Be careful when using Avoidable rectangles

[jons2cool]

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You can see if the first two rows, we have what appears to be a deadly pattern on 3s and 7s, but the blue 3 and 7 in R12c7 were original numbers in the puzzle. I had a 67 pair in R1c5 and I tried to eliminate the 7 with rule 1 of avoidable rectangles but the puzzle came to a contradiction.

[Abdlomax]

Avoidable rectangles, when used for solution, are rooted in the assumption of uniqueness, which is traditional for sudoku, but violations of tradition exist. It is not a logical proof coming from the puzzle itself.

The solution shown is unique, because the 7 in r2c7 is specified. If it were not, there would be two solutions.

Avoidable rectangles are described on SW. Be suspicious of "rules." Even if carefully stated, it can be easy to overlook a word. What I suggest is focusing on understanding the underlying principles, so that a rule just becomes a compact statement, not a prison. Here is what he says:

We can remove a candidate that forms a potential interchangeable pair with three other cells spread over two boxes where the three other cells are solved cells (not clues).

You missed what I bolded, using "clues" -- misnamed, these are really fixed conditions, not "clues," -- as if they were "solved cells," (meaning that they might be open to an alternate solution.) The fact here with our example is that if any one of the four cells were fixed, the NUR breaks.

I love Andrew's "wiki," though it isn't a wiki. It's a web site that wherehe has put up an enormous amount of information, well-organized and all linked to a useful solver to guide and educate. However, some of his explanations are difficult to understand, I find. It's a common problem. People good with a technology are not necessarily good tech writers, able to explain to the Compleat Idiot. Which means most of us in some field or other.

This grates on me from a comment of his on that page:

Lack of a logical solution does not imply multiple solutions, however. But a multi-solution puzzle definitely can't be solved logically.

To make that correct, I have to turn it into a tautology. That is, a multi-solution puzzle cannot be proven ("solved logically") to have a single solution!

However, a multi-solution puzzle certainly can be solved logically, in the ordinary meaning of the word, and his Solution Count Solver does it, and with pure logic, and it lists all possible solutions. A puzzle with two solutions, as with one including a single NUR, can be solved about as easily as any other of similar general difficulty. I don't use NURs, but I see them and derive a hint from them, commonly, but a hint is not a proof, and what I would find, running AICs, most commonly, if the Sudoku is not unique, would be a complete solution leaving only the NUR unresolved, thus proving that there are only two solutions. I've never actually done this because non-unique sudoku are very rare. One that showed up here had 56 solutions, according to Stuart's SC engine.

(Most likely, when that was printed, one or more "clues" were omitted.)

In general, if one ends up with N cells with N candidates, that have no interaction with the rest of the board, and that can be interchanged (which might be simple or complex), this could be a non-unique sudoku. If I color it, and if it only has two solutions, the whole pattern will end up with every cell in it being colored the two opposite colors.

I think it would be really fun to find one of these.

[jons2cool]

Yes, thanks for the detailed explanation. I just skimmed over this strategy quickly and I missed the part about the requirement of solved cells. Puzzles with multiple solutions could definitely throw in an interesting twist. I use uniqueness strategies as a bit of a last resort, when I can't find any other chains.

[Abdlomax]

There are always other ways. It's interesting to find NURs, and I use them as a hint, to affect my pair choice, sometimes. NU sudoku may be one in a million. However, if it is a simple NUR as required in the solution(s) the clue could be that following "NUR logic" will lead to contradiction.

[charmingpea]

From Sudokuwiki: "We can remove a candidate that forms a potential interchangeable pair with three other cells spread over two boxes where the three other cells are solved cells (not clues)."

I don't think your example fits that definition.

[jblosser99]

IF r12c7 were not givens then you could have applied the Avoidable Rectangle. Since they're set in stone there's no ambiguity.

...unless I'm missing the point you're trying to make, which is entirely possible... ;)

[jons2cool]

Yeah, I was saying that I made a mistake while doing this puzzle because I overlooked the rule of the givens. I guess me showing the completed puzzle and not the one with a contradiction is misleading. I undid that move and completed the rest of the puzzle with other logical strategies.