is just a naked pair {46} in R17C5 locked for C5.

I missed the {46} 'conjugates' I think they're called? - and, on further study I realize I STILL didn't know how to accomplish ALS.

There's a skyscraper on digit 2 that eliminates 2 from R2C1. After that it is just singles.
R6C1(2) = R6C7(2) - R1C7(2) = R2C8(2) -> R2C1 <> 2

Para

I'm having a hard time understanding your skyscraper. The two references I've found so far didn't help me much either:

http://www.sudopedia.org/wiki/Skyscraper
provides no example
and this one:
http://www.sudoku.org.uk/SudokuThread.asp?fid=7&sid=1666&p1=1&p2=1#post14000
still doesn't make it clear for me.

Could you write a bit more about your skyscraper, like I'm a real beginner sudokuist, which, in fact, I am?

[daj95376]

Code: Select all

# Sashimi X-Wing in [c28]                                 => [r2c1]<>2
# Skyscraper     in [r26] => [r1c7],[r5c8]<>2 => [r2c8]=2 => [r2c1]<>2
+-----------------------------------+
|  .  .  .  |  2  .  2  |  2  .  .  |
|  2  .  .  |  .  .  .  |  .  2  .  |
|  2  2  .  |  .  .  2  |  .  .  .  |
|-----------+-----------+-----------|
|  .  .  .  |  .  2  .  |  .  .  .  |
|  2  2  .  |  .  .  .  |  2  2  2  |
|  2  .  .  |  .  .  .  |  2  .  .  |
|-----------+-----------+-----------|
|  .  .  2  |  .  .  .  |  .  .  .  |
|  .  .  .  |  .  .  2  |  2  .  2  |
|  .  .  .  |  2  .  2  |  .  .  .  |
+-----------------------------------+


Para's chain described an Empty Rectangle in [r6b3].

Code: Select all

Sashimi X-Wing  as AIC: [r2c8]=2=[r5c8]-2-[r5c2]=2=[r3c2]-2-[r2c1]
Skyscraper      as AIC: [r2c8]=2=[r2c1]-2-[r6c1]=2=[r6c7]-2-[r1c7],[r5c8]
Empty Rectangle as AIC: [r6c1]=2=[r6c7]-2-[r1c7]=2=[r2c8]-2-[r2c1]


[stumble]

Thank you daj95376, but I find your explanation too dense for me. Let's try it this way:
I found another definition of a Skyscraper:
http://www.sudopedia.org/wiki/Turbot_Fish

"A Turbot Fish is a single-digit solving technique that uses a loop of odd length. The minimum length is 5, forming a fish-shaped pattern that gave the technique its name. Longer loops are possible, but they are better known as Fishy Cycles.

Here is an example of a Turbot Fish:

.-------.-------.-------.
| . | . | . | . | . . . |
| . X . | . X . | . . . |
| . | . | . | . | . . . |
:-------+-------+-------:
| . | . | . | . | . . . |
| . X . | . | * | . . . |
| . | . | . X . | . . . |
:-------+-------+-------:
| . | . | . | . | . . . |
| . | . | . | . | . . . |
| . | . | . | . | . . . |
'-------'-------'-------'
The candidates in columns 2 and 5 are strongly linked. The pattern causes either r5c2 or r6c5 to be true. Since r5c6 can see both these candidates, it can be eliminated.

This pattern is also known as Skyscraper."

Now let me try to adapt that explanation to Paras Skyscraper and you guys tell me if I'm wrong, or if my phrasing not true, or any other criticisms.
Here's my picture of Para's Skyscraper drawn as a chain.

Here's my take on Para's Skyscraper:
There are 2 weak links and 1 strong one on the digit 2. Each end of the chain can 'see', is strongly linked to the target cell r2c1. The rule says that, therefore, I can eliminate the 2 in r2c1. The binary conjugates {12} {12} are meaningless and could just as well have been {128} {126}, as long as they contained the digit 2 and were strong links in the chain.
Am I somewhere in the ball park?

[Sudtyro]

Am I somewhere in the ball park?

Code: Select all

| .  . . | 2 . 2 | 2c . . |
| 2e . . | . . . | .  2d .|
| 2  2 . | . . 2 | .  . . |
:--------+-------+--------:
| .  . . | . . . | .  . . |
| 2  2 . | . . . | 2  2 2 |
| 2a . . | . . . | 2b . . |
:--------+-------+--------:
| .  . . | . . . | .  . . |
| .  . . | . . 2 | 2  . 2 |
| .  . . | 2 . 2 | .  . . |

OK, let’s back up a notch and look closely at Para’s AIC in the grid above where the chain’s components are labeled with letters from a to e. There may be some confusion about weak/strong links and weak/strong inference, and you can read up about these in Sudopedia.

Here’s a quick review. The linking symbols in the AIC refer to strong(=) and weak(-) inference between two candidates. Strong inference means they can’t both be false. Weak inference means they can’t both be true.

Links are different: a strong LINK can have either strong or weak inference, while a weak LINK can have only weak inference. There is a STRONG link between the two (different) digits in a bivalue cell, and between the two (same) digits forming a conjugate pair (the case of only two of those digits left in a house).
There is a WEAK link between any two (different) digits in the same cell, and between any two of the same digits in a house.

So, in the grid above, we have the following strong/weak inferences to work with:
2a = 2b (from the conjugate pair’s strong link)
2b – 2c (from their weak link)
2c = 2d (another conjugate pair)

Note also that we have:
2e – 2d (from the conjugate pair’s strong link, but we use weak inference)
2e – 2a (from their weak link)

Now form Para’s AIC:
(2): r6c1 = r6c7 – r1c7 = r2c8 => r2c1 <> 2

The chain starts and ends with a strong link. So any candidate that can “see” (meaning weakly link to) both ends of the chain can be eliminated. In other words, the two candidates at the ends of the chain have a derived strong inference, meaning that at least one of them must be true. Since 2e can “see” both 2d and 2a, it must therefore be false. And you’re done!

Note also that daj95376’s Sashimi X-Wing reveals yet another simple AIC in this grid:
(2): r5c2 = r3c2 – r2c1 = r2c8 => r5c8 <> 2.

Sometimes in an AIC you will see the eliminated candidate explicitly included in the chain along with its two weak links. For example:
(2): r2c1 - r6c1 = r6c7 – r1c7 = r2c8 – r2c1 => r2c1 <> 2.
The chain now forms a closed loop, which is a key requirement when working with Nice loops. Nice loops, however, use a different kind of notation and have a special set of rules for forming the chains and eliminations.

And finally, there’s always the naming nightmare:
Para’s AIC is known variously as a Turbot Fish, a Turbot Chain, and an X-Chain; it may also be equivalent to certain variants of an Empty Rectangle, a Skyscraper, or a 2-String Kite. The names of these patterns can be a bit unnerving and often counter-intuitive, for historical reasons. For example, a Turbot Fish is not a fish at all, while an X-Wing is not really a wing but rather a 2-Fish that never got a “fishy” name. And (hold on), a Sashimi anything is a sort of fish that isn’t all there.

Welcome to Sudoku!

p.s. Keep checking the basics. That conjugate pair in b9 => r8c6 <> 2.
See also if you can find a different AIC and elimination using Para's same five candidates.

[stumble]

Am I somewhere in the ball park?

There may be some confusion about weak/strong links and weak/strong inference,

Gee, ya think?

Thanks very much for explaining this stuff. I will now study your explanation for a while. For me, reading sudopedia is a lot like trying to read computer explanations. Once you understand them, they read crystal clear, but if you don't, there seems to be a great deal of ways to misinterpret. I have a growing file of sudopedia definitions and examples but I either misunderstand their sudokuese or forget what I've read and get it wrong AGAIN.