## Minimum steps to a trivial puzzle?

Everything about Sudoku that doesn't fit in one of the other sections

### Minimum steps to a trivial puzzle?

So I was looking at this pattern shown below.
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` *-----------* |1..|6..|8..| |.2.|.7.|.9.| |..3|..8|..1| |---+---+---| |2..|4..|9..| |.3.|.5.|.1.| |..4|..6|..2| |---+---+---| |3..|5..|7..| |.4.|.6.|.8.| |..5|..7|..9| *-----------*`

By itself, it isn't a valid sudoku. However, by replacing one of the givens many valid puzzles can be made.
Hidden Text: Show
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` *-----------* |1..|2..|8..| |.2.|.7.|.9.| |..3|..8|..1| |---+---+---| |2..|4..|9..| |.3.|.5.|.1.| |..4|..6|..2| |---+---+---| |3..|5..|7..| |.4.|.6.|.8.| |..5|..7|..9| *-----------*SE=8.9`

Hidden Text: Show
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` *-----------* |1..|6..|8..| |.2.|.7.|.9.| |..7|..8|..1| |---+---+---| |2..|4..|9..| |.3.|.5.|.1.| |..4|..6|..2| |---+---+---| |3..|5..|7..| |.4.|.6.|.8.| |..5|..7|..9| *-----------*SE=8.9`

Hidden Text: Show
Code: Select all
` *-----------* |1..|6..|8..| |.2.|.7.|.9.| |..3|..8|..1| |---+---+---| |2..|4..|9..| |.3.|.8.|.1.| |..4|..6|..2| |---+---+---| |3..|5..|7..| |.4.|.6.|.8.| |..5|..7|..9| *-----------*SE=7.3`

Hidden Text: Show
Code: Select all
` *-----------* |1..|6..|8..| |.2.|.7.|.9.| |..3|..8|..1| |---+---+---| |2..|4..|5..| |.3.|.5.|.1.| |..4|..6|..2| |---+---+---| |3..|5..|7..| |.4.|.6.|.8.| |..5|..7|..9| *-----------*SE=8.9`

I've listed a few of them above. There are many more, but I forgot to keep a list, and I don't have code to quickly find them again.

One interesting aspect of this set is the lack of trivial puzzles. (I define trivial as puzzles which can be solved through direct or hidden singles/doubles/triples/quads, pointing, and box/line reduction.)

So the question is, if two replacements were allowed instead of one, would there still be no trivial puzzles? If not what about three? If not three what is the fewest replacements required.

If it turns out that there are trivial puzzles possible with two replacements, are there other patterns for which there aren't? And what pattern would allow for the highest number of replacements without creating a trivial puzzle, but produce non-trivial puzzles for any number of replacements which are done, up to the number which produces a trivial puzzle?

I don't know, but I think it's worth looking into.
ixsetf

Posts: 50
Joined: 11 May 2014

### Re: Minimum steps to a trivial puzzle?

all guys having contributed to the pattern game faced that problem.

Starting from a non valid seed is one way among others to work in the vicinity of a set of given.

I don't think anybody as a general response, but we have many clues.

Hard puzzles are often grouped in small areas. When you catch one of them, you have good chances to find others in the near vicinity
High ed rating (diamond with high ratings for example) have more hard puzzles in their vicinity than other seeds

You can also look for specific families of puzzles if the pattern have the corresponding potential (SK loop, JExocet)

Your pattern has a high potential for hard puzzles, this can explain the rating of your 3 puzzles.
In most cases, such ratings can be found for a very small percentage of puzzles.

To answer more directly to your question, a vicinity +- 3 in most cases produces 100 to 1000 times the number of puzzles in the seed
champagne
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Location: France Brittany

### Re: Minimum steps to a trivial puzzle?

Running vicinity 1 to 3 I got 252 new puzzles (no morphing check)

Here 6 puzzle "easy"

Code: Select all
`1..9..8...5..7..9...3..8..12..4..9...3..5..1...4..6..26..5..7...4..6..8...5..7..9 ;2.3;1.2;1.21..9..8...2..7..9...3..8..12..4..3...3..8..1...4..6..23..5..7...4..6..8...5..7..9 ;2.3;1.2;1.21..6..8...2..7..9...9..8..12..4..9...3..5..1...4..6..23..5..7...1..6..8...5..7..6 ;2.3;1.2;1.21..6..8...2..7..9...3..8..12..4..9...3..5..7...4..3..23..5..7...4..6..8...5..7..6 ;2.3;1.2;1.21..6..8...2..7..9...3..8..12..4..3...3..5..1...4..6..26..5..7...4..6..2...5..7..9 ;2.3;1.2;1.21..6..8...2..4..9...9..8..12..4..9...3..5..1...4..3..23..5..7...4..6..8...5..7..9 ;2.3;1.2;1.2`
champagne
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Location: France Brittany

### Re: Minimum steps to a trivial puzzle?

ixsetf wrote:One interesting aspect of this set is the lack of trivial puzzles. (I define trivial as puzzles which can be solved through direct or hidden singles/doubles/triples/quads, pointing, and box/line reduction.)

This pattern, with a maximum number of mini-diagonals, has been discussed many years ago for just this reason, either in this or in the Programmers' Forum (no time for a search now). A slight variation was used in Patterns Game 150 (and maybe others), which reached SE 10.7.

Regards,

Mike Metcalf

m_b_metcalf
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Location: Berlin

### Re: Minimum steps to a trivial puzzle?

ixsetf wrote:If it turns out that there are trivial puzzles possible with two replacements, are there other patterns for which there aren't?

Here's a pattern for which I've been unable to find a trivial puzzle:
Code: Select all
` . 1 . . 2 . . 3 . 4 . . 5 . . 6 . . . . . . . . . . . . 3 . . 7 . . 1 . 6 . . 4 . . 8 . . . . . . . . . . . . 2 . . 1 . . 7 . 5 . . 8 . . 4 . . . . . . . . . . 5   ED=7.1/1.2/1.2`

Regards,

Mike Metcalf

m_b_metcalf
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Location: Berlin

### Re: Minimum steps to a trivial puzzle?

m_b_metcalf wrote:
Here's a pattern for which I've been unable to find a trivial puzzle:
...
Mike Metcalf

Hi Mike,

Another simple way is to consider the file of 17 clues puzzles.
In many cases, you have only one puzzle per pattern

I am sure that you have many examples of a pattern without a trivial puzzle in that file.
champagne
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Posts: 6709
Joined: 02 August 2007
Location: France Brittany

### Re: Minimum steps to a trivial puzzle?

m_b_metcalf wrote:Here's a pattern for which I've been unable to find a trivial puzzle:
Code: Select all
` . 1 . . 2 . . 3 . 4 . . 5 . . 6 . . . . . . . . . . . . 3 . . 7 . . 1 . 6 . . 4 . . 8 . . . . . . . . . . . . 2 . . 1 . . 7 . 5 . . 8 . . 4 . . . . . . . . . . 5   ED=7.1/1.2/1.2`

This pattern has 379 puzzles, all minimal.
The lowest ER is 7.1, and the highest ED is 3.8.
This was the highest rated puzzle:

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`. 1 . . 2 . . 3 .4 . . 5 . . 6 . .. . . . . . . . .. 2 . . 3 . . 1 .6 . . 7 . . 4 . .. . . . . . . . .. 8 . . 9 . . 2 .7 . . 6 . . 5 . .. . . . . . . . 9   ED=9.9/9.9/3.8`

Best Regards,
Blue.
blue

Posts: 702
Joined: 11 March 2013

### Re: Minimum steps to a trivial puzzle?

champagne wrote:Here 6 puzzle "easy"

Code: Select all
`1..9..8...5..7..9...3..8..12..4..9...3..5..1...4..6..26..5..7...4..6..8...5..7..9 ;2.3;1.2;1.21..9..8...2..7..9...3..8..12..4..3...3..8..1...4..6..23..5..7...4..6..8...5..7..9 ;2.3;1.2;1.21..6..8...2..7..9...9..8..12..4..9...3..5..1...4..6..23..5..7...1..6..8...5..7..6 ;2.3;1.2;1.21..6..8...2..7..9...3..8..12..4..9...3..5..7...4..3..23..5..7...4..6..8...5..7..6 ;2.3;1.2;1.21..6..8...2..7..9...3..8..12..4..3...3..5..1...4..6..26..5..7...4..6..2...5..7..9 ;2.3;1.2;1.21..6..8...2..4..9...9..8..12..4..9...3..5..1...4..3..23..5..7...4..6..8...5..7..9 ;2.3;1.2;1.2`

Are these all the easy puzzles within 3 vicinity of the pattern? If so that is a very small fraction of the original set.

blue wrote:This was the highest rated puzzle:

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`. 1 . . 2 . . 3 .4 . . 5 . . 6 . .. . . . . . . . .. 2 . . 3 . . 1 .6 . . 7 . . 4 . .. . . . . . . . .. 8 . . 9 . . 2 .7 . . 6 . . 5 . .. . . . . . . . 9   ED=9.9/9.9/3.8`

Best Regards,
Blue.

I notice that this puzzle has some similarities with the pattern in my original post. It might not be immediately obvious how, so I flipped the two leftmost columns of each box and reassigned the numbers.
Hidden Text: Show
Code: Select all
`+---+---+---+|1..|7..|4..||.2.|.8.|.6.||...|...|...|+---+---+---+|7..|4..|1..||.6.|.5.|.2.||...|...|...|+---+---+---+|3..|9..|7..||.5.|.6.|.8.||...|...|..9|+---+---+---+`

This has exactly the same properties as blue's puzzle.

If you think of the set as three bands running diagonally and wrapping around the edges, numbered sequentially along each band, you can adjust the start of each band so that 1's are placed at r1c1, r4c7 and r7c4 you are within vicinity 5 of a subset of the pattern, and if you discount the band starting at r7c4 you are in vicinity 1.
ixsetf

Posts: 50
Joined: 11 May 2014

### re: diagonality

m_b_metcalf wrote:
This pattern, with a maximum number of mini-diagonals,
has been discussed many years ago---

A slight variation was used in Patterns Game 150 (and maybe others)

this was one of the discussions

some examples in the Patterns Game --game 98 was also highly-diagonal, with 27 + 1;
and see discussion starting here at the end of that game

Pat

Posts: 3674
Joined: 18 July 2005

### Re: Minimum steps to a trivial puzzle?

ixsetf wrote:Are these all the easy puzzles within 3 vicinity of the pattern? If so that is a very small fraction of the original set.

yes they are
champagne
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Joined: 02 August 2007
Location: France Brittany

puzzles which only need "basic" moves
(what ixsetf calls "trivial") --

906 only need "singles" {not attached}
1,806 "basic" moves, beyond "singles"
out of total 14,442
* i did not check for minimality

* surely includes many isomorphs
Attachments
BHT.txt
1,806

Pat

Posts: 3674
Joined: 18 July 2005

### Re:

Pat wrote:

puzzles which only need "basic" moves
(what ixsetf calls "trivial") --

906 only need "singles" {not attached}
1,806 "basic" moves, beyond "singles"
out of total 14,442
* i did not check for minimality

* surely includes many isomorphs

I appreciate the effort, everything I've tested so far does appear to be simple valid and related to the original. However, I noticed there are a number of duplicates in your file. For example,

1..6..8...2..7..9...3..8..12..4..9...3..5..1...4..6..26..5..7...4..6..2...5..7..9

appears 22 times in the attachment.
ixsetf

Posts: 50
Joined: 11 May 2014

sorry !!
quick-and-dirty
was too dirty

did not check for minimality,
never looked for isomorphs,
but duplicates ??
that's just carelessness
EDIT : got a little better (no duplicates),
but came up short,
reporting total "6,219" (mutating any 1-3 of the given cells),
should be 12 + 237 + 7,824 = 8,073
thus --

1. mutate any 1 of the given cells:
the 4 puzzles you posted + 8 more = 12 (all are non-minimal): Show
Code: Select all
`1..6..8...5..7..9...3..8..12..4..9...3..5..1...4..6..23..5..7...4..6..8...5..7..9   _r2c2   7.31..6..8...2..4..9...3..8..12..4..9...3..5..1...4..6..23..5..7...4..6..8...5..7..9   _r2c5   7.31..6..8...2..7..9...3..8..12..4..9...3..8..1...4..6..23..5..7...4..6..8...5..7..9   _r5c5   7.31..6..8...2..7..9...3..8..12..4..9...3..5..7...4..6..23..5..7...4..6..8...5..7..9   _r5c8   7.31..6..8...2..7..9...3..8..12..4..9...3..5..1...4..6..23..5..7...1..6..8...5..7..9   _r8c2   7.31..6..8...2..7..9...3..8..12..4..9...3..5..1...4..6..23..5..7...4..6..2...5..7..9   _r8c8   7.31..2..8...2..7..9...3..8..12..4..9...3..5..1...4..6..23..5..7...4..6..8...5..7..9   _r1c4   8.91..6..8...2..7..9...7..8..12..4..9...3..5..1...4..6..23..5..7...4..6..8...5..7..9   _r3c3   8.91..6..8...2..7..9...3..8..12..4..5...3..5..1...4..6..23..5..7...4..6..8...5..7..9   _r4c7   8.91..6..8...2..7..9...3..8..12..4..9...3..5..1...4..1..23..5..7...4..6..8...5..7..9   _r6c6   8.91..6..8...2..7..9...3..8..12..4..9...3..5..1...4..6..28..5..7...4..6..8...5..7..9   _r7c1   8.91..6..8...2..7..9...3..8..12..4..9...3..5..1...4..6..23..5..7...4..6..8...5..7..4   _r9c9   8.9`

2. mutate any 2 of the given cells:
237 more puzzles (here too, all are non-minimal):
• 12 only need "singles": Show
Code: Select all
`1..3..8...2..7..9...3..8..52..4..9...3..5..1...4..6..23..5..7...4..6..8...5..7..9   _r1c4_r3c95..6..4...2..7..9...3..8..12..4..9...3..5..1...4..6..23..5..7...4..6..8...5..7..9   _r1c1_r1c71..6..4...2..7..9...6..8..12..4..9...3..5..1...4..6..23..5..7...4..6..8...5..7..9   _r1c7_r3c31..6..8...2..7..9...3..4..52..4..9...3..5..1...4..6..23..5..7...4..6..8...5..7..9   _r3c6_r3c91..6..8...2..7..9...3..8..17..8..9...3..5..1...4..6..23..5..7...4..6..8...5..7..9   _r4c1_r4c41..6..8...2..7..9...3..8..17..4..9...3..5..1...4..9..23..5..7...4..6..8...5..7..9   _r4c1_r6c61..6..8...2..7..9...3..8..12..4..6...3..5..1...8..6..23..5..7...4..6..8...5..7..9   _r4c7_r6c31..6..8...2..7..9...3..8..12..4..9...3..5..1...8..6..73..5..7...4..6..8...5..7..9   _r6c3_r6c91..6..8...2..7..9...3..8..12..4..9...3..5..1...4..6..29..5..7...4..6..8...5..2..9   _r7c1_r9c61..6..8...2..7..9...3..8..12..4..9...3..5..1...4..6..23..1..2...4..6..8...5..7..9   _r7c4_r7c71..6..8...2..7..9...3..8..12..4..9...3..5..1...4..6..23..1..7...4..6..8...5..7..3   _r7c4_r9c91..6..8...2..7..9...3..8..12..4..9...3..5..1...4..6..23..5..7...4..6..8...1..2..9   _r9c3_r9c6`
• 42 solved by "basic" moves (beyond "singles"): Show
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`1..9..8...5..7..9...3..8..12..4..9...3..5..1...4..6..23..5..7...4..6..8...5..7..9   _r1c4_r2c21..3..8...2..7..9...3..8..12..4..9...3..5..1...4..6..23..9..7...4..6..8...5..7..9   _r1c4_r7c41..6..3...2..7..9...3..8..12..4..6...3..5..1...4..6..23..5..7...4..6..8...5..7..9   _r1c7_r4c71..6..8...9..7..5...3..8..12..4..9...3..5..1...4..6..23..5..7...4..6..8...5..7..9   _r2c2_r2c81..6..8...2..9..4...3..8..12..4..9...3..5..1...4..6..23..5..7...4..6..8...5..7..9   _r2c5_r2c81..6..8...2..4..9...9..8..12..4..9...3..5..1...4..6..23..5..7...4..6..8...5..7..9   _r2c5_r3c31..6..8...2..7..9...6..8..12..4..9...3..5..1...9..6..23..5..7...4..6..8...5..7..9   _r3c3_r6c31..6..8...2..7..9...3..8..62..4..9...3..5..1...4..6..23..5..7...4..6..8...5..7..3   _r3c9_r9c91..6..8...2..7..9...3..8..16..4..9...3..5..1...4..6..29..5..7...4..6..8...5..7..9   _r4c1_r7c11..6..8...2..7..9...3..8..12..4..3...3..8..1...4..6..23..5..7...4..6..8...5..7..9   _r4c7_r5c51..6..8...2..7..9...3..8..12..4..9...8..3..1...4..6..23..5..7...4..6..8...5..7..9   _r5c2_r5c51..6..8...2..7..9...3..8..12..4..9...7..5..3...4..6..23..5..7...4..6..8...5..7..9   _r5c2_r5c81..6..8...2..7..9...3..8..12..4..9...3..5..7...4..3..23..5..7...4..6..8...5..7..9   _r5c8_r6c61..6..8...2..7..9...3..8..12..4..9...3..5..1...4..9..23..5..7...4..6..8...5..3..9   _r6c6_r9c61..6..8...2..7..9...3..8..12..4..9...3..5..1...4..6..26..5..7...4..6..2...5..7..9   _r7c1_r8c81..6..8...2..7..9...3..8..12..4..9...3..5..1...4..6..23..5..7...6..1..8...5..7..9   _r8c2_r8c51..6..8...2..7..9...3..8..12..4..9...3..5..1...4..6..23..5..7...1..6..8...5..7..6   _r8c2_r9c91..6..8...2..7..9...3..8..12..4..9...3..5..1...4..6..23..5..7...4..2..6...5..7..9   _r8c5_r8c89..6..8...2..7..9...3..8..12..4..9...3..5..1...9..6..23..5..7...4..6..8...5..7..9   _r1c1_r6c31..9..8...2..7..9...3..8..12..4..9...3..5..7...4..6..23..5..7...4..6..8...5..7..9   _r1c4_r5c81..6..3...2..7..9...3..8..12..4..9...3..5..1...4..6..33..5..7...4..6..8...5..7..9   _r1c7_r6c91..6..8...5..7..9...3..8..12..4..9...3..5..1...4..3..23..5..7...4..6..8...5..7..9   _r2c2_r6c61..6..8...2..4..9...3..8..12..4..9...3..5..1...4..6..26..5..7...4..6..8...5..7..9   _r2c5_r7c11..6..8...2..7..9...9..8..12..4..9...3..5..1...4..6..23..5..7...4..6..2...5..7..9   _r3c3_r8c81..6..8...2..7..9...3..9..12..4..9...3..5..1...4..6..23..9..7...4..6..8...5..7..9   _r3c6_r7c41..6..8...2..7..9...3..8..62..4..9...3..5..1...4..6..23..5..6...4..6..8...5..7..9   _r3c9_r7c71..6..8...2..7..9...3..8..16..4..9...3..5..1...4..6..23..5..7...4..6..8...6..7..9   _r4c1_r9c31..6..8...2..7..9...3..8..12..3..9...3..5..1...4..6..23..5..7...4..6..8...5..3..9   _r4c4_r9c61..6..8...2..7..9...3..8..12..4..3...3..5..1...4..6..23..5..7...1..6..8...5..7..9   _r4c7_r8c21..6..8...2..7..9...3..8..12..4..9...3..8..1...4..6..23..5..7...4..6..8...5..7..6   _r5c5_r9c97..6..8...2..7..9...3..8..11..4..9...3..5..1...4..6..23..5..7...4..6..8...5..7..9   _r1c1_r4c11..6..7...2..7..9...3..8..12..4..9...3..5..1...4..6..23..5..4...4..6..8...5..7..9   _r1c7_r7c71..6..8...2..7..9...3..2..12..4..9...3..5..1...4..6..23..5..7...4..6..8...5..8..9   _r3c6_r9c61..6..8...2..7..9...3..8..22..4..9...3..5..1...4..6..53..5..7...4..6..8...5..7..9   _r3c9_r6c91..6..8...2..7..9...3..8..12..1..9...3..5..1...4..6..23..4..7...4..6..8...5..7..9   _r4c4_r7c41..6..8...2..7..9...3..8..12..4..9...3..5..1...5..6..23..5..7...4..6..8...8..7..9   _r6c3_r9c31..6..3...2..7..9...3..8..12..4..9...3..5..1...4..6..23..5..7...4..6..8...5..2..9   _r1c7_r9c61..6..4...2..7..9...3..8..12..4..9...3..5..1...4..6..23..5..7...4..6..8...5..3..9   _r1c7_r9c61..6..8...2..7..9...3..8..56..4..9...3..5..1...4..6..23..5..7...4..6..8...5..7..9   _r3c9_r4c11..6..8...2..7..9...3..8..67..4..9...3..5..1...4..6..23..5..7...4..6..8...5..7..9   _r3c9_r4c11..6..8...2..7..9...3..8..12..4..9...3..5..1...8..6..23..9..7...4..6..8...5..7..9   _r6c3_r7c41..6..8...2..7..9...3..8..12..4..9...3..5..1...9..6..23..1..7...4..6..8...5..7..9   _r6c3_r7c4`
• 6 solved by simple fish: Show
Code: Select all
`1..6..8...2..7..6...3..8..12..4..5...3..5..1...4..6..23..5..7...4..6..8...5..7..9   _r2c8_r4c71..2..8...2..7..9...3..8..12..4..9...3..5..1...4..6..23..5..7...4..3..8...5..7..9   _r1c4_r8c51..6..8...2..7..3...3..8..12..4..9...3..5..1...4..6..23..5..7...4..6..8...5..7..4   _r2c8_r9c91..6..8...2..7..9...7..8..12..4..9...6..5..1...4..6..23..5..7...4..6..8...5..7..9   _r3c3_r5c21..6..8...2..7..9...3..8..12..4..9...9..5..1...4..6..28..5..7...4..6..8...5..7..9   _r5c2_r7c11..6..8...2..7..9...3..8..12..4..9...3..5..1...4..1..23..5..7...4..9..8...5..7..9   _r6c6_r8c5`
• 177 tougher
3. mutate any 3 of the given cells:
7,824 more puzzles (including 252 minimal, as already reported by champagne)