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Jigsaw 32

 
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nj3h
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PostPosted: Mon Apr 16, 2007 1:29 am    Post subject: Jigsaw 32 Reply with quote

The subject puzzle, after solving as far as I could caused me to hit a stumbling block.

I got a hint of Law of Leftovers eliminations in Row 1-2.

Uing this puzzle as an example can someone help me work through this technique? I have read the links listed in the other messages, but I do not understand how to apply in this situation.

Thanks,
George

.---------.---------.---------.---------.---------.---------.---------.---------.---------.
| 23578 | 358 | 4 | 1 | 2356789 | 236 | 25789 | 235678 | 235678 |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 235789 | 358 | 258 | 25689 | 25689 | 4 | 1 | 23578 | 23578 |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 129 | 6 | 3 | 58 | 58 | 7 | 29 | 129 | 4 |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 2578 | 4 | 1 | 2678 | 2678 | 9 | 3 | 2567 | 2567 |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 123589 | 7 | 258 | 23468 | 123468 | 236 | 2589 | 123569 | 23568 |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 6 | 9 | 278 | 2378 | 12378 | 5 | 4 | 1237 | 2378 |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 345 | 2 | 9 | 3457 | 3567 | 8 | 57 | 34567 | 1 |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 3458 | 358 | 27 | 234578 | 234578 | 1 | 6 | 234578 | 9 |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 3458 | 1 | 6 | 279 | 279 | 23 | 2578 | 234578 | 23578 |
'---------'---------'---------'---------'---------'---------'---------'---------'---------'
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Glyn
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PostPosted: Mon Apr 16, 2007 10:38 am    Post subject: Reply with quote

Hi George.

Rows 1 and 2 must contain a 2 complete sets of the digits 1 to 9. That means they must contain two digit 7s.

There are three jigsaw pieces (Nonets) which have cells in rows number 1 and 2. Only the left and right hand Nonets (N1 and N3) can provide the two digit 7s we require, as the middle Nonet (N2) already contains a 7 in R3C6.

In N1 and N3 we must take a 7 from either row 1 or row 2, which excludes R4C1 from being a 7.

Hope this helps.

Glyn
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Para
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PostPosted: Mon Apr 16, 2007 10:50 am    Post subject: Re: Jigsaw 32 Reply with quote

nj3h wrote:
The subject puzzle, after solving as far as I could caused me to hit a stumbling block.

I got a hint of Law of Leftovers eliminations in Row 1-2.

Uing this puzzle as an example can someone help me work through this technique? I have read the links listed in the other messages, but I do not understand how to apply in this situation.

Thanks,
George

Code:
.---------.---------.---------.---------.---------.---------.---------.---------.---------.
| 23578   | 358     | 4       | 1       | 2356789 | 236     | 25789   | 235678  | 235678  |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 235789  | 358     | 258     | 25689   | 25689   | 4       | 1       | 23578   | 23578   |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 129     | 6       | 3       | 58      | 58      | 7       | 29      | 129     | 4       |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 2578    | 4       | 1       | 2678    | 2678    | 9       | 3       | 2567    | 2567    |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 123589  | 7       | 258     | 23468   | 123468  | 236     | 2589    | 123569  | 23568   |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 6       | 9       | 278     | 2378    | 12378   | 5       | 4       | 1237    | 2378    |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 345     | 2       | 9       | 3457    | 3567    | 8       | 57      | 34567   | 1       |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 3458    | 358     | 27      | 234578  | 234578  | 1       | 6       | 234578  | 9       |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 3458    | 1       | 6       | 279     | 279     | 23      | 2578    | 234578  | 23578   |
'---------'---------'---------'---------'---------'---------'---------'---------'---------'


Hi

This is just the explanation for this particular move. I haven't checked if this solves the puzzle.

Code:

Nonet numbers

111233333
112222333
114442263
144555266
148555269
448555669
748866699
777888899
777778999


We draw a line Between Row 2 and 3. There then are 5 cells under the line from nonets 1 and 3 and 5 cells above the line from nonet 2.

R345C1 + R3C29 = R2C3456 + R1C4

R2C3456 + R1C4(the cells above the line) contain the digits 1,2,4,5,6,8 and 9. Or to say it otherwise, they don't contain the digits 3 or 7. So there can't be a 3 or a 7 in the cells under the line so R4C1 can't contain a 7 and R5C1 can't contain a 3.

greetings

Para
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nj3h
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PostPosted: Tue Apr 17, 2007 1:26 am    Post subject: Reply with quote

Para and Glyn,

Thanks so much for taking the time to help me out with this puzzle.

I have to admit that I still am lost for the next hint from sumoCue. I used the same method to check Rows 8-9 and Column 1-2 and 8-9 in a similat fashion to the explation of Rows 1-2 above. I saw nothing that eliminated any other candidates.

The way I read the hint comment is that the elimination are in Rows 1-2. I guess the comment means that using Rows 1-2 there are candidate eliminations elsewhere. There were no eliminations in Rows 1-2.

Now I get a message that says "Law of Leftovers eliminations in row 8-9". As I indicated above, I can find no eliminations using Rows 8-9. I drew a line between Rows 7 and 8. How does one know where to draw such a line.

Below is the current state of the grid that yielded the LoL elim in Rows 8-9 message.

Please help me to grasp this technique.

Again, thanks for the help.

Regards,
George

.---------.---------.---------.---------.---------.---------.---------.---------.---------.
| 23578 | 358 | 4 | 1 | 2356789 | 236 | 25789 | 235678 | 235678 |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 235789 | 358 | 258 | 25689 | 25689 | 4 | 1 | 23578 | 23578 |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 129 | 6 | 3 | 58 | 58 | 7 | 29 | 129 | 4 |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 258 | 4 | 1 | 2678 | 2678 | 9 | 3 | 2567 | 2567 |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 12589 | 7 | 258 | 23468 | 123468 | 236 | 2589 | 123569 | 23568 |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 6 | 9 | 278 | 2378 | 12378 | 5 | 4 | 1237 | 2378 |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 345 | 2 | 9 | 3457 | 3567 | 8 | 57 | 34567 | 1 |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 3458 | 358 | 27 | 234578 | 234578 | 1 | 6 | 234578 | 9 |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 3458 | 1 | 6 | 279 | 279 | 23 | 2578 | 234578 | 23578 |
'---------'---------'---------'---------'---------'---------'---------'---------'---------'
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sudokuEd
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PostPosted: Tue Apr 17, 2007 8:27 am    Post subject: Reply with quote

nj3h wrote:
Now I get a message that says "Law of Leftovers eliminations in row 8-9". ...I drew a line between Rows 7 and 8. How does one know where to draw such a line.
Thanks for bringing up LoL George. I'm new to LoL and find SumoCue's hint confusing but think I can make sense of it.

Drawing the line after row 7 there are at least 3 different ways to divide up the 3 nonets into above and below the line so that the SAME number of INCOMPLETE nonet cells are above and below (the most important bit for where to draw the line it seems).

I can see two 5-cell sets and one 8-cell set of leftover cells for r89. All of them could potentially give LoL eliminations since they must all have the same digits.
1)8 cells above the line are r567c39 + r7c48. 8 cells below the line are r8c123+r9c12345
2)5 cells above the line are r567c3 + r7c14. 5 cells below the line are r8c89 + r9c789
3)5 cells above the line are r7c189 + r56c9. 5 cells below the line are r8c4567 + r9c6

Unfortunately, I can't see that any of them give LoL eliminations the way that Glyn and Para did.

How to make sense of the hint? Any of these left-over sets requires a 6 below the line -> there must be a 6 'above the line'. This is only available in r7c8 or r5c9 -> no 6 in r5c8 since it can 'see' both those cells. So the 'LoL elimination in r89' is 6 from r5c8.

Of course, a much easier way (for us!) to get this same elimination is to notice that these are the only 2 6's in nonet 9 -> no 6 in r5c8.

Hope this makes sense of SumoCue.

Good Luck!
Ed



Code:
.---------.---------.---------.---------.---------.---------.---------.---------.---------.
| 23578   | 358     | 4       | 1       | 2356789 | 236     | 25789   | 235678  | 235678  |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 235789  | 358     | 258     | 25689   | 25689   | 4       | 1       | 23578   | 23578   |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 129     | 6       | 3       | 58      | 58      | 7       | 29      | 129     | 4       |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 258     | 4       | 1       | 2678    | 2678    | 9       | 3       | 2567    | 2567    |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 12589   | 7       | 258     | 23468   | 123468  | 236     | 2589    | 123569  | 23568   |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 6       | 9       | 278     | 2378    | 12378   | 5       | 4       | 1237    | 2378    |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 345     | 2       | 9       | 3457    | 3567    | 8       | 57      | 34567   | 1       |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 3458    | 358     | 27      | 234578  | 234578  | 1       | 6       | 234578  | 9       |
:---------+---------+---------+---------+---------+---------+---------+---------+---------:
| 3458    | 1       | 6       | 279     | 279     | 23      | 2578    | 234578  | 23578   |
'---------'---------'---------'---------'---------'---------'---------'---------'---------'
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Para
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PostPosted: Tue Apr 17, 2007 8:58 am    Post subject: Reply with quote

Hi

What Ed just explained is the second way to use LoL. This time the 6's are conveniently both in the same nonet but they don't have to be.
The first eliminations you can make with LoL. That is by simply comparing which digits are in the cells above and under the line.
The second type of eliminations are eliminations made by digits that are already placed above or under the line. If one digit is already placed in a cell above the line, we know it must also be in the cells under the line or vice versa. Then we determine which cells this digit can be placed in. Then we check if there are any cells that can see all the cells(above or under the line) and we can eliminate this digit from these cells(that see all cells which can contain the digit in question).

There's some nice examples of this second way in this thread:

http://www.sudocue.net/forum/viewtopic.php?t=653

greetings

Para
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Para
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PostPosted: Tue Apr 17, 2007 9:15 am    Post subject: Reply with quote

Hi

I don't know if there's an easy way to explain where to check. I usually suggest to check every set of rows and columns. Just like when looking for singles or naked subsets.
For this puzzle i would first suggest Law of Leftovers on Column 789. This gives the best eliminations. Which gives you 4 cells left and 4 cells right of the line.
The problem with the hints from SumoCue is that it seems to be checking first all rows and then all columns and it shows all possible eliminations, even the more difficult ones. Usually there are easier eliminatinions somewhere else.

greetings

Para
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Glyn
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PostPosted: Tue Apr 17, 2007 4:00 pm    Post subject: Reply with quote

The elimination here seems to be based on a 6 being required in the 5 cells of Nonets 8 and 9 that lie above the line above row 8 which mirror the 5 cells of Nonet 7 which lie below that line.

Code:

The Nonets are defined thus:

111233333
112222333
114442253
144666255
147666258
447666558
947755588
999777788
999997888


The two candidate positions for the 6 are in cells R5C9 and R7C8. These form a pointing pair on 6 with R5C8 being the shared peer of both so R5C8<>6.

We could have got the elimination plus one more if we had used the strong link between digit 6 at R7C5 and R7C8, here there are two shared peers R45C8<>6. If you do this there is another pointing pair on 6 to look for straight away.

Have fun.
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Para
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PostPosted: Tue Apr 17, 2007 4:29 pm    Post subject: Reply with quote

Glyn wrote:
The elimination here seems to be based on a 6 being required in the 5 cells of Nonets 8 and 9 that lie above the line above row 8 which mirror the 5 cells of Nonet 7 which lie below that line.



The two candidate positions for the 6 are in cells R5C9 and R7C8. These form a pointing pair on 6 with R5C8 being the shared peer of both so R5C8<>6.

We could have got the elimination plus one more if we had used the strong link between digit 6 at R7C5 and R7C8, here there are two shared peers R45C8<>6. If you do this there is another pointing pair on 6 to look for straight away.

Have fun.


Sumocue calls a lot eliminations LoL eliminations because they can be made using Law of Leftovers as well (as Ed showed).

Quote:
Code:

The Nonets are defined thus:

111233333
112222333
114442253
144666255
147666258
447666558
947755588
999777788
999997888


Is there a general way to number nonets? I generally like to number nonets to their relative position to eachother, mostly because that way the numbering looks like the numbering of nonets in a regular sudoku.

That maybe is just a personal thing.

greetings

Para
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nj3h
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PostPosted: Tue Apr 17, 2007 6:36 pm    Post subject: Reply with quote

Again thanks to all the folks that are helping myself and others to understand the LoL concept.

SudokuEd: I can see the 3 sets above and below the line (8 cell, 5 cell, and another 5 cell). What I am having trouble visualizing is why the above and below the line cells are complimentary to each other. I have to be dense, but I haven't hit the aha moment yet.

Further, can you please provide more explanation for the discussion in the "How to make..." paragraph. For some reason I can not see what is going on there. I just do not understand the logic in this paragraph. I think this must tie back to the fact that I am not grasping the above and below the line concept.

The next paragraph make sense, though, since the elimination in r5c8 of the 6 can be made without LoL techniques. I missed this logical exclusion going through the puzzle.

Thanks for your comments and future explanations, Ed.

Para: Thanks again for your help. I will check out the link that you provided. I suspect if I could understand the above and below the line theory, I could understand the concept. For some reason, this technique just isn't clicking yet.

Regards to all,
George
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Para
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PostPosted: Tue Apr 17, 2007 9:31 pm    Post subject: Reply with quote

nj3h wrote:
What I am having trouble visualizing is why the above and below the line cells are complimentary to each other. I have to be dense, but I haven't hit the aha moment yet.


Hi

Ok, this is basically based on the elementary rules of sudoku. I'll use these pictures to help explain.


Basic rules of Sudoku:
- Every Row, Column and Nonet has to contain the numbers 1-9 exactly once.



In this example we have 3 nonets almost completely in Row 1, 2 and 3. Nonets coloured yellow. In each of these nonets we have to place numbers 1-9 exactly once.



When we fill these nonets with 1-9, we have filled Row 1, 2 and 3 almost completely with 1-9. There are 2 cells empty in row 3. The missing digits are 1 and 3, which are also the digits under the line.

We can explain this by the fact that the 3 nonets contain 1-9 once each, so in total three times.
Also row 1, 2, and 3 contain the digits 1-9 once each, again in total three times.
So row 1, 2 and 3 contain exactly the same digits as 3 nonets. So all the digits that are in the 3 nonets but aren't in row 1, 2 and 3 are missing from row 1, 2 and 3.
In this case the a digit 1 and 3 in R4C19 are in the 3 nonets but not in row 1, 2 or 3. But they have to be in row 1, 2 and 3. So they go in the cells that are in row 1, 2 and 3 but aren't in the 3 nonets. In this case R3C28.

I hope this is clear. I always feel my explanations lack some clarity.

greetings

Para
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Glyn
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PostPosted: Tue Apr 17, 2007 9:56 pm    Post subject: Reply with quote

Hi Para

I think this algorithm would label the nonets in the same way that Ruud does for this puzzle.

For each nonet find the left edge of its' top row. If you then order the corners firstly by ascending row number, then by ascending column number then everything should work out.

The corners are at R1C1,R1C4,R1C5,R3C3,R3C8,R4C4,R5C3,R5C9,R7C1 and these define nonets 1 to 9 respectively.

I guess it can be confusing because one instinctively wants to call the bottom corner nonet 9, but the symmetry here scuppers that. I'm not sure if Jean Christophe uses the same system.

By the way great notes on LOL, perhaps Ruud will make it a 'sticky'.
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Para
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PostPosted: Tue Apr 17, 2007 10:51 pm    Post subject: Reply with quote

Hi

Yeah i figured that was the way you numbered your nonets. It just seem a bit unnatural to me. But it is the easiest way to program. Wink
That is why i always tend to add a nonet-numbering so people know which nonets i am talking about compared to what programs use.
I am just going to be stubborn. (must be a human thing)

greetings

Para
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