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`. . ab|. . ab`

. . ab|. . ab

. . . |. . .

I'd also like to call this a "closed system". Closed implies that no patterns outside this system could affect the numbers within the system, and no numbers within the system could affect any numbers outside the system. If you solve the rest of the puzzle, you are left with the rectangle and it doesn't matter what you do inside the rectangle, the rest of the puzzle remains the same.

Next, let's try to verify that this puzzle has an unique solution:

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`*-----------*`

|5..|..3|7..|

|436|.7.|2..|

|.87|.4.|3..|

|---+---+---|

|...|.62|...|

|9..|...|..3|

|...|43.|...|

|---+---+---|

|..4|.5.|83.|

|..1|.2.|965|

|..3|7..|..4|

*-----------*

With basic techniques we would get this far:

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`*-----------------------------------------------------------*`

| 5 2 9 | 168 18 3 | 7 4 168 |

| 4 3 6 | 189 7 *159 | 2 *1589 189 |

| 1 8 7 | 2 4 *569 | 3 *59 69 |

|-------------------+-------------------+-------------------|

| 3 14 58 | 189 6 2 | 45 7 189 |

| 9 46 2 | 5 18 7 | 46 18 3 |

| 7 16 58 | 4 3 19 | 56 189 2 |

|-------------------+-------------------+-------------------|

| 2 9 4 | 16 5 16 | 8 3 7 |

| 8 7 1 | 3 2 4 | 9 6 5 |

| 6 5 3 | 7 9 8 | 1 2 4 |

*-----------------------------------------------------------*

Here you can see an uniqueness rectangle in r23c68 that would let us remove candidate 9 from r2c6. However, we cannot use the uniqueness rectangle, as we don't know yet that this puzzle has only one solution.

There is also a possible reduction to be found by coloring the 9s:

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`*-----------------------------------------------------------*`

| 5 2 9 | 168 18 3 | 7 4 168 |

| 4 3 6 |+189 7 159 | 2 *1589 189 |

| 1 8 7 | 2 4 569 | 3 59 69 |

|-------------------+-------------------+-------------------|

| 3 14 58 |-189 6 2 | 45 7 +189 |

| 9 46 2 | 5 18 7 | 46 18 3 |

| 7 16 58 | 4 3 +19 | 56 -189 2 |

|-------------------+-------------------+-------------------|

| 2 9 4 | 16 5 16 | 8 3 7 |

| 8 7 1 | 3 2 4 | 9 6 5 |

| 6 5 3 | 7 9 8 | 1 2 4 |

*-----------------------------------------------------------*

This let's us remove the 9 from r2c8. Now, remember what I said about closed systems: "no patterns outside this system could affect the numbers within the system" - but we just found a pattern that allowed us to remove candidate 9 from one of the involved cells...? This tell's us that, multiple solutions or not, these particular 4 cells can not form an deadly pattern => we may remove candidate 9 from r2c6 and proceed with the x-wing in r36c68 to eliminate 9 from r3c9.

The numerical proof for this reduction can be found in the coloring-grid:

If r2c6=9 (can see a +cell) => r3c6=5 => r3c8=9 (can see a -cell)

This rule applies to all almost-deadly patterns:

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`If a reduction based on numeral logic prevents us from forming one solution of the deadly pattern, it will also prevent us from forming the other solution.`

Let's have another look at the uniqueness rectangle:

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`. . ab|. . ab`

. . ab|. . ab

. . . |. . .

There is three different ways to interfere with the pattern from the outside:

1. Place an A somewhere else in one of the involved units, has to remove both candidates a in that unit and prevents both of the possible solutions in the deadly pattern.

2. Place an B somewhere else in one of the involved units, has to remove both candidates b in that unit and prevents both of the possible solutions in the deadly pattern.

3. Place a third number C on any of the four corners, removes both candidates a and b in that cell and prevents both of the possible solutions in the deadly pattern.

The real implication that the coloring grid gave us was:

either r2c4=9 => r2c6<>9 and r2c8<>9

or r6c8=9 => r2c8<>9 and r3c8<>9

Neither of these would allow us to construct the uniqueness pattern in any form.

There is of course one more way to remove candidates: T&E.

I'll quote myself again: "no numbers within the system could affect any numbers outside the system." => If an implication made by one of the numbers inside the possible deadly pattern causes a contradiction outside the pattern, it cannot be a deadly pattern. If you have a look at the uniqueness rectangle you will see that there is no way that any implication could end up with a contradiction. The definition of a deadly pattern is that it may end up in two possible correct solutions and may not end up in a contradiction.

To summarize:

-A uniqueness reduction is made because if that reduction was not true, a deadly pattern would form (multiple solutions).

-A uniqueness reduction cannot be made if you don't know that the puzzle has an unique solution, there might be a deadly pattern in the puzzle.

-If you are able to remove a candidate inside a possible deadly pattern by use of numeral logic, the deadly pattern cannot be there, and you may make uniqueness reduction assosiated to that particular deadly pattern, even if you don't know if the puzzle has an unique solution.

Long explanation for something that doesn't have any great effect on how we solve the puzzles... but I thought I'd share it with you anyway.

Comments?

RW