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For fans of Killer Sudoku, Samurai Sudoku and other variants

Re: Singles backdoor size 6

Postby tarek » Tue Apr 15, 2014 8:44 am

Minimal singles backdoor size 6 puzzles

These were Minimal puzzles created from the same seed puzzle (So all share the same solution). That seed puzzle was special in that gsf's Sudoku confirmed that it has a singles backdoor size 5 and an FNBTHWXYKO backdoor size of 4.

I could not confirm how big the FNBTHWXYKO backdoor size for the minimal forms as it took forever on my limited hardware but they will all be >=4. All should have a Singles backdoor size 6. So IMO these are the most difficult Sudoku puzzles EVER posted to date in the FNBTHWXYKO world :D.

Code: Select all
.-----------------------------------------------------------------------------------------------.
| 35678     13456789  123456789 | 23456789  46789     12456789  | 123456789 13456789  145789    |
| 12356789  137       123457    | 12389     12346789  1248      | 123689    1         12345789  |
| 123456789 1234567   12345678  | 12345689  123456789 12345678  | 1235689   1234569   123456789 |
:-----------------------------------------------------------------------------------------------:
| 23456789  46789     12456789  | 1356789   13456789  145789    | 23456789  146789    12456789  |
| 23689     12346789  123456789 | 12356789  1379      12345789  | 12356789  12346789  12478     |
| 12345689  12346     123456789 | 123569    12345679  12347     | 12345689  12346     12345678  |
:-----------------------------------------------------------------------------------------------:
| 123456789 156789    12456789  | 123456789 12345678  1245678   | 2356789   13456789  123456789 |
| 12356789  179       1245789   | 123689    12346789  12478     | 2356789   1379      12345789  |
| 123569    12345679  123456789 | 12345689  12346     12345678  | 123456789 12345679  123457    |
:-----------------------------------------------------------------------------------------------:
.-----------------------------------------------------------------------------------------------.
| 35678     13456789  123456789 | 23456789  46789     12456789  | 123456789 13456789  145789    |
| 123456789 137       123478    | 123689    12346789  1248      | 123689    1         12345789  |
| 123456789 1234567   12345678  | 12345689  23456789  12345678  | 1235689   1234569   123456789 |
:-----------------------------------------------------------------------------------------------:
| 23456789  46789     12456789  | 356789    13456789  145789    | 23456789  146789    12456789  |
| 123689    12346789  123456789 | 12356789  1379      12345789  | 12356789  12346789  12478     |
| 12345689  12346     12345678  | 123569    12345679  12347     | 123456789 12346     12345678  |
:-----------------------------------------------------------------------------------------------:
| 123456789 156789    12456789  | 23456789  12345678  1245678   | 2356789   13456789  123456789 |
| 12356789  179       12456789  | 23689     12346789  12478     | 2356789   1379      12345789  |
| 123569    12345679  123456789 | 12345689  12346     12345678  | 123456789 12345679  123457    |
:-----------------------------------------------------------------------------------------------:
.-----------------------------------------------------------------------------------------------.
| 35678     13456789  123456789 | 23456789  46789     12456789  | 123456789 13456789  145789    |
| 12356789  137       1234578   | 12389     12346789  12478     | 123689    1         12345789  |
| 123456789 1234567   12345678  | 12345689  23456789  12345678  | 1235689   1234569   123456789 |
:-----------------------------------------------------------------------------------------------:
| 23456789  146789    12456789  | 356789    13456789  145789    | 23456789  146789    12456789  |
| 23689     12346789  123456789 | 12356789  1379      12345789  | 12356789  12346789  12478     |
| 12345689  12346     123456789 | 123569    12345679  12347     | 12345689  12346     12345678  |
:-----------------------------------------------------------------------------------------------:
| 123456789 156789    12456789  | 123456789 12345678  1245678   | 2356789   13456789  123456789 |
| 12356789  179       1245789   | 23689     12346789  12478     | 2356789   1379      12345789  |
| 123569    12345679  123456789 | 12345689  12346     12345678  | 123456789 12345679  123457    |
:-----------------------------------------------------------------------------------------------:
.-----------------------------------------------------------------------------------------------.
| 35678     13456789  123456789 | 23456789  46789     12456789  | 123456789 13456789  145789    |
| 123456789 137       12347     | 1289      1234689   1248      | 123689    1         12345789  |
| 123456789 1234567   12345678  | 12345689  123456789 123456789 | 1235689   12345689  123456789 |
:-----------------------------------------------------------------------------------------------:
| 23456789  146789    12456789  | 356789    13456789  145789    | 23456789  146789    12456789  |
| 23689     12346789  123456789 | 12356789  1379      12345789  | 12356789  12346789  12478     |
| 12345689  12346     123456789 | 123569    12345679  123457    | 12345689  12346     12345678  |
:-----------------------------------------------------------------------------------------------:
| 123456789 156789    12456789  | 23456789  12345678  1245678   | 2356789   13456789  123456789 |
| 12356789  179       1245789   | 23689     12346789  12478     | 2356789   1379      12345789  |
| 1235689   12345679  123456789 | 12345689  12346     12345678  | 123456789 12345679  123457    |
:-----------------------------------------------------------------------------------------------:
.-----------------------------------------------------------------------------------------------.
| 345678    13456789  123456789 | 23456789  46789     12456789  | 123456789 13456789  145789    |
| 123456789 137       1347      | 123689    12346789  12478     | 123689    1         12345789  |
| 123456789 1234567   12345678  | 12345689  123456789 12345678  | 1235689   12345689  123456789 |
:-----------------------------------------------------------------------------------------------:
| 23456789  46789     12456789  | 356789    13456789  145789    | 23456789  146789    12456789  |
| 23689     12346789  123456789 | 12356789  1379      12345789  | 12356789  12346789  12478     |
| 12345689  12346     123456789 | 123569    12345679  123457    | 12345689  12346     12345678  |
:-----------------------------------------------------------------------------------------------:
| 123456789 156789    12456789  | 123456789 12345678  1245678   | 2356789   13456789  123456789 |
| 12356789  179       1245789   | 23689     12346789  12478     | 2356789   1379      12345789  |
| 123569    12345679  123456789 | 12345689  12346     12345678  | 123456789 12345679  123457    |
:-----------------------------------------------------------------------------------------------:
.-----------------------------------------------------------------------------------------------.
| 356789    13456789  123456789 | 23456789  46789     12456789  | 123456789 13456789  145789    |
| 12356789  137       123457    | 123689    12346789  12478     | 123689    1         12345789  |
| 123456789 1234567   12345678  | 12345689  12456789  12345678  | 1235689   12345689  123456789 |
:-----------------------------------------------------------------------------------------------:
| 23456789  46789     12456789  | 1356789   13456789  145789    | 23456789  146789    12456789  |
| 23689     12346789  123456789 | 12356789  1379      12345789  | 12356789  12346789  12478     |
| 12345689  12346     123456789 | 123569    12345679  12347     | 12345689  12346     12345678  |
:-----------------------------------------------------------------------------------------------:
| 123456789 156789    12456789  | 123456789 12345678  1245678   | 2356789   13456789  123456789 |
| 12356789  179       1245789   | 23689     12346789  12478     | 2356789   1379      12345789  |
| 123569    12345679  123456789 | 12345689  12346     12345678  | 123456789 12345679  123457    |
:-----------------------------------------------------------------------------------------------:
.-----------------------------------------------------------------------------------------------.
| 356789    13456789  123456789 | 23456789  46789     12456789  | 123456789 13456789  145789    |
| 12356789  137       123457    | 123689    12346789  12478     | 123689    1         12345789  |
| 123456789 1234567   12345678  | 12345689  12456789  12345678  | 1235689   1234569   123456789 |
:-----------------------------------------------------------------------------------------------:
| 23456789  46789     12456789  | 1356789   13456789  145789    | 23456789  146789    12456789  |
| 23689     12346789  123456789 | 12356789  1379      12345789  | 12356789  12346789  12478     |
| 12345689  12346     12345678  | 123569    12345679  12347     | 123456789 12346     12345678  |
:-----------------------------------------------------------------------------------------------:
| 123456789 156789    12456789  | 123456789 12345678  1245678   | 2356789   13456789  123456789 |
| 12356789  179       1245789   | 123689    12346789  12478     | 2356789   1379      12345789  |
| 123569    12345679  123456789 | 12345689  12346     12345678  | 123456789 12345679  123457    |
:-----------------------------------------------------------------------------------------------:
.-----------------------------------------------------------------------------------------------.
| 356789    13456789  123456789 | 23456789  46789     12456789  | 123456789 13456789  145789    |
| 12356789  137       12345678  | 123689    12346789  12478     | 123689    1         12345789  |
| 123456789 1234567   12345678  | 12345689  23456789  12345678  | 1235689   1234569   123456789 |
:-----------------------------------------------------------------------------------------------:
| 23456789  46789     12456789  | 356789    13456789  145789    | 23456789  146789    12456789  |
| 23689     12346789  123456789 | 12356789  1379      12345789  | 12356789  12346789  12478     |
| 12345689  12346     123456789 | 123569    12345679  123457    | 12345689  12346     12345678  |
:-----------------------------------------------------------------------------------------------:
| 123456789 156789    12456789  | 123456789 12345678  1245678   | 2356789   13456789  123456789 |
| 12356789  179       1245789   | 23689     12346789  12478     | 2356789   1379      12345789  |
| 123569    12345679  123456789 | 12345689  12346     12345678  | 123456789 12345679  123457    |
:-----------------------------------------------------------------------------------------------:
.-----------------------------------------------------------------------------------------------.
| 35678     13456789  123456789 | 23456789  46789     12456789  | 123456789 13456789  145789    |
| 12356789  137       123457    | 123689    12346789  12478     | 1235689   1         12345789  |
| 123456789 1234567   12345678  | 12345689  123456789 12345678  | 1235689   12345689  123456789 |
:-----------------------------------------------------------------------------------------------:
| 23456789  46789     12456789  | 356789    13456789  145789    | 23456789  146789    12456789  |
| 23689     12346789  123456789 | 12356789  1379      12345789  | 12356789  12346789  12478     |
| 12345689  12346     12345678  | 123569    12345679  123457    | 123456789 12346     12345678  |
:-----------------------------------------------------------------------------------------------:
| 123456789 156789    12456789  | 123456789 12345678  1245678   | 2356789   13456789  123456789 |
| 12356789  179       1245789   | 23689     12346789  12478     | 2356789   1379      12345789  |
| 123569    12345679  123456789 | 12345689  12346     12345678  | 123456789 12345679  123457    |
:-----------------------------------------------------------------------------------------------:
.-----------------------------------------------------------------------------------------------.
| 356789    13456789  123456789 | 23456789  46789     12456789  | 123456789 13456789  145789    |
| 12356789  137       123457    | 12689     12346789  1248      | 1235689   1         12345789  |
| 123456789 1234567   12345678  | 12345689  123456789 12345678  | 1235689   12345689  123456789 |
:-----------------------------------------------------------------------------------------------:
| 23456789  46789     12456789  | 356789    13456789  145789    | 23456789  146789    12456789  |
| 23689     12346789  123456789 | 12356789  1379      12345789  | 12356789  12346789  12478     |
| 12345689  12346     12345678  | 123569    12345679  123457    | 123456789 12346     12345678  |
:-----------------------------------------------------------------------------------------------:
| 123456789 156789    12456789  | 123456789 12345678  1245678   | 2356789   13456789  123456789 |
| 12356789  179       1245789   | 123689    12346789  12478     | 2356789   1379      12345789  |
| 123569    12345679  123456789 | 12345689  12346     12345678  | 123456789 12345679  123457    |
:-----------------------------------------------------------------------------------------------:
.-----------------------------------------------------------------------------------------------.
| 356789    13456789  123456789 | 23456789  46789     12456789  | 123456789 13456789  145789    |
| 12356789  137       123457    | 123689    12346789  1248      | 123689    1         12345789  |
| 123456789 1234567   12345678  | 12345689  123456789 12345678  | 1235689   1234569   123456789 |
:-----------------------------------------------------------------------------------------------:
| 23456789  146789    12456789  | 1356789   13456789  145789    | 23456789  146789    12456789  |
| 23689     12346789  123456789 | 12356789  1379      12345789  | 12356789  12346789  12478     |
| 12345689  12346     12345678  | 123569    12345679  12347     | 123456789 12346     12345678  |
:-----------------------------------------------------------------------------------------------:
| 123456789 156789    12456789  | 123456789 12345678  1245678   | 2356789   13456789  123456789 |
| 12356789  179       1245789   | 123689    12346789  12478     | 2356789   1379      12345789  |
| 123569    12345679  123456789 | 12345689  12346     12345678  | 123456789 12345679  123457    |
:-----------------------------------------------------------------------------------------------:


To the purists who need the cells to have more than 1 candidate, I've modified these puzzles to another minimal form which shares the same singles outcome:
Hidden Text: Show
Code: Select all
.-----------------------------------------------------------------------------------------------.
| 35678     13456789  123456789 | 23456789  46789     12456789  | 23456789  3456789   45789     |
| 12356789  137       123457    | 12389     12346789  1248      | 23689     123456789 2345789   |
| 123456789 1234567   12345678  | 12345689  123456789 12345678  | 235689    234569    23456789  |
:-----------------------------------------------------------------------------------------------:
| 23456789  46789     12456789  | 1356789   13456789  145789    | 23456789  146789    12456789  |
| 23689     12346789  123456789 | 12356789  1379      12345789  | 12356789  12346789  12478     |
| 12345689  12346     123456789 | 123569    12345679  12347     | 12345689  12346     12345678  |
:-----------------------------------------------------------------------------------------------:
| 123456789 156789    12456789  | 123456789 12345678  1245678   | 2356789   13456789  123456789 |
| 12356789  179       1245789   | 123689    12346789  12478     | 2356789   1379      12345789  |
| 123569    12345679  123456789 | 12345689  12346     12345678  | 123456789 12345679  123457    |
:-----------------------------------------------------------------------------------------------:
.-----------------------------------------------------------------------------------------------.
| 35678     13456789  123456789 | 23456789  46789     12456789  | 23456789  3456789   45789     |
| 123456789 137       123478    | 123689    12346789  1248      | 23689     123456789 2345789   |
| 123456789 1234567   12345678  | 12345689  23456789  12345678  | 235689    234569    23456789  |
:-----------------------------------------------------------------------------------------------:
| 23456789  46789     12456789  | 356789    13456789  145789    | 23456789  146789    12456789  |
| 123689    12346789  123456789 | 12356789  1379      12345789  | 12356789  12346789  12478     |
| 12345689  12346     12345678  | 123569    12345679  12347     | 123456789 12346     12345678  |
:-----------------------------------------------------------------------------------------------:
| 123456789 156789    12456789  | 23456789  12345678  1245678   | 2356789   13456789  123456789 |
| 12356789  179       12456789  | 23689     12346789  12478     | 2356789   1379      12345789  |
| 123569    12345679  123456789 | 12345689  12346     12345678  | 123456789 12345679  123457    |
:-----------------------------------------------------------------------------------------------:
.-----------------------------------------------------------------------------------------------.
| 35678     13456789  123456789 | 23456789  46789     12456789  | 23456789  3456789   45789     |
| 12356789  137       1234578   | 12389     12346789  12478     | 23689     123456789 2345789   |
| 123456789 1234567   12345678  | 12345689  23456789  12345678  | 235689    234569    23456789  |
:-----------------------------------------------------------------------------------------------:
| 23456789  146789    12456789  | 356789    13456789  145789    | 23456789  146789    12456789  |
| 23689     12346789  123456789 | 12356789  1379      12345789  | 12356789  12346789  12478     |
| 12345689  12346     123456789 | 123569    12345679  12347     | 12345689  12346     12345678  |
:-----------------------------------------------------------------------------------------------:
| 123456789 156789    12456789  | 123456789 12345678  1245678   | 2356789   13456789  123456789 |
| 12356789  179       1245789   | 23689     12346789  12478     | 2356789   1379      12345789  |
| 123569    12345679  123456789 | 12345689  12346     12345678  | 123456789 12345679  123457    |
:-----------------------------------------------------------------------------------------------:
.-----------------------------------------------------------------------------------------------.
| 35678     13456789  123456789 | 23456789  46789     12456789  | 23456789  3456789   45789     |
| 123456789 137       12347     | 1289      1234689   1248      | 23689     123456789 2345789   |
| 123456789 1234567   12345678  | 12345689  123456789 123456789 | 235689    2345689   23456789  |
:-----------------------------------------------------------------------------------------------:
| 23456789  146789    12456789  | 356789    13456789  145789    | 23456789  146789    12456789  |
| 23689     12346789  123456789 | 12356789  1379      12345789  | 12356789  12346789  12478     |
| 12345689  12346     123456789 | 123569    12345679  123457    | 12345689  12346     12345678  |
:-----------------------------------------------------------------------------------------------:
| 123456789 156789    12456789  | 23456789  12345678  1245678   | 2356789   13456789  123456789 |
| 12356789  179       1245789   | 23689     12346789  12478     | 2356789   1379      12345789  |
| 1235689   12345679  123456789 | 12345689  12346     12345678  | 123456789 12345679  123457    |
:-----------------------------------------------------------------------------------------------:
.-----------------------------------------------------------------------------------------------.
| 345678    13456789  123456789 | 23456789  46789     12456789  | 23456789  3456789   45789     |
| 123456789 137       1347      | 123689    12346789  12478     | 23689     123456789 2345789   |
| 123456789 1234567   12345678  | 12345689  123456789 12345678  | 235689    2345689   23456789  |
:-----------------------------------------------------------------------------------------------:
| 23456789  46789     12456789  | 356789    13456789  145789    | 23456789  146789    12456789  |
| 23689     12346789  123456789 | 12356789  1379      12345789  | 12356789  12346789  12478     |
| 12345689  12346     123456789 | 123569    12345679  123457    | 12345689  12346     12345678  |
:-----------------------------------------------------------------------------------------------:
| 123456789 156789    12456789  | 123456789 12345678  1245678   | 2356789   13456789  123456789 |
| 12356789  179       1245789   | 23689     12346789  12478     | 2356789   1379      12345789  |
| 123569    12345679  123456789 | 12345689  12346     12345678  | 123456789 12345679  123457    |
:-----------------------------------------------------------------------------------------------:
.-----------------------------------------------------------------------------------------------.
| 356789    13456789  123456789 | 23456789  46789     12456789  | 23456789  3456789   45789     |
| 12356789  137       123457    | 123689    12346789  12478     | 23689     123456789 2345789   |
| 123456789 1234567   12345678  | 12345689  12456789  12345678  | 235689    2345689   23456789  |
:-----------------------------------------------------------------------------------------------:
| 23456789  46789     12456789  | 1356789   13456789  145789    | 23456789  146789    12456789  |
| 23689     12346789  123456789 | 12356789  1379      12345789  | 12356789  12346789  12478     |
| 12345689  12346     123456789 | 123569    12345679  12347     | 12345689  12346     12345678  |
:-----------------------------------------------------------------------------------------------:
| 123456789 156789    12456789  | 123456789 12345678  1245678   | 2356789   13456789  123456789 |
| 12356789  179       1245789   | 23689     12346789  12478     | 2356789   1379      12345789  |
| 123569    12345679  123456789 | 12345689  12346     12345678  | 123456789 12345679  123457    |
:-----------------------------------------------------------------------------------------------:
.-----------------------------------------------------------------------------------------------.
| 356789    13456789  123456789 | 23456789  46789     12456789  | 23456789  3456789   45789     |
| 12356789  137       123457    | 123689    12346789  12478     | 23689     123456789 2345789   |
| 123456789 1234567   12345678  | 12345689  12456789  12345678  | 235689    234569    23456789  |
:-----------------------------------------------------------------------------------------------:
| 23456789  46789     12456789  | 1356789   13456789  145789    | 23456789  146789    12456789  |
| 23689     12346789  123456789 | 12356789  1379      12345789  | 12356789  12346789  12478     |
| 12345689  12346     12345678  | 123569    12345679  12347     | 123456789 12346     12345678  |
:-----------------------------------------------------------------------------------------------:
| 123456789 156789    12456789  | 123456789 12345678  1245678   | 2356789   13456789  123456789 |
| 12356789  179       1245789   | 123689    12346789  12478     | 2356789   1379      12345789  |
| 123569    12345679  123456789 | 12345689  12346     12345678  | 123456789 12345679  123457    |
:-----------------------------------------------------------------------------------------------:
.-----------------------------------------------------------------------------------------------.
| 356789    13456789  123456789 | 23456789  46789     12456789  | 23456789  3456789   45789     |
| 12356789  137       12345678  | 123689    12346789  12478     | 23689     123456789 2345789   |
| 123456789 1234567   12345678  | 12345689  23456789  12345678  | 235689    234569    23456789  |
:-----------------------------------------------------------------------------------------------:
| 23456789  46789     12456789  | 356789    13456789  145789    | 23456789  146789    12456789  |
| 23689     12346789  123456789 | 12356789  1379      12345789  | 12356789  12346789  12478     |
| 12345689  12346     123456789 | 123569    12345679  123457    | 12345689  12346     12345678  |
:-----------------------------------------------------------------------------------------------:
| 123456789 156789    12456789  | 123456789 12345678  1245678   | 2356789   13456789  123456789 |
| 12356789  179       1245789   | 23689     12346789  12478     | 2356789   1379      12345789  |
| 123569    12345679  123456789 | 12345689  12346     12345678  | 123456789 12345679  123457    |
:-----------------------------------------------------------------------------------------------:
.-----------------------------------------------------------------------------------------------.
| 35678     13456789  123456789 | 23456789  46789     12456789  | 23456789  3456789   45789     |
| 12356789  137       123457    | 123689    12346789  12478     | 235689    123456789 2345789   |
| 123456789 1234567   12345678  | 12345689  123456789 12345678  | 235689    2345689   23456789  |
:-----------------------------------------------------------------------------------------------:
| 23456789  46789     12456789  | 356789    13456789  145789    | 23456789  146789    12456789  |
| 23689     12346789  123456789 | 12356789  1379      12345789  | 12356789  12346789  12478     |
| 12345689  12346     12345678  | 123569    12345679  123457    | 123456789 12346     12345678  |
:-----------------------------------------------------------------------------------------------:
| 123456789 156789    12456789  | 123456789 12345678  1245678   | 2356789   13456789  123456789 |
| 12356789  179       1245789   | 23689     12346789  12478     | 2356789   1379      12345789  |
| 123569    12345679  123456789 | 12345689  12346     12345678  | 123456789 12345679  123457    |
:-----------------------------------------------------------------------------------------------:
.-----------------------------------------------------------------------------------------------.
| 356789    13456789  123456789 | 23456789  46789     12456789  | 23456789  3456789   45789     |
| 12356789  137       123457    | 12689     12346789  1248      | 235689    123456789 2345789   |
| 123456789 1234567   12345678  | 12345689  123456789 12345678  | 235689    2345689   23456789  |
:-----------------------------------------------------------------------------------------------:
| 23456789  46789     12456789  | 356789    13456789  145789    | 23456789  146789    12456789  |
| 23689     12346789  123456789 | 12356789  1379      12345789  | 12356789  12346789  12478     |
| 12345689  12346     12345678  | 123569    12345679  123457    | 123456789 12346     12345678  |
:-----------------------------------------------------------------------------------------------:
| 123456789 156789    12456789  | 123456789 12345678  1245678   | 2356789   13456789  123456789 |
| 12356789  179       1245789   | 123689    12346789  12478     | 2356789   1379      12345789  |
| 123569    12345679  123456789 | 12345689  12346     12345678  | 123456789 12345679  123457    |
:-----------------------------------------------------------------------------------------------:
.-----------------------------------------------------------------------------------------------.
| 356789    13456789  123456789 | 23456789  46789     12456789  | 23456789  3456789   45789     |
| 12356789  137       123457    | 123689    12346789  1248      | 23689     123456789 2345789   |
| 123456789 1234567   12345678  | 12345689  123456789 12345678  | 235689    234569    23456789  |
:-----------------------------------------------------------------------------------------------:
| 23456789  146789    12456789  | 1356789   13456789  145789    | 23456789  146789    12456789  |
| 23689     12346789  123456789 | 12356789  1379      12345789  | 12356789  12346789  12478     |
| 12345689  12346     12345678  | 123569    12345679  12347     | 123456789 12346     12345678  |
:-----------------------------------------------------------------------------------------------:
| 123456789 156789    12456789  | 123456789 12345678  1245678   | 2356789   13456789  123456789 |
| 12356789  179       1245789   | 123689    12346789  12478     | 2356789   1379      12345789  |
| 123569    12345679  123456789 | 12345689  12346     12345678  | 123456789 12345679  123457    |
:-----------------------------------------------------------------------------------------------:


Here are the original 11 puzzles in the 729 line format
Hidden Text: Show
Code: Select all
..3.5678.1.3456789123456789.23456789...4.678912.4567891234567891.34567891..45.789123.567891.3...7..12345.7..123....891234.678912.4...8.123..6.891........12345.7891234567891234567..12345678.123456.8912345678912345678.123.56.89123456..9123456789.23456789...4.678912.4567891.3.567891.34567891..45.789.234567891..4.678912.456789.23..6.891234.6789123456789123.567891.3...7.912345.789123.567891234.678912.4..78.123456.891234.6...123456789123.56..91234567.91234..7..123456.891234.6...12345678.1234567891...5678912.45678912345678912345678.12.45678..23.567891.3456789123456789123.567891.....7.912.45.789123..6.891234.678912.4..78..23.567891.3...7.912345.789123.56..91234567.9123456789123456.891234.6...12345678.1234567891234567.912345.7..
..3.5678.1.3456789123456789.23456789...4.678912.4567891234567891.34567891..45.7891234567891.3...7..1234..78.123..6.891234.678912.4...8.123..6.891........12345.7891234567891234567..12345678.123456.89.2345678912345678.123.56.89123456..9123456789.23456789...4.678912.456789..3.567891.34567891..45.789.234567891..4.678912.456789123..6.891234.6789123456789123.567891.3...7.912345.789123.567891234.678912.4..78.123456.891234.6...12345678.123.56..91234567.91234..7..1234567891234.6...12345678.1234567891...5678912.456789.2345678912345678.12.45678..23.567891.3456789123456789123.567891.....7.912.456789.23..6.891234.678912.4..78..23.567891.3...7.912345.789123.56..91234567.9123456789123456.891234.6...12345678.1234567891234567.912345.7..
..3.5678.1.3456789123456789.23456789...4.678912.4567891234567891.34567891..45.789123.567891.3...7..12345.78.123....891234.678912.4..78.123..6.891........12345.7891234567891234567..12345678.123456.89.2345678912345678.123.56.89123456..9123456789.234567891..4.678912.456789..3.567891.34567891..45.789.234567891..4.678912.456789.23..6.891234.6789123456789123.567891.3...7.912345.789123.567891234.678912.4..78.123456.891234.6...123456789123.56..91234567.91234..7..123456.891234.6...12345678.1234567891...5678912.45678912345678912345678.12.45678..23.567891.3456789123456789123.567891.....7.912.45.789.23..6.891234.678912.4..78..23.567891.3...7.912345.789123.56..91234567.9123456789123456.891234.6...12345678.1234567891234567.912345.7..
..3.5678.1.3456789123456789.23456789...4.678912.4567891234567891.34567891..45.7891234567891.3...7..1234..7..12.....891234.6.8912.4...8.123..6.891........12345.7891234567891234567..12345678.123456.89123456789123456789123.56.89123456.89123456789.234567891..4.678912.456789..3.567891.34567891..45.789.234567891..4.678912.456789.23..6.891234.6789123456789123.567891.3...7.912345.789123.567891234.678912.4..78.123456.891234.6...123456789123.56..91234567.912345.7..123456.891234.6...12345678.1234567891...5678912.456789.2345678912345678.12.45678..23.567891.3456789123456789123.567891.....7.912.45.789.23..6.891234.678912.4..78..23.567891.3...7.912345.789123.56.891234567.9123456789123456.891234.6...12345678.1234567891234567.912345.7..
..345678.1.3456789123456789.23456789...4.678912.4567891234567891.34567891..45.7891234567891.3...7..1.34..7..123..6.891234.678912.4..78.123..6.891........12345.7891234567891234567..12345678.123456.8912345678912345678.123.56.89123456.89123456789.23456789...4.678912.456789..3.567891.34567891..45.789.234567891..4.678912.456789.23..6.891234.6789123456789123.567891.3...7.912345.789123.567891234.678912.4..78.123456.891234.6...123456789123.56..91234567.912345.7..123456.891234.6...12345678.1234567891...5678912.45678912345678912345678.12.45678..23.567891.3456789123456789123.567891.....7.912.45.789.23..6.891234.678912.4..78..23.567891.3...7.912345.789123.56..91234567.9123456789123456.891234.6...12345678.1234567891234567.912345.7..
..3.567891.3456789123456789.23456789...4.678912.4567891234567891.34567891..45.789123.567891.3...7..12345.7..123..6.891234.678912.4..78.123..6.891........12345.7891234567891234567..12345678.123456.8912.45678912345678.123.56.89123456.89123456789.23456789...4.678912.4567891.3.567891.34567891..45.789.234567891..4.678912.456789.23..6.891234.6789123456789123.567891.3...7.912345.789123.567891234.678912.4..78.123456.891234.6...123456789123.56..91234567.91234..7..123456.891234.6...12345678.1234567891...5678912.45678912345678912345678.12.45678..23.567891.3456789123456789123.567891.....7.912.45.789.23..6.891234.678912.4..78..23.567891.3...7.912345.789123.56..91234567.9123456789123456.891234.6...12345678.1234567891234567.912345.7..
..3.567891.3456789123456789.23456789...4.678912.4567891234567891.34567891..45.789123.567891.3...7..12345.7..123..6.891234.678912.4..78.123..6.891........12345.7891234567891234567..12345678.123456.8912.45678912345678.123.56.89123456..9123456789.23456789...4.678912.4567891.3.567891.34567891..45.789.234567891..4.678912.456789.23..6.891234.6789123456789123.567891.3...7.912345.789123.567891234.678912.4..78.123456.891234.6...12345678.123.56..91234567.91234..7..1234567891234.6...12345678.1234567891...5678912.45678912345678912345678.12.45678..23.567891.3456789123456789123.567891.....7.912.45.789123..6.891234.678912.4..78..23.567891.3...7.912345.789123.56..91234567.9123456789123456.891234.6...12345678.1234567891234567.912345.7..
..3.567891.3456789123456789.23456789...4.678912.4567891234567891.34567891..45.789123.567891.3...7..12345678.123..6.891234.678912.4..78.123..6.891........12345.7891234567891234567..12345678.123456.89.2345678912345678.123.56.89123456..9123456789.23456789...4.678912.456789..3.567891.34567891..45.789.234567891..4.678912.456789.23..6.891234.6789123456789123.567891.3...7.912345.789123.567891234.678912.4..78.123456.891234.6...123456789123.56..91234567.912345.7..123456.891234.6...12345678.1234567891...5678912.45678912345678912345678.12.45678..23.567891.3456789123456789123.567891.....7.912.45.789.23..6.891234.678912.4..78..23.567891.3...7.912345.789123.56..91234567.9123456789123456.891234.6...12345678.1234567891234567.912345.7..
..3.5678.1.3456789123456789.23456789...4.678912.4567891234567891.34567891..45.789123.567891.3...7..12345.7..123..6.891234.678912.4..78.123.56.891........12345.7891234567891234567..12345678.123456.8912345678912345678.123.56.89123456.89123456789.23456789...4.678912.456789..3.567891.34567891..45.789.234567891..4.678912.456789.23..6.891234.6789123456789123.567891.3...7.912345.789123.567891234.678912.4..78.123456.891234.6...12345678.123.56..91234567.912345.7..1234567891234.6...12345678.1234567891...5678912.45678912345678912345678.12.45678..23.567891.3456789123456789123.567891.....7.912.45.789.23..6.891234.678912.4..78..23.567891.3...7.912345.789123.56..91234567.9123456789123456.891234.6...12345678.1234567891234567.912345.7..
..3.567891.3456789123456789.23456789...4.678912.4567891234567891.34567891..45.789123.567891.3...7..12345.7..12...6.891234.678912.4...8.123.56.891........12345.7891234567891234567..12345678.123456.8912345678912345678.123.56.89123456.89123456789.23456789...4.678912.456789..3.567891.34567891..45.789.234567891..4.678912.456789.23..6.891234.6789123456789123.567891.3...7.912345.789123.567891234.678912.4..78.123456.891234.6...12345678.123.56..91234567.912345.7..1234567891234.6...12345678.1234567891...5678912.45678912345678912345678.12.45678..23.567891.3456789123456789123.567891.....7.912.45.789123..6.891234.678912.4..78..23.567891.3...7.912345.789123.56..91234567.9123456789123456.891234.6...12345678.1234567891234567.912345.7..
..3.567891.3456789123456789.23456789...4.678912.4567891234567891.34567891..45.789123.567891.3...7..12345.7..123..6.891234.678912.4...8.123..6.891........12345.7891234567891234567..12345678.123456.8912345678912345678.123.56.89123456..9123456789.234567891..4.678912.4567891.3.567891.34567891..45.789.234567891..4.678912.456789.23..6.891234.6789123456789123.567891.3...7.912345.789123.567891234.678912.4..78.123456.891234.6...12345678.123.56..91234567.91234..7..1234567891234.6...12345678.1234567891...5678912.45678912345678912345678.12.45678..23.567891.3456789123456789123.567891.....7.912.45.789123..6.891234.678912.4..78..23.567891.3...7.912345.789123.56..91234567.9123456789123456.891234.6...12345678.1234567891234567.912345.7..
User avatar
tarek
 
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Singles backdoor size 9

Postby 999_Springs » Tue Apr 15, 2014 2:51 pm

because the way tarek's last post was written clearly indicates he does not believe me when i tell him EVER SO VOCIFEROUSLY that there is a very easily found singles backdoor size NINE in the world of pencilmark sudoku i'll just post it here

so you start with the canonical grid

Code: Select all
123|456|789
456|789|123
789|123|456
---+---+---
231|564|897
564|897|231
897|231|564
---+---+---
312|645|978
645|978|312
978|312|645


and then you regard that grid as a concatenation of nine 3x3 latin squares. easy enough right?

then you replace each 3x3 latin square with this (replace 1,2,3 with 4,5,6 or 7,8,9 as appropriate)

Code: Select all
123  12   123
12   23   13
13   123  12


or this

Code: Select all
1    23   23
23   13   12
23   12   123


to get a grid that looks something like this, which is what you'd get if you only used the first one:

Code: Select all
123  12   123 |456  45   456 |789  78   789
456  45   456 |789  78   789 |123  12   123
789  78   789 |123  12   123 |456  45   456
--------------+--------------+-------------
12   23   13  |45   56   46  |78   89   79 
45   56   46  |78   89   79  |12   23   13 
78   89   79  |12   23   13  |45   56   46 
--------------+--------------+-------------
13   123  12  |46   456  45  |79   789  78 
46   456  45  |79   789  78  |13   123  12 
79   789  78  |13   123  12  |46   456  45 

voila

you may now bow down in eternal worship to me
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Re: Singles backdoor size 9

Postby blue » Fri Apr 18, 2014 5:29 pm

999_Springs wrote:
Code: Select all
123  12   123 |456  45   456 |789  78   789
456  45   456 |789  78   789 |123  12   123
789  78   789 |123  12   123 |456  45   456
--------------+--------------+-------------
12   23   13  |45   56   46  |78   89   79 
45   56   46  |78   89   79  |12   23   13 
78   89   79  |12   23   13  |45   56   46 
--------------+--------------+-------------
13   123  12  |46   456  45  |79   789  78 
46   456  45  |79   789  78  |13   123  12 
79   789  78  |13   123  12  |46   456  45 

voila

Nine completely independent (3x3) puzzles, each with backdoor size of one. Very clever :!:
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Re: Singles backdoor size 9

Postby tarek » Fri Apr 18, 2014 10:21 pm

blue wrote:
999_Springs wrote:
Code: Select all
123  12   123 |456  45   456 |789  78   789
456  45   456 |789  78   789 |123  12   123
789  78   789 |123  12   123 |456  45   456
--------------+--------------+-------------
12   23   13  |45   56   46  |78   89   79 
45   56   46  |78   89   79  |12   23   13 
78   89   79  |12   23   13  |45   56   46 
--------------+--------------+-------------
13   123  12  |46   456  45  |79   789  78 
46   456  45  |79   789  78  |13   123  12 
79   789  78  |13   123  12  |46   456  45 

voila

Nine completely independent (3x3) puzzles, each with backdoor size of one. Very clever :!:


Indeed well done. And that is your nine backdoors!!!
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637 candidates

Postby dobrichev » Tue Jul 09, 2019 7:50 am

This puzzle has 637 candidates.
Code: Select all
+-------------------------------+-------------------------------+-------------------------------+
| 123456789 123456789 23456789  | 12346789  123456789 1246789   | 12346789  1234678   123456789 |
| 123456789 135       23567     | 2356789   123456789 123456789 | 123456789 237       123456789 |
| 123456789 123456789 123456789 | 1256789   123456789 1245789   | 123456789 12346789  123456789 |
+-------------------------------+-------------------------------+-------------------------------+
| 123456789 123456789 123456789 | 256789    123456789 123456789 | 123456789 123456789 123456789 |
| 123456789 14        123456789 | 1245789   123456789 123456789 | 45        123456789 123456789 |
| 123456789 123456789 123456789 | 12356789  123456789 123456789 | 123456789 123456789 123456789 |
+-------------------------------+-------------------------------+-------------------------------+
| 123456789 123456789 23456789  | 123456789 1234569   12346789  | 123456789 27        123456789 |
| 123456789 123456789 123456789 | 123456789 13459     1469      | 123456789 123456789 123456789 |
| 123456789 123456789 25        | 123456789 12345679  12345679  | 123456789 27        257       |
+-------------------------------+-------------------------------+-------------------------------+

Play it
It appears 13 years after Ruud's record of 633. It was found using JPF's seeds from 17s, then vicinity search.
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Re: Pencilmark only Sudoku

Postby tarek » Wed Jul 10, 2019 8:07 pm

I can see that you kept your interest here too dobrichev!

Well done!

The Pencilmark sudoku has many uncharted areas where records can be broken. I've already had a play with "Most difficult" and "biggest singles backdoor". I'm sure with better techniques and faster, more powerful machines these records can be easily broken


tarek
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Re: Pencilmark only Sudoku

Postby dobrichev » Thu Jul 11, 2019 9:11 pm

Hi Tarek,

IMO Pencilmark sudoku is maybe the most natural extension of the standard rules. All solvers in one or another way use pencilmarks, and I have no explanation why it isn't deeply investigated.

Аt the low-clue end, puzzles with 92 disallowed candidates + 637 allowed are maybe low-hanging fruits. At the moment my cache has 133488 such puzzles, living on at least 3 islands not connected by {-1,+1}.
For 93/636 the count is 454813 and growing, again on at least 3 islands.
This was collected form scratch for about one week, using 2 old home PC and unstable software.

High-clue end is entirely unexplored.

For the "Most difficult" I know you are far ahead and expect some high serate ratings from you. Is there something above ER 11+? Is ER applicable at all for Penclimark sudoku?
The preliminary observations show that low-clue puzzles are easier to solve, just like in the ordinary game.

Finally, if there is such animal as "biggest singles backdoor required to ensure that puzzle has multiple solutions", then records are possible.

Cheers,
MD
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Re: Pencilmark only Sudoku

Postby Mathimagics » Fri Jul 12, 2019 1:42 pm

dobrichev wrote:IMO Pencilmark sudoku is maybe the most natural extension of the standard rules. All solvers in one or another way use pencilmarks, and I have no explanation why it isn't deeply investigated.

I agree 100%.

It's just an alternate form of presenting the puzzle, and rich with possibilities that the standard method doesn't cover …

I'd like to delve into it more, just as soon as I have some spare time .. . perhaps in 2023? 2025? :?
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Re: Pencilmark only Sudoku

Postby dobrichev » Fri Jul 12, 2019 3:08 pm

Too early. Follow the tradition and wait for 12 and a half years after my last posting on the subject.
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Re: Pencilmark only Sudoku

Postby tarek » Fri Jul 12, 2019 6:02 pm

Hi Mladen and Mathimagics,
dobrichev wrote:Is there something above ER 11+? Is ER applicable at all for Penclimark sudoku?

1to9only has kindly contributed to an SE version that handles the Sukaku. You will need to use the 729 character format. You can follow that discussion by following this link http://forum.enjoysudoku.com/help-with-sudoku-explainer-t6677-60.html?hilit=sukaku#p271963

Sadly I didn't have time dedicated to this task since although I was initially enthusiastic about it. The slow SE had a role. The idea is to have an SE defeating puzzle (i.e. 12+) but I didn't even test my Singles backdoor size 6 puzzles :oops:

Minimal singles backdoor size 6 puzzles: Show
tarek wrote:Minimal singles backdoor size 6 puzzles

Code: Select all
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..3.567891.3456789123456789.23456789...4.678912.4567891234567891.34567891..45.789123.567891.3...7..12345678.123..6.891234.678912.4..78.123..6.891........12345.7891234567891234567..12345678.123456.89.2345678912345678.123.56.89123456..9123456789.23456789...4.678912.456789..3.567891.34567891..45.789.234567891..4.678912.456789.23..6.891234.6789123456789123.567891.3...7.912345.789123.567891234.678912.4..78.123456.891234.6...123456789123.56..91234567.912345.7..123456.891234.6...12345678.1234567891...5678912.45678912345678912345678.12.45678..23.567891.3456789123456789123.567891.....7.912.45.789.23..6.891234.678912.4..78..23.567891.3...7.912345.789123.56..91234567.9123456789123456.891234.6...12345678.1234567891234567.912345.7..
..3.5678.1.3456789123456789.23456789...4.678912.4567891234567891.34567891..45.789123.567891.3...7..12345.7..123..6.891234.678912.4..78.123.56.891........12345.7891234567891234567..12345678.123456.8912345678912345678.123.56.89123456.89123456789.23456789...4.678912.456789..3.567891.34567891..45.789.234567891..4.678912.456789.23..6.891234.6789123456789123.567891.3...7.912345.789123.567891234.678912.4..78.123456.891234.6...12345678.123.56..91234567.912345.7..1234567891234.6...12345678.1234567891...5678912.45678912345678912345678.12.45678..23.567891.3456789123456789123.567891.....7.912.45.789.23..6.891234.678912.4..78..23.567891.3...7.912345.789123.56..91234567.9123456789123456.891234.6...12345678.1234567891234567.912345.7..
..3.567891.3456789123456789.23456789...4.678912.4567891234567891.34567891..45.789123.567891.3...7..12345.7..12...6.891234.678912.4...8.123.56.891........12345.7891234567891234567..12345678.123456.8912345678912345678.123.56.89123456.89123456789.23456789...4.678912.456789..3.567891.34567891..45.789.234567891..4.678912.456789.23..6.891234.6789123456789123.567891.3...7.912345.789123.567891234.678912.4..78.123456.891234.6...12345678.123.56..91234567.912345.7..1234567891234.6...12345678.1234567891...5678912.45678912345678912345678.12.45678..23.567891.3456789123456789123.567891.....7.912.45.789123..6.891234.678912.4..78..23.567891.3...7.912345.789123.56..91234567.9123456789123456.891234.6...12345678.1234567891234567.912345.7..
..3.567891.3456789123456789.23456789...4.678912.4567891234567891.34567891..45.789123.567891.3...7..12345.7..123..6.891234.678912.4...8.123..6.891........12345.7891234567891234567..12345678.123456.8912345678912345678.123.56.89123456..9123456789.234567891..4.678912.4567891.3.567891.34567891..45.789.234567891..4.678912.456789.23..6.891234.6789123456789123.567891.3...7.912345.789123.567891234.678912.4..78.123456.891234.6...12345678.123.56..91234567.91234..7..1234567891234.6...12345678.1234567891...5678912.45678912345678912345678.12.45678..23.567891.3456789123456789123.567891.....7.912.45.789123..6.891234.678912.4..78..23.567891.3...7.912345.789123.56..91234567.9123456789123456.891234.6...12345678.1234567891234567.912345.7..


dobrichev wrote:if there is such animal as "biggest singles backdoor required to ensure that puzzle has multiple solutions", then records are possible.

Now then ... You'll love this.. Your Minimum clue puzzle is not necessarily a locally minimal one. If adding any one possible candidate to your puzzle results in multiple solutions then that is a locally minimal puzzle. So your record can be for the Minimum clue locally minimal Pencil-mark sudoku ... Interesting isn't it?!

dobrichev wrote:Too early. Follow the tradition and wait for 12 and a half years after my last posting on the subject.
I agree
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tarek
 
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Re: Pencilmark only Sudoku

Postby dobrichev » Fri Jul 12, 2019 7:33 pm

I used 1to9only's extension of serate in GUI mode just to verify independently that the published puzzle has single solution. I will try it in batch mode.
tarek wrote:
dobrichev wrote:if there is such animal as "biggest singles backdoor required to ensure that puzzle has multiple solutions", then records are possible.

Now then ... You'll love this.. Your Minimum clue puzzle is not necessarily a locally minimal one. If adding any one possible candidate to your puzzle results in multiple solutions then that is a locally minimal puzzle. So your record can be for the Minimum clue locally minimal Pencil-mark sudoku ... Interesting isn't it?!

If this is a joke, I did not get it. Of course the published puzzle, as well as other 133487 ED puzzles of the same size, ARE locally minimal.

My point was that while doing {-1, +1} at pencilmarks level, some puzzles required more than ten millions guesses before finding a secondary solution. In singles. Changing guessing strategy reduced the guessing 50 times, but only after including the line-block eliminations the guesses were reduced to a practical value of hundreds. Locked candidates aren't considered singles, right?
dobrichev
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Re: Pencilmark only Sudoku

Postby tarek » Sat Jul 13, 2019 9:51 am

dobrichev wrote:I used 1to9only's extension of serate in GUI mode just to verify independently that the published puzzle has single solution. I will try it in batch mode.
tarek wrote:
dobrichev wrote:if there is such animal as "biggest singles backdoor required to ensure that puzzle has multiple solutions", then records are possible.

Now then ... You'll love this.. Your Minimum clue puzzle is not necessarily a locally minimal one. If adding any one possible candidate to your puzzle results in multiple solutions then that is a locally minimal puzzle. So your record can be for the Minimum clue locally minimal Pencil-mark sudoku ... Interesting isn't it?!

If this is a joke, I did not get it. Of course the published puzzle, as well as other 133487 ED puzzles of the same size, ARE locally minimal.

My point was that while doing {-1, +1} at pencilmarks level, some puzzles required more than ten millions guesses before finding a secondary solution. In singles. Changing guessing strategy reduced the guessing 50 times, but only after including the line-block eliminations the guesses were reduced to a practical value of hundreds. Locked candidates aren't considered singles, right?

No joke intended but I also didn't mean YOUR puzzles either. It is the concept of minimality which I found easier to understand with our regular vanilla sudokus. With Pencilmark sudokus you can see how minimal means more candidates not less.

With the candidate space getting closer to 729, vicinity searches even at {-1 +1} level is tedious ... I'm hoping that you, Mathimagics and others will optimise and find quicker ways of doing this.

tarek
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Re: Pencilmark only Sudoku

Postby dobrichev » Sat Jul 13, 2019 10:17 am

Yes, minimal and maximal are confusing. Don't ask me how many times I renamed the variables, functions, and parameters in my code.

In my current process a combined {-1,+1} and {-2,+1} mutation takes in average a bit more than one second per puzzle. Rarely some puzzles require 15+ seconds. Puzzles with more restrictions (farther from the low-end) take 1.5 seconds or more.
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Re: Pencilmark only Sudoku

Postby HATMAN » Mon Jul 29, 2019 9:00 am

On Dobrichev's

"why it isn't deeply investigated"

I would surmise that while they are interesting mathematically (combinatorics and group theory I guess) or computationally, they are a long way from what people want to solve hence naturally of less interest.

To deal with this by avoiding piles of raw pencil-marks, various ways have been tried. Early on there were Phrrhon's double clue ones, I've done a few myself and I recently did negative clue ones posted here.

A while ago para did a killer with adjacent cage totals i.e. they were wrong by +/-1. So here is a vanilla (not plain) on the same basis:
The actual cell values are adjacent to the number given. This cried out to me to be NC, so it is. It is also X.

Not too difficult: one small fish and some NC moves of similar level.

I'll post more if anyone likes them.

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Re: Pencilmark only Sudoku

Postby HATMAN » Wed Jul 31, 2019 8:03 am

Sukaku Twins 2

Another Twins Sukaku: NC and X and the givens are wrong by one.

I'm posting this because it is unique with just the three red 2 clues. I knew that the solution space was restrained* as for some reason X and NC interfere with each other, but I was surprised to find it unique with just three clues.

* this word usage is from Ruud long ago which I have kept up - tariq will no doubt remember.

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