The New Sudoku Players' Forum

by **philvo** » Sat Jan 21, 2023 7:54 am

Hello,
As continuation on the sub-thread on thread “3-row x-sudokus”, I wonder if it is possible to create
an X-sudoku with only clues on the last band, ie with 54 empty leading cells.
phil

by **Leren** » Sun Jan 22, 2023 9:18 pm

Well, I think it's polite to at least respond to this question. I think the answer is no. One reason is that if it was possible there would have to be a solution grid with every hitset in it requiring at least 7 rows. Doesn't sound right to me.

More practically, this is the best I have come up with:

Code: Select all: ......... ......... ......... ......... ......... ....432.. 913684752 468527913 527139468

This has just 19 solutions, and if you put 7 in r5c9 you get a unique solution, and there is a 12 clue version which is unique, so the most leading empty cells is 44.

I suppose you could answer the question by setting Row 7 to 123456789, and then checking every combination in Rows 8 and 9 so that they are both full and don't conflict with Row 7.

If none of the full band 3 cases leads to a unique solution then your answer is no, which is what I'm expecting. I'll leave it to others to check it, or at least beat the 44 leading empty cells case.

Leren

by **philvo** » Mon Jan 23, 2023 9:34 am

Hi Leren,
Thanks for your answer. I agree with your method to be sure there is no solution.
For me, there are 929,280 grids to scan, but my solver does not allow to parse these grids in reasonable time.
From my recent random tests, I also think there is probably no solution.
phil

by **m_b_metcalf** » Mon Jan 23, 2023 1:47 pm

Leren wrote:More practically, this is the best I have come up with:
Code: Select all
......... ......... ......... ......... ......... ....432.. 913684752 468527913 527139468

This has just 19 solutions,

urhegyi wrote:My record at the moment is 51 empty leading cells:
Code: Select all
...................................................9871.25.4..673.18..424.8.3.7.9

Based on urhegyi's puzzle, I found this, with one more empty cell, and just six solutions:

Code: Select all: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 1 3 4 5 6 7 8 9 2 5 9 2 1 8 4 7 3 6 7 6 8 9 2 3 5 4 1 6 solutions

Mike

by **Leren** » Mon Jan 23, 2023 7:35 pm

Code: Select all: ......... ......... ......... ......... ......... ....4329. 913684752 468527913 527139468

This one has one extra clue added in Row 6 and just 2 solutions. Also if you add 1 to r6c7 to Mike's puzzle - like so

Code: Select all: ......... ......... ......... ......... ......... ......123 134567892 592184736 768923541

The solution is unique, with 51 leading empty cells.

Leren

by **m_b_metcalf** » Tue Jan 24, 2023 11:55 am

Leren wrote:
Code: Select all
......... ......... ......... ......... ......... ....4329. 913684752 468527913 527139468

This one has one extra clue added in Row 6 and just 2 solutions.

And it can be twice tweaked, yielding another 51-empty-clue puzzle (with one solution):

Code: Select all: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 9 7 . 1 3 . 8 4 . 5 . . 8 6 5 2 . . . 3 5 . 7 . 3 9 . 6 8 ED=7.2/1.2/1.2, 50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 7 . 1 3 . 8 4 . 5 . . 8 6 5 2 . . . 3 2 . 7 . 3 9 . 6 8 ED=2.9/1.2/1.2, 51

and two cousins:

Code: Select all: ...................................................927.13.86.5..84..7..325..39.68 ED=2.0/1.2/1.2 ...................................................526.13.84....8..27.3925..39.68 ED=8.3/1.2/1.2

Mike

by **Leren** » Fri Feb 03, 2023 5:51 am

Found these two puzzles with band 3 full and 52 empty leading cells. No doubt many sub-puzzles can be found in these.

Code: Select all: ....................................................12345671829281934765967528431 ....................................................12234561789871923654956487321

I've also got a collection of about 3300 puzzles with band 3 full and r6c789 with clues

Leren

by **m_b_metcalf** » Fri Feb 03, 2023 10:46 am

Leren wrote:Found these two puzzles with band 3 full and 52 empty leading cells. No doubt many sub-puzzles can be found in these.

Brilliant. Here are some sub-puzzles, all rated at ED=7.1/1.2/1.2:

Hidden Text: Show

Mike

P.S. I think 'many' may be a slight exaggeration.

by **Leren** » Fri Feb 03, 2023 8:24 pm

Hi Mike,

All of your puzzles solve band 3 completely with singles. This is not surprising as it is obvious that you can remove at least 5 cells in band 3, one in each row of one box and one in any cell of the other 2 boxes, and achieve this.

I suspect that there are a lot of ways of doing this. You can then try removing extra band 3 cells to see if band 3 still solves with singles and the puzzle is minimal, which is what you appear to have done.

Here is my first manual go at this sort of thing for the first puzzle.

Hidden Text: Show

Leren

by **m_b_metcalf** » Sat Feb 04, 2023 9:46 am

Leren wrote:All of your puzzles solve band 3 completely with singles. This is not surprising as it is obvious that you can remove at least 5 cells in band 3, one in each row of one box and one in any cell of the other 2 boxes, and achieve this.

I suspect that there are a lot of ways of doing this. You can then try removing extra band 3 cells to see if band 3 still solves with singles and the puzzle is minimal, which is what you appear to have done.
[snip]

Perhaps the next challenge is to find a sub puzzle which does not solve band 3 with just singles, but the sub puzzle is ultimately solvable.

Leren,
I don't deliberately look for puzzles that solve with just singles in band 3. I'm using modified Patterns Game code, and that's what it delivers.

I tried tweaking your new puzzles (the first of which isn't minimal), and got only 4 more:

Code: Select all: ....................................................12.3456.7...718.3.54.56.2739. 19 ....................................................12.3456.78..719.3.54.56.873.. 19 ....................................................12.34.51.676.1.235.4.85.69.21 20 ....................................................12.3456.78..719.3.54.56.873.. 19

all with the same rating as before.

I'll call it a day now.

Regards,

Mike

by **Leren** » Sat Feb 04, 2023 7:34 pm

Hi Mike,

I'm also calling it a day on this topic.

To summarise what we found, there are no Unique Solution Puzzles (USPs) with all clues in Band 3.

The maximum number of empty leading leading cells for a USP is 52, with just 2 Essentially Different USPs with Band 3 full, with both cells 53 and 54 occupied.

With 51 empty leading cells and cells 52, 53 & 54 occupied, unless I have made an error, there are 3318 Essentially Different USPs with Band 3 full.

Obviously none of these USPs is minimal. We have found some minimal USPs by removing redundant clues from Band 3. A full listing of minimal USPs would be both complex and beyond the scope of the original question.

Leren

The New Sudoku Players' Forum

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