## Hard 5*5 Latin Squares

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

### Hard 5*5 Latin Squares

5*5 is the smallest size for Latin Squares to contain chains, and I think it is small enough for us to find out all puzzles requiring chains.
I used JSudoku to generate such puzzles(setting 1+ guess) and got several non-equivalent puzzles. I wonder if there are more.
qiuyanzhe

Posts: 89
Joined: 21 August 2017
Location: China

### Re: Hard 5*5 Latin Squares

As rows, columns, digits can be switched each other to make isomorphic puzzles that "look quite different", only one of each equivalence class is shown.
If no extra marks, the puzzles can be solved by Pairs, X-wings and 5-step chains.
When a grid has multiple automorphisms, number of automorphisms is marked(titled "Symmetry"), including identity transformation.

The puzzle grids are after singles and normalization(shape first, then digits)
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`pattern 1: 7 givens12...2.3...4......1.....5//Uniqueness Allowed: Gurth's_Symmetrical_Placement//Symmetry:6//Uniqueness Disabled: (unknown)`

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`pattern 2: 8 givens123..34.....5.....1.....2`

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`pattern 3: 8 givens12...34.....12...4......5//Symmetry: 2`

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`pattern 4: 10 givens123..45.3...42...5......3`

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`pattern 5: 12 givens1234531...4.2..2..3.5....//Symmetry: 2`
qiuyanzhe

Posts: 89
Joined: 21 August 2017
Location: China

### Re: Hard 5*5 Latin Squares

qiuyanzhe wrote:5*5 is the smallest size for Latin Squares to contain chains, and I think it is small enough for us to find out all puzzles requiring chains.
I used JSudoku to generate such puzzles(setting 1+ guess) and got several non-equivalent puzzles. I wonder if there are more.

I assume by "puzzles requiring chains" you mean puzzles that can't be solved with singles alone? In Latin Square parlance, these would be uniquely completable partial LS's (ie. valid puzzles) that are not strong?

I take it also that we are only interested in minimal puzzles (ie. critical sets). We can certainly find all such puzzles in reasonable time. Using one representative from each of the 2 main classes of 5x5 Latin Squares, I obtained the following results:

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`  Grid CSZ    NCS     SCS    TW    PW  -----------------------------------   5.1  6      50      50   5.1  7    1000    1000   5.1  8   30900   30900   5.1  9   18800   18800   5.1 10    2500    2500   5.2  7     600     588    12   5.2  8   21588   20904    36   648   5.2  9   23718   23718   5.2 10    2340    2268    36    36   5.2 11     216     216`

CSZ = Critical set size, SCS = # of strong CS (singles only), TW = # of totally weak CS (no singles), PW = partially weak (these all had 2 or 4 singles).

The 7-clue case is interesting, the only puzzles that aren't strong, are totally weak.

The 12 x 7-clue TW cases might all be the "same" puzzle by your definition. I have listed them here just in case:
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`12....14......2...3.5....12....1.5.3......2......412....1..3..5..4.......2.1......45.....24.2...3...1......4.33...2.5......2.1.......53.4.....2..5..2..2.....45.3........15..2..2.....4.3...1.4.2..5.....2......533...24......1....3...1.5.....24....5..2....4..1..33...2..2..5........5.14..3....4.2.....2.`
Last edited by Mathimagics on Thu Dec 27, 2018 6:26 am, edited 1 time in total.

Mathimagics
2017 Supporter

Posts: 1585
Joined: 27 May 2015
Location: Canberra

### Re: Hard 5*5 Latin Squares

Wow, thanks for yor work!
These 7-clue puzzles are isomorphic, so we've got that it is the only weak 7-clue puzzle!
As for 8-clue, it turns out that my second grid is not minimal(r1c2 redundant), and according to the symmetry group size(72),.we can conclude that patterns 2&3 are the only weak 8-clue puzzles.
Patterns 4&5 are not minimal, and seems there are so many ways to erase the redundant givens. Also I didn't expect the 11s. Sharing a few of your results may help knowing the whole thing.
Cheers!
qiuyanzhe

Posts: 89
Joined: 21 August 2017
Location: China

### Re: Hard 5*5 Latin Squares

qiuyanzhe wrote:Wow, thanks for your work!

You're welcome! I just can't figure out why I missed your original post (but I was not alone it seems).

qiuyanzhe wrote:Sharing a few of your results may help knowing the whole thing.

Happy to do that, just tell me what you need?

Mathimagics
2017 Supporter

Posts: 1585
Joined: 27 May 2015
Location: Canberra

### Re: Hard 5*5 Latin Squares

Hi Mathimagics,
I think some examples for weak critical sets of other size would be useful, by isomorphs we may know all such critical sets.
Your new post is quite informative to me. From the first file I got that adding some digits to the 7-clue puzzle can make more weak puzzles.
(two more:(+r1c3+r4c35+r5c1) (+r1c3+r5c4))
A chart could be found in Pg.6 of the second file.
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`LS Size #CS #Iso#Main#NS#Strong#TW5.1 6    50    1   1   1   1    0    7  1000   10   4   4   4    0    8 30900  312  57  57  57    0    9 18800  188  37  37  37    0   10  2500   25   6   6   6    05.2 7   600   50  11  10  10    1    8 21588 1802 322 311 311    1    9 23718 1981 348 348 348    0   10  2340  198  39  38  36    2   11   216   18   4   4   4    0`

No 11-cell TW critical set was reported there.. did I misread something?
qiuyanzhe

Posts: 89
Joined: 21 August 2017
Location: China

### Re: Hard 5*5 Latin Squares

qiuyanzhe wrote:No 11-cell TW critical set was reported there.. did I misread something?

No, I just transcribed the results from my logs incorrectly when making my table. I have corrected this above, the weak counts for 5.2.11 should have been for 5.2.10.

Thanks for spotting that!

Here are my weak (TW or PW) lists. I did not reduce the minimal puzzles by isomorphism (as Adams et al did), but you can do that. You should find that these reductions lead to counts that match their table exactly (for TW anyway).

Each puzzle is tagged with the number of singles (0 for totally weak).

5.2.7
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[/code]
0: 12....14......2...3.5....
0: 12....1.5.3......2......4
0: 12....1..3..5..4.......2.
0: 1......45.....24.2...3...
0: 1......4.33...2.5......2.
0: 1.......53.4.....2..5..2.
0: .2.....45.3........15..2.
0: .2.....4.3...1.4.2..5....
0: .2......533...24......1..
0: ..3...1.5.....24....5..2.
0: ...4..1..33...2..2..5....
0: ....5.14..3....4.2.....2.
[/code]

5.2.8
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` 4: 123...1.5.....24.....3... 4: 123...1.5.....24......1.. 4: 123.....5..4..24.....3... 4: 123.....5.....24...1..1.. 4: 12.4..1..3.4.....2..5.... 4: 12.4..1..3...1...2..5.... 4: 12.4.....3.4....52..5.... 4: 12.4.....3...1...2..5.1.. 4: 12..5.14..3.....5......2. 4: 12..5.14..3........1...2. 4: 12..5..4..3..1.....1...2. 4: 12..5..4..3.....5....3.2. 4: 12....14..3.....5..1...2. 4: 12....1.5.....24.....31.. 4: 12....1..3.4.1...2..5.... 4: 12.....4..34.......1.3.2. 4: 12.....4..3..1..5..1...2. 4: 12.....4..3..1..5.....12. 4: 12.....4..3.....5..1.3.2. 2: 12.....4.....12.5.3.5.... 0: 12.....4.....12...3.53... 2: 12.....4......2...3153... 0: 12......5.34.....2.1....4 2: 12......5.34.....2....1.4 2: 12......5.3......2.1.3..4 4: 12......5..4..245.....1.. 4: 12......5..4..24.....31.. 4: 12......5....124...1.3... 4: 12......5.....24...1.31.. 2: 12.......3.45..4......12. 4: 12.......3.4.1..52..5.... 4: 12.......3.4.1...2..5.1.. 4: 12.......3.4.....2.15.1.. 2: 12.......3..51.45......2. 0: 12.......3..5..45.....12. 4: 12.......3...1..52..53... 4: 1.34..14..3.....5......2. 4: 1.34..1.5.....24.....3... 4: 1.34..1.5.....24......1.. 4: 1.34...4..3.....5....3.2. 4: 1.34....5..4..24.....3... 4: 1.34....5.....24...1..1.. 4: 1.34.....3.45..4.......2. 4: 1.34.....3.4....52..5.... 4: 1.34.....3..5..4.......24 4: 1.3.5.14..3.....5......2. 4: 1.3.5.14..3........1...2. 4: 1.3.5.1..3.4.....2..5.... 4: 1.3.5..4..3..1.....1...2. 4: 1.3.5..4..3.....5....3.2. 4: 1.3.5...5.3......23.....4 4: 1.3.5...5.3......2...3..4 4: 1.3.5...5..4..24.....3... 4: 1.3.5....3.4....52..5.... 2: 1.3...14......2.5.3.5.... 4: 1.3...1.5..4..245........ 4: 1.3...1.5.....245....3... 0: 1.3...1..3.45..4.......2. 2: 1.3...1..3..5..45......2. 2: 1.3....45.....245....3... 4: 1.3....4.3.45..4.......2. 4: 1.3....4.3.45..........24 4: 1.3....4.3..5..45......2. 4: 1.3....4.3..5...5.3....2. 4: 1.3....4.3.....45.....12. 4: 1.3....4.3......5.3...12. 4: 1.3.....53.4.1...23...... 4: 1.3.....53.4.1...2...3... 4: 1.3.....53.4..2..23...... 4: 1.3.....5.34.1...2...3... 4: 1.3.....5.34..2..2...3... 4: 1.3.....5.3...2..23.....4 4: 1.3.....5.3...2..2...3..4 4: 1.3.....5..45.2..2...3... 4: 1.3.....5..4.12..2...3... 4: 1.3.....5..4..245....3... 0: 1.3.....5..4..24.2...3... 4: 1.3.....5..4..2..23..3... 4: 1.3.....5...512..2...3... 2: 1.3.....5.....24.2...3..4 4: 1.3.....5.....2..23..3..4 4: 1.3......3.45..4......12. 4: 1.3......3.45..4.......24 4: 1.3......3.45....2.....24 4: 1.3......3.4...4.2....12. 4: 1.3......3.4.....2....124 4: 1.3......3..5..45.....12. 4: 1..45.1.5.....24.....3... 4: 1..45.1..3.4.....2..5.... 4: 1..45.1..3...1...2..5.... 4: 1..45..4..3.....5....3.2. 4: 1..45..4....5.2...3.5.... 4: 1..45..4......2.5.3.5.... 4: 1..45...5..4..24.....3... 4: 1..45....3.4....52..5.... 4: 1..45....3...1...2..5.1.. 0: 1..4..14......2.5.3.5.... 2: 1..4..14......2...3.53... 2: 1..4..1.5.3......2...3..4 4: 1..4..1..3.4.....2..53... 4: 1..4..1..3......52..53... 4: 1..4...45...5.2.5.3...... 4: 1..4...45....12.....53... 4: 1..4...45....12......3..4 4: 1..4...45.....2.5.3.5.... 4: 1..4...45.....2...3.53... 4: 1..4...45.....2...3..3..4 4: 1..4...4.3.4....5..1...2. 4: 1..4...4.3......52.....24 4: 1..4...4.3......5..1...24 4: 1..4...4....512.5......2. 4: 1..4...4....5.2.5.3.5.... 4: 1..4...4....5.2.5.3....2. 4: 1..4...4.....12.5.3.5.... 4: 1..4...4.....12.5...5..2. 4: 1..4...4.....12...3.53... 2: 1..4....53.4.....2..53... 4: 1..4.....3.45..4.2.....2. 2: 1..4.....3.45....2..5..2. 4: 1..4.....3.45....2.....24 4: 1..4.....3.4...452.....2. 4: 1..4.....3.4...45..1...2. 4: 1..4.....3.4....523....2. 4: 1..4.....3.4....52.1...2. 4: 1..4.....3.4....52..53... 0: 1..4.....3.4....52..5..2. 4: 1..4.....3.4....52.....24 4: 1..4.....3.4.....231...2. 4: 1..4.....3..5..4.2.....24 4: 1...5.14..34....5......2. 4: 1...5.14..34.........3.2. 2: 1...5.1.5.34.....2......4 0: 1...5.1.5.3......2...3..4 2: 1...5.1..3.45..4.......2. 4: 1...5..45...5.2......31.. 4: 1...5..45...5.2......3.2. 4: 1...5..45.....2.5....31.. 2: 1...5..4.334....5......2. 4: 1...5..4..34....5....3.2. 2: 1...5..4..3...2.5.3....2. 0: 1...5..4..3...2.5....3.2. 4: 1...5..4....5.2.5.3....2. 4: 1...5..4....5.2.5....3.2. 4: 1...5..4....5.2...3.5..2. 4: 1...5..4......2.5.3.5..2. 4: 1...5..4......2.5...531.. 4: 1...5..4......2.5...53.2. 4: 1...5..4......2.5....312. 4: 1...5..4......2.5....3.24 4: 1...5..4......2.5.....124 4: 1...5...5334.....2.1..... 4: 1...5...5334.....2......4 4: 1...5...533......2...3..4 4: 1...5...53.45....2.1..... 4: 1...5...53.45....2......4 4: 1...5...53.......23..3..4 4: 1...5...5.34.....2.1....4 4: 1...5...5.3...2..2.1.3... 4: 1...5...5.3......23..3..4 4: 1...5...5.3......2.1.3..4 4: 1...5...5.....2..231.3... 4: 1...5...5.....2..23..3..4 4: 1.....14..34....5..1...2. 4: 1.....1.5.....245....31.. 4: 1.....1..3.4.1...2..53... 2: 1......45..4..2..23..3... 4: 1......45...5.2.5.3....2. 4: 1......45...5.2.5....31.. 4: 1......45...5.2.5....3.2. 4: 1......45....12...3..3..4 0: 1......45.....245....31.. 2: 1......45.....24..3..31.. 2: 1......45.....2.52...3..4 4: 1......45.....2.5.3.5..2. 4: 1......45.....2.5...531.. 4: 1......45.....2.5...53.2. 4: 1......45.....2.5....31.4 4: 1......45.....2.5....3.24 4: 1......45.....2.5.....124 0: 1......45.....2..23..3..4 4: 1......45.....2...3.531.. 4: 1......45.....2...3.53.2. 4: 1......45.....2...3.5.1.4 4: 1......45.....2...3..31.4 4: 1......45.....2...3..3.24 0: 1......4.334....5..1...2. 2: 1......4.33.5...5..1...2. 4: 1......4.3.45..4.2.....2. 4: 1......4.3.45....2.....24 2: 1......4.3.4..2.5.3....2. 4: 1......4.3.4...452.....2. 4: 1......4.3.4...45..1...2. 4: 1......4.3.4....523....2. 4: 1......4.3.4....52.....24 4: 1......4.3.4....5.31...2. 4: 1......4.3.4....5..1...24 4: 1......4.3.4.....231...2. 0: 1......4.3..5.2.5.3....2. 2: 1......4.3..5.2.5....3.2. 4: 1......4.3..5..452.....2. 4: 1......4.3..5..45..1...2. 4: 1......4.3..5..4..31...2. 4: 1......4.3..5...523....2. 4: 1......4.3..5...5.31...2. 4: 1......4.3..5...5.3...12. 4: 1......4..34.1..5..1...2. 4: 1......4..34.1..5.....12. 4: 1......4..34....5..1.3.2. 4: 1......4..34....5..1..12. 2: 1......4..3...2.5..1.3.2. 4: 1......4....512.5.3....2. 4: 1......4.....12.5.3.5..2. 4: 1......4.....12...3.531.. 4: 1......4.....12...3.53.2. 4: 1......4.....12...3.53..4 4: 1......4.....12...3.5.1.4 4: 1......4.....12...3..3.24 4: 1.......5334.1...2...3... 4: 1.......5334.1...2......4 4: 1.......5334..2..2...3... 4: 1.......5334..2..2......4 4: 1.......533.51...2......4 4: 1.......533...2..2...3..4 4: 1.......53.451...2...3... 4: 1.......53.451...2......4 4: 1.......53.45.2..2...3... 4: 1.......53.45.2..2......4 4: 1.......53.45....2.1....4 2: 1.......53.45....2...3.2. 0: 1.......53.45....2.....24 4: 1.......53.4.1...23..3... 0: 1.......53.4.1...2..53... 2: 1.......53.4.1...2..5...4 4: 1.......53.4..2..23..3... 2: 1.......53.4....52.....24 4: 1.......53..512..2...3... 4: 1.......53....2..23..3..4 4: 1.......5.345....2.1....4 4: 1.......5.34.1...2.1....4 4: 1.......5.34..2..2.1....4 4: 1.......5.3.51...2.1....4 4: 1.......5.3...2..2.1.3..4 4: 1.......5..45.2..2.1....4 4: 1.......5..4..245....31.. 2: 1.......5..4..24.2...31.. 4: 1.......5....1245..1.3... 4: 1.......5....1245....31.. 4: 1.......5.....245..1.31.. 4: 1.......5.....2..231.3..4 4: 1........3.45..4.2....12. 4: 1........3.45....2....124 4: 1........3.4.1..52..53... 2: 1........3.4.1..52..5..2. 4: 1........3.4.1...2.153... 4: 1........3.4.1...2..531.. 4: 1........3.4.....2.1531.. 4: 1........3..5..452....12. 4: 1........3..5..45.3...12. 4: 1........3..5..45..1..12. 4: 1........3..5..4..31..12. 4: 1........3..5...523...12. 4: .234..14..3........1...2. 4: .234..1..3.4.....2..5.... 4: .234..1..3...1...2..5.... 4: .234...4..3..1.....1...2. 4: .234....5.3.5....2......4 4: .234....5.3......2....1.4 4: .234....5.....24...1..1.. 4: .234.....3.4....52..5.... 4: .234.....3...1...2..5.1.. 4: .23.5.1.5.....24.....3... 4: .23.5.1.5.....24......1.. 4: .23.5.1..3...1...2..5.... 4: .23.5..4..3..1.....1...2. 4: .23.5..4......2...315.... 4: .23.5..4......2...3.5...4 4: .23.5...5..4..24.....3... 4: .23.5...5.....24...1..1.. 4: .23.5....3...1...2..5.1.. 2: .23...14.....12...3.5.... 0: .23...14......2...315.... 4: .23...1.5....124...1..... 4: .23...1.5....124......1.. 2: .23...1..3..51.4.......2. 4: .23....45.3.5..4...1..... 4: .23....45.3.5...5..1..... 4: .23....45.3.....5..1..1.. 4: .23....4.3..51....3.5.... 4: .23....4.3..51......53... 4: .23....4.3...12...3.5.... 4: .23....4.3...12.....53... 4: .23....4.3....2...315.... 4: .23....4.3........315...4 4: .23....4..3...2....153... 4: .23....4..3.......315...4 4: .23....4..3........153..4 4: .23....4.....12...3.53... 4: .23....4......2...3153... 4: .23....4......2...315...4 2: .23.....53...124......1.. 4: .23.....5.3.5..4.2......4 4: .23.....5.3.5..4...1..1.. 4: .23.....5.3.5..4......1.4 0: .23.....5.3...24...1..1.. 2: .23.....5.3...24......1.4 4: .23.....5.3....45.3...1.. 4: .23.....5.3....45..1..1.. 4: .23.....5.3....4.2.1..1.. 4: .23.....5.3....4.2....1.4 4: .23.....5.3....4..31..1.. 4: .23.....5.3.....52.1..1.. 4: .23.....5....124...1..1.. 4: .2.45.14..3.....5......2. 4: .2.45.14..3........1...2. 4: .2.45.1.5.....24......1.. 4: .2.45..4..3..1.....1...2. 4: .2.45..4..3.....5....3.2. 4: .2.45...5.....24...1..1.. 4: .2.45....3..51.4.......2. 4: .2.45....3..5..4..3....2. 4: .2.45....3...1...2..5.1.. 2: .2.4..14......2...315.... 2: .2.4..1.5.3......2.1....4 0: .2.4..1.5.3......2....1.4 4: .2.4..1..3...1...2.15.... 4: .2.4..1..3.......2.15.1.. 4: .2.4...45.34.....2.1..... 4: .2.4...45.34......31..... 4: .2.4...45.3.5.........1.4 4: .2.4...45.3......2.1....4 4: .2.4...45.3......2....1.4 4: .2.4...45.3.......31....4 2: .2.4...4.3...1...2.15.... 4: .2.4....53...1.4.....31.. 4: .2.4....53.....4..3.5.1.. 4: .2.4....53.....4..3..31.. 4: .2.4....5.345..4......1.. 4: .2.4....5.34...4.2....1.. 4: .2.4....5.34.....2.1....4 4: .2.4....5.34.....2....1.4 4: 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5.2.10
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Mathimagics
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### Re: Hard 5*5 Latin Squares

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`Min/After Singles      Code    Symmetry Count Comment7/7:   12....14......2...3.5....    6     128/8:   12.....4.....12...3.53...    2     3610/10: 123..214..3..........312.    3     24       123..214...4...4.2.1.....    6     12  The two 10/10s are only different in a UR in the givens.TOTAL-------------------------------------848/10:  12.......3..51.45......2.    1     7210/12: 12.45.1.5..4..245........    2     36  These /12's are essentially the same.8/12:  123...1.5.....24.....3...    2     36       .23.5...5..4..24.....3...    2     36       123.....5..4..24.....3...    1     72       1.34....5..4..24.....3...    1     72       1.34..1.5.....24.....3...    1     72       1.3.5...5..4..24.....3...    1     72       1..45.1.5.....24.....3...    1     72       1..45...5..4..24.....3...    1     72       ..345.1.5.....24.....3...    1     72TOTAL------------------------------------684More:(Puzzles that cannot be deduced from a minimal puzzle)11     123...14....512.5.3.5....   3    249      123...14......2...315....   2    368      123...14......2...3.5....   1    72`

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qiuyanzhe

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