72 size-4 unavoidable sets

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

72 size-4 unavoidable sets

Postby tdillon » Fri Aug 20, 2021 7:12 pm

Here is a puzzle whose chief interest lies in the properties of its solution grid:

Code: Select all
...4.67......8.2.........14.9..3..6.3.4.....8..59....75..8.3.....2.9....81...76..

This grid has the distinction of containing 72 size-4 unavoidable sets: more than any other canonical grid.

I don't know if this property has any influence on whether we find more interesting or less interesting puzzles on the grid, but the one above is pretty hard.

Behind this there are two grids that contain 66 size-4 unavoidable sets and one that contains 62. The lowest N for which no grid contains N size-4 unavoidable sets is 55.
Hidden Text: Show
##### 72
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##### 57
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##### 56
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##### 55
##### 54
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Here is (roughly) the distribution of counts of size-4 unavoidable sets across canonical grids, so this is really quite an extreme value and a special grid!

Image
tdillon
 
Posts: 60
Joined: 14 June 2019

Re: 72 size-4 unavoidable sets

Postby yzfwsf » Fri Aug 20, 2021 8:51 pm

tdillon wrote: but the one above is pretty hard.

Gurth's symmetry placement.Detected using solution grid, It may be impossible for manual solvers.
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Posts: 511
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Re: 72 size-4 unavoidable sets

Postby Serg » Fri Aug 20, 2021 9:50 pm

Hi, tdillon!
tdillon wrote:...
This grid has the distinction of containing 72 size-4 unavoidable sets: more than any other canonical grid.

I don't know if this property has any influence on whether we find more interesting or less interesting puzzles on the grid, but the one above is pretty hard.

Behind this there are two grids that contain 66 size-4 unavoidable sets and one that contains 62. The lowest N for which no grid contains N size-4 unavoidable sets is 55.

Do you mean any Sudoku solution grid has not less than 55 U4 unavidable sets (UA with 4 cells)? But MC grid, for example, has no U4.
Code: Select all
      MC grid

+-----+-----+-----+
|1 2 3|4 5 6|7 8 9|
|4 5 6|7 8 9|1 2 3|
|7 8 9|1 2 3|4 5 6|
+-----+-----+-----+
|2 3 1|5 6 4|8 9 7|
|5 6 4|8 9 7|2 3 1|
|8 9 7|2 3 1|5 6 4|
+-----+-----+-----+
|3 1 2|6 4 5|9 7 8|
|6 4 5|9 7 8|3 1 2|
|9 7 8|3 1 2|6 4 5|
+-----+-----+-----+

Here is an old discussion at this forum about minimal/maximal numbers of U4 per solution grid - Unbiased grid generation. They said minimal number of U4 per grid - 0, maximal number of U4 per grid - 36.

Serg

[Edited. I corrected the link.]
Serg
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Location: Russia

Re: 72 size-4 unavoidable sets

Postby tdillon » Fri Aug 20, 2021 10:51 pm

Hi Serg,

Serg wrote:Do you mean any Sudoku solution grid has not less than 55 U4 unavidable sets (UA with 4 cells)? But MC grid, for example, has no U4.

No, I meant that for every N: 0 <= N < 55 there's at least one of the 5472730538 canonical grids that has N U4's.

Serg wrote:Here is an old discussion at this forum about minimal/maximal numbers of U4 per solution grid - Unbiased grid generation. They said minimal number of U4 per grid - 0, maximal number of U4 per grid - 36.

Thanks for the link! At a quick glance I'm not sure what the disconnect is, so I'll try to look more closely later.

For reference, here is the 72U4 grid in the non-canonical form I first found it along with (r1, c1, r2, c2) for the top-left and bottom-right corners of each U4.
Code: Select all
123695847456278319789314256874951632265843791391762485647139528938526174512487963 72 644634678266
0 0 1 7
0 0 5 2
0 0 6 3
0 1 2 6
0 1 3 8
0 1 6 7
0 2 3 7
0 2 4 5
0 3 7 5
0 3 8 7
0 4 1 8
0 4 4 7
0 4 8 6
0 5 2 7
0 6 5 7
0 7 7 8
1 0 5 6
1 0 7 8
1 1 3 4
1 1 4 2
1 1 5 8
1 2 2 8
1 2 7 5
1 3 2 6
1 3 3 8
1 3 6 7
1 4 8 5
1 5 6 8
1 6 8 8
1 7 4 8
2 0 3 1
2 0 4 6
2 0 5 3
2 1 8 4
2 2 4 7
2 2 8 6
2 3 6 4
2 4 7 6
2 5 5 6
2 5 7 8
2 6 3 8
2 6 6 7
3 0 5 7
3 0 8 4
3 1 4 6
3 1 5 3
3 1 6 2
3 1 7 7
3 1 8 5
3 2 4 4
3 3 6 5
3 5 7 6
4 0 5 5
4 0 8 2
4 1 7 5
4 2 5 8
4 2 6 6
4 2 7 3
4 3 6 8
4 3 8 4
5 0 6 4
5 0 7 1
5 0 8 8
5 2 6 3
5 4 7 5
5 4 8 7
6 0 7 5
6 1 8 3
6 3 7 6
6 4 8 8
7 2 8 4
7 5 8 7
tdillon
 
Posts: 60
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Re: 72 size-4 unavoidable sets

Postby tdillon » Fri Aug 20, 2021 10:57 pm

Argh ... I see the mistake. I was counting rectangles that span 4 boxes instead of just 2. Never mind. :-)
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