The New Sudoku Players' Forum

[Nick67]

Imagine that you are solving a valid puzzle, and
find this situation:

Code: Select all

 123 . . |123 . . |123   . .
 123 . . |123 . . |123   . .
 123 . . |123 . . |1234  . .
 --------+--------+---------
 .   . . | .  . . | .    . .
 .   . . | .  . . | .    . .
 .   . . | .  . . | .    . .
 --------+--------+---------
 .   . . | .  . . | .    . .
 .   . . | .  . . | .    . .
 .   . . | .  . . | .    . .


You could apply a variation of the usual uniqueness logic here.
Assume 4 is not the final value of r3c7. Then, in a solution to the puzzle,
each of the cells c1r1-3, c4r1-3, and c7r1-3 will contain a 1, 2 or 3.
After finding a solution, in those 9 cells you could change
each 1 to a 2, each 2 to a 3, and each 3 to a 1, and
you'd have a second solution. But this contradicts
the assumption that the puzzle is valid. So the
assumption that 4 is not the final value of r3c7
must be wrong.

So, you could enter 4 as the final value
for that cell, and then continue solving the puzzle.

The trouble is, I can't seem to create a puzzle with
a unique solution that has the candidate configuration
pictured above.

So, I thought I'd ask the forum: can anyone come up with such a puzzle?
Or one in which there is a different "uniqueness pattern"
in which each cell in the pattern has at least 3 candidates?

Thanks in advance,
Nick

P.S. I got the idea for the puzzle above directly from posts by tso
and Myth Jellies. (So ... not really that much creative thinking
on my part!)

[Myth Jellies]

Seems that anything like this will lead to that uniqueness pattern

Code: Select all

 .   .   . | .   .   . | .   .   .
 .   .   . | .   .   . | .   .   .
 .   .   . | .   .   . | .   .   .
-----------+-----------+-----------
 .   1   . | .   2   . | .   3   .
 .   2   . | .   3   . | .   1   .
 .   3   . | .   1   . | .   2   .
-----------+-----------+-----------
 .   .   1 | .   .   2 | .   .   3
 .   .   2 | .   .   3 | .   .   1
 .   .   3 | .   .   1 | .   .   2


Therefore you would need something like this to escape it:

Code: Select all

 .   .   . | .   .   . | .   .   .
 .   .   . | .   .   . | .   .   .
 .   .   . | .   .   . | .   .   .
-----------+-----------+-----------
 .   1X  . | .   2   . | .   3   .
 .   2   . | .   3   . | .   1   .
 .   3   . | .   1   . | .   2   .
-----------+-----------+-----------
 .   .   1 | .   .   2 | .   .   3
 .   .   2 | .   .   3 | .   .   1
 .   .   3 | .   .   1 | .   .   2


Haven't worked out exactly how many NX choices (where N = 1, 2, or 3; and X is not N) you need to actually get valid relief from the uniqueness pattern result.

[Lummox JR]

All "uniqueness grid" tests require N digits, N columns, N rows, N boxes. The loop techniques allow you to branch out and just have up to N choices in each column, row, and box in the loop. What Nick posted is a 3x3 grid.

[Myth Jellies]

To my way of thinking, the uniqueness pattern in question is really

Code: Select all

 123 . . |123 . . |123   . .
 123 . . |123 . . |123   . .
 123 . . |123 . . |123   . .
 --------+--------+---------
 .   . . | .  . . | .    . .
 .   . . | .  . . | .    . .
 .   . . | .  . . | .    . .
 --------+--------+---------
 .   . . | .  . . | .    . .
 .   . . | .  . . | .    . .
 .   . . | .  . . | .    . .


There are several different ways that you can use uniqueness patterns in solving puzzles. One way is to perturb a uniqueness pattern with additions, and then realize that the ultimate solution comes from those additions, which is what Nick67 did with his extra 4 in r3c7.

Another way of using a uniqueness pattern is by recognizing a puzzle state which forces the uniqueness pattern, and then perturbing that forcing puzzle state by offering options that no longer force the uniqueness pattern. Thus the solution comes from the non-forcing options. That is what I was attempting.

As the uniqueness patterns get more complex, sometimes the second method of use may become easier to spot, and it may also become easier to design into a puzzle.

[Nick67]

Unfortunately, I found that I can prove that the candidate
configuration in my example puzzle (shown again below)
is impossible. The proof is below.

But! Maybe someone can find a different uniqueness pattern
in which each cell in the pattern has at least 3 candidates,
and a valid puzzle in which the pattern occurs?

Basically, I am wondering if anyone can find the 3-digit
analogue of the "unique rectangle."

Anyway, here is that proof:

Consider any puzzle with this candidate configuration:

Code: Select all

 123 . . |123 . . |123   . .
 123 . . |123 . . |123   . .
 123 . . |123 . . |1234  . .
 --------+--------+---------
 .   . . | .  . . | .    . .
 .   . . | .  . . | .    . .
 .   . . | .  . . | .    . .
 --------+--------+---------
 .   . . | .  . . | .    . .
 .   . . | .  . . | .    . .
 .   . . | .  . . | .    . .


Let's start in box 3. In the final solution, there will be
a 1,2, or 3 in r1c7 and r2c7. There can't be a 1,2, or 3 in
r1c8, r1c9, r2c8, or r2c9 due to the naked triples in
rows 1 and 2. So, inside box 3, there will be a 1, 2, or 3
in each of rows 1, 2, and 3.

Then, no matter the order of the 1,2, and 3 in box 3
in the final solution, we can swap r1c1 with r1c4, r2c1 with r2c4,
and r3c1 with r3c4, and we will have a second solution.

So ... we can conclude that there is no puzzle with a unique
solution that has the candidate configuration shown above.

[Myth Jellies]

In other words, your single 4 was not enough of a perturbation of the uniqueness grid to allow a unique solution. It might be interesting to find out how many 4's, and perhaps other digits, you would need, and where to place them to allow for a unique solution.

[Nick67]

Myth Jellies,

I think that is an excellent way to look at the situation. I will
think about it more, in this way.

Also, I agree with your comments about the use of the phrase
"uniqueness pattern". Based on your comments, here is
a potential definition for the phrase:

uniqueness pattern - a pattern of candidates such that, if the associated
cells contain only the candidates in the pattern, then the puzzle must
have multiple solutions.

[Edit: I should add that MadOverlord has used the phrase "deadly pattern"
to describe the same thing.]

[Myth Jellies]

Heh heh, yeah, that's the way I have used it. After seeing your definition, though, it seems to be misleading. Perhaps it should remain a multi-solution (or deadly) pattern and we can leave uniqueness patterns for the various types of perturbations built onto the multi-solution pattern as defined previously.