What is the smallest minimal puzzle?

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

What is the smallest minimal puzzle?

Just to be clear, this puzzle:
Code: Select all
`+---+---+---+|...|..2|..8||..2|7..|1..||.6.|.9.|.2.|+---+---+---+|.1.|4..|..9||..9|.6.|7..||3..|...|.5.|+---+---+---+|.2.|.5.|.4.||..4|..8|5..||5..|1..|..3|+---+---+---+`
has the Lex representation:
Code: Select all
`000002008002700100060090020010400009009060700300000050020050040004008500500100003`

or:
Code: Select all
`.....2..8..27..1...6..9..2..1.4....9..9.6.7..3......5..2..5..4...4..85..5..1....3`

where . stands for 0.

What is the smallest (valid) minimal puzzle? i.e. having the minimum Lex representation?

JPF
JPF
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Re: What is the smallest minimal puzzle?

quick try
Code: Select all
`+---+---+---+|...|...|...||...|...|..1||...|..2|...|+---+---+---+|..1|.3.|..4||..5|...|..6||.4.|7..|1.8|+---+---+---+|.69|..7|..5||.2.|..9|.8.||1.4|82.|.93|+---+---+---+`

Code: Select all
`.................1.....2.....1.3...4..5.....6.4.7..1.8.69..7..5.2...9.8.1.482..93`
coloin

Posts: 2004
Joined: 05 May 2005

Re: What is the smallest minimal puzzle?

Hi coloin,

I don't know the exact answer, but I will try to post better ones as long as I can...

Code: Select all
`.................1.....2.....1.3...4..5.....6.4.7..1.8.69..7..5.2...9.8.1.482..93    coloin   .................1.....2.....1.....3..2..3.4...4..562..1..57.8..39.84..6.47.2...9    JPF      `

JPF
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Re: What is the smallest minimal puzzle?

JPF wrote:What is the smallest (valid) minimal puzzle? i.e. having the minimum Lex representation?

So we are just looking at the set of all minimal puzzles in minlex form, without any requirement for them to solve to minlex grids?

Mathimagics
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Re: What is the smallest minimal puzzle?

Mathimagics wrote:So we are just looking at the set of all minimal puzzles in minlex form, without any requirement for them to solve to minlex grids?

right!

JPF
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Re: What is the smallest minimal puzzle?

Ok, how's this?

Code: Select all
`.................1.....1.2...1.....3..2..4.....4..56...5..7..4..6.3..5...8.9.....`

Mathimagics
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Re: What is the smallest minimal puzzle?

edited

Code: Select all
`.................1.....1.2...1.....3..2..4.....4..56...5..7..4..6.3..5...8.9.....      Mathimagics.................1.....1..2..1.3...4.5..6..7..768......18.5..2..9..8.3..5..2.361.      JPF1.................1.....1..2..1..2..3..2....4...567.....18.23.54.24..6....3..18.96      JPF2`

Code: Select all
`+---+---+---+|...|...|...||...|...|..1||...|..1|..2|+---+---+---+|..1|..2|..3||..2|...|.4.||..5|67.|...|+---+---+---+|.18|.23|.54||.24|..6|...||.3.|.18|.96|    JPF2+---+---+---+`

JPF
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Re: What is the smallest minimal puzzle?

a new one:
Code: Select all
`.................1.....1..2..1..2..3..2....4...3.561.7.14..3..8.25.18.7..39.2.5..`

Code: Select all
`+---+---+---+|...|...|...||...|...|..1||...|..1|..2|+---+---+---+|..1|..2|..3||..2|...|.4.||..3|.56|1.7|+---+---+---+|.14|..3|..8||.25|.18|.7.||.39|.2.|5..|   JPF3+---+---+---+`

JPF
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Re: What is the smallest minimal puzzle?

last improvement :
Code: Select all
`.................1.....1..2..1..2..3..2....4...3.45..6.15.7..8..26.5..9..3.8.6.1.`

Code: Select all
`+---+---+---+|...|...|...||...|...|..1||...|..1|..2|+---+---+---+|..1|..2|..3||..2|...|.4.||..3|.45|..6|+---+---+---+|.15|.7.|.8.||.26|.5.|.9.||.3.|8.6|.1.|    JPF4+---+---+---+`

JPF
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Re: What is the smallest minimal puzzle?

Its funny - that we havent done this exercise before ....and it depends on the way you fill in the first 2 bands ...
JPF wrote:last improvement : ....

And by that comment I thought it was definitive - and that you had done an exhaustive search after the first 12 clues.... !
Gridchecker checked ...
Maybe it is the smallest minlex puzzle in "2.55 × 10^25 non-essentially-equivalent minimal puzzles".

Code: Select all
`.................1.....1..2..1..2..3..2....4...3.45..6.1...3.8..25.7..9..36.54.1.`

Code: Select all
`+---+---+---+|...|...|...||...|...|..1||...|..1|..2|+---+---+---+|..1|..2|..3||..2|...|.4.||..3|.45|..6|+---+---+---+|.1.|..3|.7.||.25|.8.|.9.||.36|.54|.1.|+---+---+---+   `
coloin

Posts: 2004
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Re: What is the smallest minimal puzzle?

you wrote:And by that comment I thought it was definitive - and that you had done an exhaustive search after the first 12 clues.... !

Actually, I did it "by hand" for all the bands.
I still don't see obvious reasons why the following patterns would be impossible:
Code: Select all
`+---+---+---+   +---+---+---+|...|...|...|   |...|...|...||...|...|..1|   |...|...|..1||...|..1|..2|   |...|..1|..2|+---+---+---+   +---+---+---+|...|xxx|xxx|   |..1|...|xxx||xxx|xxx|xxx|   |xxx|xxx|xxx||xxx|xxx|xxx|   |xxx|xxx|xxx|+---+---+---+   +---+---+---+|xxx|xxx|xxx|   |xxx|xxx|xxx||xxx|xxx|xxx|   |xxx|xxx|xxx||xxx|xxx|xxx|   |xxx|xxx|xxx|+---+---+---+   +---+---+---+`
hidden UAs ?

JPF
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Re: What is the smallest minimal puzzle?

Hi, JPF!
JPF wrote:I still don't see obvious reasons why the following patterns would be impossible:
Code: Select all
`+---+---+---+   +---+---+---+|...|...|...|   |...|...|...||...|...|..1|   |...|...|..1||...|..1|..2|   |...|..1|..2|+---+---+---+   +---+---+---+|...|xxx|xxx|   |..1|...|xxx||xxx|xxx|xxx|   |xxx|xxx|xxx||xxx|xxx|xxx|   |xxx|xxx|xxx|+---+---+---+   +---+---+---+|xxx|xxx|xxx|   |xxx|xxx|xxx||xxx|xxx|xxx|   |xxx|xxx|xxx||xxx|xxx|xxx|   |xxx|xxx|xxx|+---+---+---+   +---+---+---+`

The first pattern is subset of P123 maximal invalid pattern:
Code: Select all
`        P123+-----+-----+-----+|. . .|. x x|x x x||. . .|. x x|x x x||. . .|. x x|x x x|+-----+-----+-----+|. . .|x x x|x x x||. . .|x x x|x x x||x x x|x x x|x x x|+-----+-----+-----+|. . .|x x x|x x x||. . .|x x x|x x x||x x x|x x x|x x x|+-----+-----+-----+`

(See List of 40 maximal pattern.)

Serg
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Re: What is the smallest minimal puzzle?

Yes ....that would explain that !!!

But a better one has emerged ...

Code: Select all
`.................1.....1..2..1..2..3..2....4...3.45..6.1...3.7..25.6..8..367.4.19`

Code: Select all
`+---+---+---+  |...|...|...|        |...|...|..1|        |...|..1|..2|        +---+---+---+        |..1|..2|..3|        |..2|...|.4.|        |..3|.45|..6|        +---+---+---+        |.1.|..3|.7.|        |.25|.6.|.8.|        |.36|7.4|.19|        +---+---+---+ `
coloin

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Joined: 05 May 2005