On the Confluence of Uniqueness Theories

Advanced methods and approaches for solving Sudoku puzzles

Postby David P Bird » Mon Sep 22, 2008 5:31 am

[Edited post]

[Original Statement]
If we assumed uniqueness in making our first exclusion in an AUR, we can't continue as if we've locally proved it, which is what the "un-deadly pattern" deduction requires.

[Edit]
In fact, as RW's response below details, this is of no consequence because as the solution progresses, identical exclusions will follow regardless of whether uniqueness was originally assumed or not.

The rest of the original post has therefore been deleted as being misleading.
Last edited by David P Bird on Mon Sep 22, 2008 9:24 am, edited 1 time in total.
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Postby RW » Mon Sep 22, 2008 10:54 am

David P Bird wrote:If we assumed uniqueness in making our first exclusion in an AUR, we can't continue as if we've locally proved it, which is what the "un-deadly pattern" deduction requires.

Yes, this is correct. However, if uniqueness is assumed when making the first elimination, I don't see why that assumption should be challenged at a later stage of solving the puzzle. If you suddenly suspect that the puzzle might have multiple solutions, then you'd have to go back and undo all uniqueness eliminations anyway.

My previous post was purely a theoretical answer to ronks question "Can this pattern be the sole reason for multiple solutions". To answer that question, we'll have to examine all possible ways 'b' might have been eliminated. In practise, I don't think we need to consider this issue, because the dangers in the AR are no greater than any potential risks we've taken before. If b is eliminated by uniqueness technique, then eliminating a is also an uniqeness elimination. If b wasn't eliminated by uniqueness technique, then eliminating a is not an uniqueness elimination.

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Postby David P Bird » Mon Sep 22, 2008 1:26 pm

RW Thanks for the correction. You're absolutely right. I've therefore pulled most of my previous post as I don't want to lead the innocent astray.
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Postby ronk » Tue Sep 23, 2008 2:03 pm

udosuk wrote:Scenario 2: the puzzle has multiple solutions. However suppose the 4 cells are in the pattern [ab]+[ba]. And this pattern exists in one of the possible solutions, let's call it solution grid S1. Without changing the other 77 cells of S1, we change these 4 cells into [ba]+[ab]. With regard to the original given clues, this new solution grid, S2, must also be a legitimate solution to the original puzzle.

But remember earlier we have used other logical means to eliminate candidate b from the 4th cell. Thus S2 must not be a legal solution to the original puzzle.

In my book, since you've based S2 on the unavoidable set in S1, you've based the deduction on uniqueness. It's not the uniqueness test known as UR Type 1, but the logic definitely uses a uniqueness property.
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Postby udosuk » Tue Sep 23, 2008 11:20 pm

ronk wrote:In my book, since you've based S2 on the unavoidable set in S1, you've based the deduction on uniqueness. It's not the uniqueness test known as UR Type 1, but the logic definitely uses a uniqueness property.

In your book, the terms "unavoidable set" & "deadly pattern" must be tightly glued to the word "uniqueness", so for any technique/logic that makes use of similar settings of these concepts, you'll refer it as "uniqueness technique".

In my book, you're using a "uniqueness technique" if you assume the puzzle has a unique solution. If you don't assume uniqueness you're not using a "uniqueness technique", no matter how similar your logic is to a uniqueness-based move.

I don't see in anywhere in my scenario 2 that I based my logic on the puzzle having a unique solution, nor did I use any "uniqueness property" there.
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Postby Myth Jellies » Wed Sep 24, 2008 12:06 am

Perhaps another way of putting it is both deduction patterns rely on properties of multi-solution patterns; but only one of them requires the additional assumption that the puzzle has a unique solution in order to eliminate candidates.
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Postby Pat » Thu Sep 25, 2008 8:09 am

on page 3, RW wrote:You don't even need a solved cell,
it's enough is one of the candidates in the potential deadly pattern is eliminated --

Code: Select all
 ab  |  ab
 ab  |  axyz


b is already eliminated from the 4th cell
=> we may eliminate a as well
without assuming the puzzle has one answer



my way of proving it, is by asking the question --
why should we retain the a possibility in the 4th cell ?
    we'd retain it
    in case there's an answer with a in the 4th cell

    but such an answer
    would solve the 4 cells thus --

    Code: Select all
     a  |  b
     b  |  a


    -- and since none of those 4 cells was a Given (clue),
    we could swap the arrangement
    (leaving all other cells unchanged),
    and get an extra answer
    with this arrangement in the 4 cells --

    Code: Select all
     b  |  a
     a  |  b


    -- therefore our earlier exclusion of b in the 4th cell was wrong
      this probably means
      that the puzzle has more than one answer
      and we had earlier chosen to assume Uniqueness-Of-Answer,
      thus making some unjustified exclusions

      or possibly we had made some mistake
    makes no difference how it happened --
    our earlier exclusion of b in the 4th cell was wrong,
    we're already in trouble
    and heading for some disaster,
    might as well exclude the a in the 4th cell
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Postby champagne » Thu Sep 25, 2008 8:18 am

Hi all,

When I introduced that small example, I would never have thought it would generate so many comments.:D


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Postby Allan Barker » Thu Sep 25, 2008 9:57 pm

I think the following may have bearing on the subject of this thread.

While working out a solution to loop 2 of the puzzle Fata Morgana Illusion of Fata Morgana, I noticed that the core of the logic seems to make a deadly/unique rectangle, core refers to the six cells in the center band that contain candidates (136). Using pure sudoku rule-1 logic, this core produces a pattern of several SIS that combine in a way that leads to the eliminations.

I guess my question is first, this is a UR/DR/AUR, right? second, how does one enumerate strong inference sets related to the UR? By my set way of thinking, this UR should have enough "uniqueness power" to force 2 candidates outside to be true, however I have little practical UR experience.
Code: Select all
Elimination logic, no eliminates included
   +--------------------------------------------------+
  |       6    6  |               |         1        |
  | 3     3       |               |                  |
  | 3          6  |  136      136 |              1   |
  +--------------------------------------------------+
  |       136     |               |                  |
  |       136  6  |  136      136 |  3     136       |
  |               |               |        136       |
  +--------------------------------------------------+
  |  1            |  136      136 |  3           6   |
  |               |               |        6     6   |
  |       1       |               |  3     3         |
  +--------------------------------------------------+

Full Pencil Grid:
  +-----------------------------------------------------------------------------+
  | 2458    247(6)  245678  | 126789  16789   1678    | 24589   248(1)  3       |
  | 2348    247(3)  1       | 23789   3789    5       | 6       248     249     |
  | 258(3)  9       258(6)  | 28(136) 4       8(136)  | 258     7       25(1)   |
  +-----------------------------------------------------------------------------+
  | 12348   (136)   2468    | 13678   13678   9       | 2347    5       12467   |
  | 7       24(136) 2469    | (136)   5       (136)   | 2349    24(136) 8       |
  | 1389    5       689     | 4       13678   2       | 379     (136)   1679    |
  +-----------------------------------------------------------------------------+
  | 45(1)   8       457     | 7(136)  2       47(136) | 457(3)  9       457(6)  |
  | 249     247     3       | 5       6789    4678    | 1       248(6)  2467    |
  | 6       247(1)  24579   | 13789   13789   13478   | 234578  248(3)  2457    |
  +-----------------------------------------------------------------------------+

Of course, the pie in the sky would be a common logical framework that smoothly integrates URs to the rest of Sudoku logic.

If this turns out to be better placed in th FM thread, I can drag it over there leaving a reference here.
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Postby StrmCkr » Thu Sep 25, 2008 10:59 pm

no allan that is right on topic.

i have been building 9-10+ er rated puzzles based on

hidden linked fish (3x same arrangment patterns occuping the same cells.

they form a deadly pattern in them selves exactly as you have dipicted.

notes:
4X fish patterns can do the same thing as well.
Some do, some teach, the rest look it up.
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Postby Myth Jellies » Fri Sep 26, 2008 1:39 am

You might wish to bone up on MUGs

Code: Select all
   +--------------------------------------------------+
  |               |               |                  |
  |               |               |                  |
  |               |  136      136 |                  |
  +--------------------------------------------------+
  |               |               |                  |
  |               |  136      136 |                  |
  |               |               |                  |
  +--------------------------------------------------+
  |               |  136      136 |                  |
  |               |               |                  |
  |               |               |                  |
  +--------------------------------------------------+

This is a permeable MUG deadly pattern that will require at least four cells in r357c1235789 to solve to 1, 3, and/or 6. Same thing can be said for cells in b258 outside of the pattern.
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