The New Sudoku Players' Forum

[hrcjcr]

I prefer the xy wing as the place to start: r8c5, r6c5, r6c8 => r8c8<>7. It is easier for me to "see".

[r.e.s.]

Yes "if the '126' reduced to '16' then.... is bad

But what if the '126' reduced to '12'? That is my mental block.
[...]
I very clearly understand we cannot have
16 16
16 16
in a one-solution valid puzzle.

But we could certainly have
16 16
16 1

Here's the point you seem to miss ...
If the candidate list for a cell is initially '126', and then it's determined that it cannot be '16' (i.e., the correct digit cannot be 1 and it cannot be 6), it follows that it must be '2'.

[QBasicMac]

Here's the point you seem to miss ...
If the candidate list for a cell is initially '126', and then it's determined that it cannot be '16' (i.e., the correct digit cannot be 1 and it cannot be 6), it follows that it must be '2'.

More mantra.

No, I don't miss that. I clearly see it cannot be "16". Zero confusion here.

But it CAN be "1" or maybe "6". It just cannot be "16". I know it cannot be "16". I am not confused on that point.

As I was unable, evidently, to explain, SINCE IT CANNOT BE 16, then 126 must be something else. Here are some possible something else's:
12, 26, 16, 1, 2, 6

Now of this set of possible cases, I am told that, magically, only "2" works.

Right! I can memorize that. But not believe it or understand it.

Mac

[tso]

Here's the point you seem to miss ...
If the candidate list for a cell is initially '126', and then it's determined that it cannot be '16' (i.e., the correct digit cannot be 1 and it cannot be 6), it follows that it must be '2'.

More mantra.

No, I don't miss that. I clearly see it cannot be "16". Zero confusion here.

But it CAN be "1" or maybe "6". It just cannot be "16". I know it cannot be "16". I am not confused on that point.

As I was unable, evidently, to explain, SINCE IT CANNOT BE 16, then 126 must be something else. Here are some possible something else's:
12, 26, 16, 1, 2, 6

Now of this set of possible cases, I am told that, magically, only "2" works.

Right! I can memorize that. But not believe it or understand it.

Mac

No -- it can't be '1' AND it can't be '6'.

You don't HAVE to understand why it works if you don't want to. If you have four cells that reside in two columns, two rows and two boxes like this:

Code: Select all

[ab ] | [ab ]
[ab ] | [abx]


... where 'a' and 'b' are each a single candidate and 'x' is one or more candidates, you can always reduce it to this:

Code: Select all

[ab ] | [ab ]
[ab ] | [  x]


But you are asking, why can't it be this:

Code: Select all

[ab ] | [ab ]
[ab ] | [a  ]


That certainly isn't ambiguous, right? But neither is this:

Code: Select all

[ab ] | [ab ]
[ab ] | [b  ]


But that's the point in the first place! Arbitrarily filling in the cell with one of two indistinguishable possibilities doesn't eliminate the ambiguity, it demonstrates it.

But it CAN be "1" or maybe "6". It just cannot be "16". I know it cannot be "16". I am not confused on that point.

Actually you are. It cannot be a 1 or a 6 because of the underlying 2-row/2-column/2-box structure. There is no additional piece of information that you will be able to get from anywhere else within the Sudoku that will have influence on just ONE of these cells. Any additional data will effect either TWO of these cells or NONE of them. (Yes, the cell *could* be a 1 or a 6 -- if and only if it were GIVEN that it were in the original set of clues! The puzzle setter has the right to make arbitrary placements, the solver does not.)



For example:

Code: Select all

[12 ] | [12]
[12 ] | [123]


If there were a '1' (or a 2) somewhere else in row two (or column 2 or box 2), it would lead to a contradiction:

Code: Select all

[1 ] | [1 ]
[2 ] | [23]


A '1' or a '2' in any OTHER row, column or box will have NO EFFECT on these cells.

[emm]

Mac - is it clear that while theoretically the cell could be 1 or 2 or 6, it's the particular alignment of the 4 cells in this puzzle - in exactly 2 rows, 2 columns and 2 boxes - that means that 1 and 6 cannot be the candidates for all 4 cells. And the only other possibility for one of those cells is 2.

Some of these ideas we do take on faith. If we use them and find them to be valid - then we believe them.

[r.e.s.]

tso,
Thanks for taking the trouble to help explain this -- I was about to post a very similar reply.

Mac,
It seems to me there's a semantics issue that may continue to bother you until it's thoroughly put to rest -- namely, the meaning of the phrase "the candidate list cannot be '16'". For a given unsolved cell in a given valid puzzle, the point is this: there is only one correct digit for the cell, and the only subsets of {1,2,3,4,5,6,7,8,9} that cannot be candidate sets are the ones that do not contain the correct digit. So, to say "the candidate list cannot be '16'" is just to say that the correct digit is not 1 and it is not 6 -- which eliminates the other candidate sets you were wanting to consider. ('Set' is more appropriate than 'list', but I don't believe that's been the source of any confusion.)

[QBasicMac]

Thanks for all your efforts, folks.

I am going on a long flight with boring waits. I will use that time to consider your inputs.

Mac

[MCC]

Good boy. Here's a choccy drop, Now fetch the stick.

cb888, people who provide solutions usually get short shrift here.

If you'd read this through this topic you would see that Mac had already solved the puzzle and that the discussion had now turned to the property of uniqueness.

If you have any thoughts on this property then please contribute.