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PostPosted: Mon May 10, 2010 12:47 pm 
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Restrained Sudoku puzzles are not new. The extra constraints usually makes it easy to require less clues for a unique solution.
The minimum number of clues for 9x9 (regional constraints only) puzzles can't go below 8.

I've introduced on the Players forum (RIP) several examples of such puzzles.

The puzzle combination of Sudoku with X diagonals, Diamond diagonals, Disjoint groups & Disjoint groups diagonals is an example of a Restrained puzzle that allows this number of clues easily.

This thread will be dedicated to such 8-clue Regional-constraints Restrained puzzles


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PostPosted: Mon May 10, 2010 1:05 pm 
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The follwing is a combo (Sudoku, X, Disjoint Groups (DG), Diamond diagonals (DD) & DG diagonals (DGD)) with 42 "Declared" regional constraints.

It is special because it starts with 68 consecutive empty cells & still achieves a single solution. It is beyond What many would consider humanly solvable so no need to attempt un-aided :naughty:

Here is the puzzle with the constraints explained:
Image

Solutin:
Image

& the above in text formats:
Code:
....................................................................6..521.849..7
 . . . | . . . | . . . 
 . . . | . . . | . . . 
 . . . | . . . | . . . 
------+-------+------
 . . . | . . . | . . . 
 . . . | . . . | . . . 
 . . . | . . . | . . . 
------+-------+------
 . . . | . . . | . . . 
 . . . | . . 6 | . . 5 
 2 1 . | 8 4 9 | . . 7

124935876857261943693478521341687259572394618986152734768513492439726185215849367
 1 2 4 | 9 3 5 | 8 7 6 
 8 5 7 | 2 6 1 | 9 4 3 
 6 9 3 | 4 7 8 | 5 2 1 
------+-------+------
 3 4 1 | 6 8 7 | 2 5 9 
 5 7 2 | 3 9 4 | 6 1 8 
 9 8 6 | 1 5 2 | 7 3 4 
------+-------+------
 7 6 8 | 5 1 3 | 4 9 2 
 4 3 9 | 7 2 6 | 1 8 5 
 2 1 5 | 8 4 9 | 3 6 7


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PostPosted: Fri May 14, 2010 7:33 pm 
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Finally with the some programming, I've landed some nice symmetric examples.

image to explain the constraints:
Image
From top Left Clockwise: X diagonals (X), Disjoint Groups (DG), Diamond Diagonals (DD) & Disjoint Groups Diagonals (DGD)

The 1st is an 8 clue puzzle that requires advanced fish
Code:
........438...9...................................................1...752........
 . . . | . . . | . . 4 
 3 8 . | . . 9 | . . . 
 . . . | . . . | . . . 
------+-------+------
 . . . | . . . | . . . 
 . . . | . . . | . . . 
 . . . | . . . | . . . 
------+-------+------
 . . . | . . . | . . . 
 . . . | 1 . . | . 7 5 
 2 . . | . . . | . . . 
162873594387459216945621738439215687521768943876394152758946321694132875213587469

The 2nd is a 9 clue puzzle that requires only "SUPERIOR" level techniques
Code:
................6...1....4........5.....9.....8........3....7...2................
 . . . | . . . | . . . 
 . . . | . . . | . 6 . 
 . . 1 | . . . | . 4 . 
------+-------+------
 . . . | . . . | . 5 . 
 . . . | . 9 . | . . . 
 . 8 . | . . . | . . . 
------+-------+------
 . 3 . | . . . | 7 . . 
 . 2 . | . . . | . . . 
 . . . | . . . | . . .
674589123358412967291736845912367458746895231583124679835241796129673584467958312

Walkthrough for 2nd puzzle:
Code:
................6...1....4........5.....9.....8........3....7...2................
 . . . | . . . | . . . 
 . . . | . . . | . 6 . 
 . . 1 | . . . | . 4 . 
------+-------+------
 . . . | . . . | . 5 . 
 . . . | . 9 . | . . . 
 . 8 . | . . . | . . . 
------+-------+------
 . 3 . | . . . | 7 . . 
 . 2 . | . . . | . . . 
 . . . | . . . | . . . 
d/: x diagonal /
d\: x diagonal \
dG: disjoint group
DGD: disjoint group diagonal
D: diamond
b: box
r: row
c: column
6 is a given clue in r2c8
[r1c789][r2c12345679][r3c79][r4c6][r5c28][r6c48][r7c38][r8c58][r9c18]<>6
1 is a given clue in r3c3
[r1c1239][r2c1236][r3c1245679][r4c34][r5c37][r6c369][r7c35][r8c348][r9c1369]<>1
4 is a given clue in r3c8
[r1c789][r2c79][r3c1245679][r56789c8][r69c5][r9c2]<>4
5 is a given clue in r4c8
[r1c2589][r24c6][r4c1234579][r56c789][r7c58][r8c48][r9c18]<>5
9 is a given clue in r5c5
[r19c159][r24c2456][r3c57][r5c12346789][r68c4568][r7c35]<>9
8 is a given clue in r6c2
[r1c29][r2c26][r3c25][r45c123][r6c13456789][r8c4][r9c1258]<>8
3 is a given clue in r7c2
[r1c258][r23459c2][r47c5][r7c134689][r89c13]<>3
7 is a given clue in r7c7
[r1c1479][r2c267][r3c57][r4c147][r5c37][r6c67][r7c1345689][r8c4789][r9c1789]<>7
2 is a given clue in r8c2
[r1c29][r2c25][r3c27][r4c26][r5c28][r68c4][r7c13][r8c1356789][r9c123]<>2
2 in r2c6 is a hidden single for D2
[r13c456][r2c13479][r5c369][r679c6]<>2
6 in r8c4 is a hidden single for D2
[r134579c4][r5c17][r79c56][r8c13679]<>6
3 in r1c9 is a hidden and naked single
[r1c13467][r23c79][r4c369][r5c19][r6c49][r89c9][r9c5]<>3
4 in r9c1 is the last digit to fill the sector
[r1c15][r245678c1][r49c6][r59c9][r69c47][r789c3]<>4
1 forms a intersection in d/ & b5
[r46c5][r5c46]<>1
7 forms a intersection in d/ & b5
[r46c5][r5c46]<>7
17 is a hidden double in b5
[r4c6]<>8 [r6c4]<>5
5 forms a intersection in d/ & D1
[r1c1][r2c4][r8c6][r9c9]<>5
8 forms a intersection in d/ & D1
[r1c1][r2c4][r8c6][r9c9]<>8
5 forms a intersection in d/ & DGD2
[r3c5][r5c3]<>5
8 forms a intersection in d/ & DGD2
[r5c7][r7c5]<>8
26 is a naked double in d\
[r1c5]<>6 [r4c2]<>6 [r4c4]<>2 [r5c1]<>2 [r5c9]<>6 [r6c6]<>6 [r6c8]<>2 [r9c5]<>2
2 forms a x-wing in b3d\ & r1c9
[r1c3][r467c9]<>2
2 in r4c3 is a hidden single for dG3
[r4c157][r6c13]<>2
2 forms a intersection in b6 & c7
[r19c7]<>2
6 forms a x-wing in b7d\ & r9c1
[r346c1][r9c7]<>6
6 in r6c7 is a hidden single for dG7
[r4c79][r6c359]<>6
2 in r5c7 is a hidden single in more than one sector
[r5c4][r7c5]<>2
2 in r6c5 is a hidden single in more than one sector
[r9c8]<>2
6 in r5c3 is a hidden single for b4
[r19c3][r3c5][r5c6]<>6
3 in r3c5 is a hidden and naked single
[r2c45][r3c146][r69c8][r8c5]<>3
4 in r7c5 is a hidden and naked single
[r14c2][r248c5][r7c469][r8c6]<>4
6 in r4c5 is a hidden single in more than one sector
[r1c2]<>6
4 in r2c4 is a hidden single for D1
[r1c46][r2c23][r45c4][r8c7]<>4
4 in r1c3 is a hidden single in more than one sector
[r4c9][r6c3]<>4
4 in r4c7 is a hidden single in more than one sector
[r6c9]<>4
4 in r5c2 is a hidden single in more than one sector
[r5c6]<>4
4 in r6c6 is a hidden single in more than one sector
r8c9 has the last 4 uncovered
5 in r2c2 is a hidden and naked single
[r2c13579][r3c12][r8c5][r9c2]<>5
5 in r9c5 is a hidden single in more than one sector
[r5c1][r7c46][r9c3467]<>5
3 in r5c8 is a hidden single for b6
[r5c46][r8c8]<>3
3 in r4c4 is a hidden single in more than one sector
[r4c1][r9c4]<>3
8 in r8c8 is a hidden and naked single
[r17c8][r28c5][r7c9][r8c137][r9c7]<>8
8 in r1c5 is a hidden single in more than one sector
[r1c467][r3c46][r5c9]<>8
8 in r4c9 is a hidden single in more than one sector
[r23c9][r7c36]<>8
8 in r3c7 is a hidden single in more than one sector
[r2c7][r3c1][r9c4]<>8
5 in r7c3 is the last digit to fill the sector
[r1c6][r68c3][r7c19][r8c1]<>5
5 in r6c1 is a hidden single in more than one sector
[r3c4]<>5
3 in r6c3 is a hidden single in more than one sector
[r2c3][r9c6]<>3
3 in r2c1 is a hidden single in more than one sector
[r8c7]<>3
8 in r2c3 is a hidden single in more than one sector
[r5c6][r9c3]<>8
5 in r5c6 is a hidden and naked single
[r3c6][r5c4]<>5
8 in r5c4 is a hidden and naked single
[r7c4]<>8
5 in r1c4 is a hidden single in more than one sector
[r1c7]<>5
5 in r3c9 is a hidden single in more than one sector
r8c7 has the last 5 uncovered
9 in r4c1 is a hidden single in more than one sector
[r1c7][r378c1][r7c4]<>9
9 in r6c9 is a hidden single in more than one sector
[r27c9][r39c6][r9c3]<>9
9 in r2c7 is a hidden single in more than one sector
[r1c8][r9c7]<>9
8 in r7c1 is a hidden single in more than one sector
r9c6 has the last 8 uncovered
6 in r1c1 is a hidden single in more than one sector
[r1c6][r3c2][r9c9]<>6
2 in r9c9 is the last digit to fill the sector
[r7c8][r9c4]<>2
2 in r1c8 is a hidden single in more than one sector
2 in r3c1 is a hidden single in more than one sector
r7c4 has the last 2 uncovered
1 in r1c7 is the last digit to fill the sector
[r1c6][r2c9][r9c7]<>1
7 in r2c9 is the last digit to fill the sector
[r2c5][r5c9][r8c36]<>7
1 in r2c5 is the last digit to fill the sector
[r8c5]<>1
7 in r8c5 is the last digit to fill the sector
[r8c1][r9c4]<>7
3 in r9c7 is the last digit to fill the sector
r8c6 has the last 3 uncovered
1 in r5c9 is a hidden and naked single
[r5c1][r6c8][r7c9]<>1
7 in r5c1 is the last digit to fill the sector
[r4c2]<>7
1 in r4c2 is the last digit to fill the sector
[r4c6][r7c8][r9c2]<>1
7 in r4c6 is the last digit to fill the sector
[r13c6][r6c4]<>7
1 in r6c4 is the last digit to fill the sector
[r9c4]<>1
7 in r6c8 is the last digit to fill the sector
[r39c2]<>7
6 in r7c9 is the last digit to fill the sector
1 in r8c1 is the last digit to fill the sector
9 in r8c3 is the last digit to fill the sector
[r9c2]<>9
7 in r9c3 is the last digit to fill the sector
6 in r3c6 is the last digit to fill the sector
6 in r9c2 is the last digit to fill the sector
7 in r1c2 is a hidden single in more than one sector
9 in r1c6 is the last digit to fill the sector
[r3c4][r7c6]<>9
9 in r3c2 is the last digit to fill the sector
[r9c8]<>9
7 in r3c4 is the last digit to fill the sector
1 in r7c6 is the last digit to fill the sector
9 in r7c8 is the last digit to fill the sector
9 in r9c4 is the last digit to fill the sector
1 in r9c8 is the last digit to fill the sector
Puzzle Solved
 Solution:
674589123358412967291736845912367458746895231583124679835241796129673584467958312


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PostPosted: Thu May 20, 2010 6:57 am 
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Another example of a constrained puzzle is the Clover X (Sudoku, Windoku, Asterisk & X diagonals), here is a very very difficult 8 clue example:

Image
Code:
.....9..........5..........2......7...........8......4..........3..........1.....
 . . . | . . 9 | . . . 
 . . . | . . . | . 5 . 
 . . . | . . . | . . . 
------+-------+------
 2 . . | . . . | . 7 . 
 . . . | . . . | . . . 
 . 8 . | . . . | . . 4 
------+-------+------
 . . . | . . . | . . . 
 . 3 . | . . . | . . . 
 . . . | 1 . . | . . .

Solution:
Code:
526349187497218356318576429263451978954867231781932564142683795635794812879125643
 5 2 6 | 3 4 9 | 1 8 7 
 4 9 7 | 2 1 8 | 3 5 6 
 3 1 8 | 5 7 6 | 4 2 9 
------+-------+------
 2 6 3 | 4 5 1 | 9 7 8 
 9 5 4 | 8 6 7 | 2 3 1 
 7 8 1 | 9 3 2 | 5 6 4 
------+-------+------
 1 4 2 | 6 8 3 | 7 9 5 
 6 3 5 | 7 9 4 | 8 1 2 
 8 7 9 | 1 2 5 | 6 4 3


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PostPosted: Fri May 21, 2010 7:26 pm 
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the Anti-King Anti-Knight Sudoku is another example where the 8-clue barrier can be easily reached. Here are 4 symmetrical examples of variable difficulties:

This is an EASY

Image
Code:
.............................3.974.............281.5.............................
 . . . | . . . | . . . 
 . . . | . . . | . . . 
 . . . | . . . | . . . 
------+-------+------
 . . 3 | . 9 7 | 4 . . 
 . . . | . . . | . . . 
 . . 2 | 8 1 . | 5 . . 
------+-------+------
 . . . | . . . | . . . 
 . . . | . . . | . . . 
 . . . | . . . | . . .

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


This is a SUPERIOR

Image
Code:
.........................5.2...9.......8.7.......1...4.3.........................
 . . . | . . . | . . . 
 . . . | . . . | . . . 
 . . . | . . . | . 5 . 
------+-------+------
 2 . . | . 9 . | . . . 
 . . . | 8 . 7 | . . . 
 . . . | . 1 . | . . 4 
------+-------+------
 . 3 . | . . . | . . . 
 . . . | . . . | . . . 
 . . . | . . . | . . .

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


This is MILDLY ADVANCED

Image
Code:
.........................5.2...9.7.............8.1...4.3.........................
 . . . | . . . | . . . 
 . . . | . . . | . . . 
 . . . | . . . | . 5 . 
------+-------+------
 2 . . | . 9 . | 7 . . 
 . . . | . . . | . . . 
 . . 8 | . 1 . | . . 4 
------+-------+------
 . 3 . | . . . | . . . 
 . . . | . . . | . . . 
 . . . | . . . | . . .

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


And this is IMPOSSIBLE
Image
Code:
....97..4..................3.........................5..................2..81....
 . . . | . 9 7 | . . 4 
 . . . | . . . | . . . 
 . . . | . . . | . . . 
------+-------+------
 3 . . | . . . | . . . 
 . . . | . . . | . . . 
 . . . | . . . | . . 5 
------+-------+------
 . . . | . . . | . . . 
 . . . | . . . | . . . 
 2 . . | 8 1 . | . . .

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


Enjoy,

tarek


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PostPosted: Sat May 22, 2010 1:03 pm 
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Hands 2

I worked on these with tarek and udosuk a while ago on sudoku.com.
Just before it closed I started looking at what you could do with just a single set of extra groups. I posted one that had only 47 sets of solutions that I was very pleased with. Nikoli_freak took up the challenge and immediately came back with a whole set of ones with low solution sets including many single setters. This is/was his favorite single setter, which he called "Hands" as it remined him of a pair of praying hands.

The following puzzle is easy - so it is to be done with no pencil marks.


Image

Part of the challenge was to make them compatible with X and Old Lace - which hands is. A slightly different and more difficult one:

Hands X OL 3
Image


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PostPosted: Sat May 22, 2010 1:29 pm 
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tarek

I enjoyed the AK-AN superior - thanks.

Maurice


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PostPosted: Sun May 23, 2010 8:49 am 
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Thanks Maurice, great puzzles ... hopefully Nikoli_Freak will re-surface again ....

I managed to find a symmetric example with an Anti-Knight top-down Rt-Lt Wrap constraint.

In this variant each cell has 8 cells which are a Knight's move away

here is a demo with cells r1c1(red), r5c2(green) & r6c6(blue) highlighted
Image

The puzzle itself is very difficult so don't solve unaided :ugeek:
Image
Text format
Code:
.........................5.......7.4...1.9...2.8.......3.........................
 . . . | . . . | . . . 
 . . . | . . . | . . . 
 . . . | . . . | . 5 . 
------+-------+------
 . . . | . . . | 7 . 4 
 . . . | 1 . 9 | . . . 
 2 . 8 | . . . | . . . 
------+-------+------
 . 3 . | . . . | . . . 
 . . . | . . . | . . . 
 . . . | . . . | . . . 

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


I have been posting these on the New Sudoku Forum ... I will try to stick to a unique topic per forum, the problem is that there are members here that are not members there and vice versa

tarek


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PostPosted: Sun May 23, 2010 10:54 pm 
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Hi Maurice,

When you made the reference to OL X, did you mean that both puzzles can be solved using the HANDS groups ONLY, and that the solution grid is OL X compatible :ugeek: ?

tarek


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PostPosted: Mon May 24, 2010 8:02 am 
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That is correct for the first one but with the second I have not tried solving it without the X and OL. It does not mean it is not solvable that way though.


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