## What is most solved cells with "n" clues?

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

### What is most solved cells with "n" clues?

I was wondering if anyone has researched what is the most cells which can be solved given "n" clues, for the range n = 4 thru 17 (I believe 4 is the fewest clues needed to solve one position, and although rare, 17 is enough to form a complete sudoku).

With the data I would be interested in a graph of "qty solved" vs. "givens". Here is what I started so far (just a quick start):

Code: Select all
givens   qty solved
-------   ------------
4          1
6          2
9          3
...
13        16
...
17        64

Here is a case where 4 givens solves one position:

Code: Select all
1 - - - - - - - -
- - - 1 - - - - -
- - - - - - 2 3 -
- - - - - - - - -
- - - - - - - - -
- - - - - - - - -
- - - - - - - - -
- - - - - - - - -
- - - - - - - - -

And here is a case where 13 givens solves 16 positions:
Code: Select all
- 4 - - - - - - -
- - - - - - - - -
- - - - 3 - - 5 -
- - - - - - 1 - -
3 - 2 - 6 - - - -
- - - - - 7 4 - 8
- 8 - - - 4 - - -
- - - - - - - - -
- - - - - - - 3 -

The data here is incomplete, and I'm not sure if best possible. Please jump in if anyone can add to it.
olimpia

Posts: 35
Joined: 14 November 2008
Location: USA

16 clues = 47 solved (2 solutions missing 18 clues)

this was explored very extensivly for the quest of finding a 16 clue 1 solution grid. so far to no avail.

but. there is a collection of 17 puzzles currently listed on here some where with all that have been found to date.

18 clues are equally as rare.
the majority of puzzles hold 21+ clues on average for 1 solution.

18 clues = 63 solved
81 clues = 0 solved :P
Some do, some teach, the rest look it up.

StrmCkr

Posts: 1088
Joined: 05 September 2006

### OK, update

OK, thanks. here's an update (also showing total complete):

Code: Select all
givens   qty solved    total
------   ------------  -----
4         1             5
5         1             6
6         2             8
7         2             9
8         2             10
9         3             12
...
13        16            29
...
16        47            63
17        64            81

If there's any remarkable (partial grids) for this data please post it also, so we can see how its constructed. thanks.
olimpia

Posts: 35
Joined: 14 November 2008
Location: USA

Code: Select all
+---+---+---+
|...|3.1|...|
|2.4|...|...|
|...|...|...|
+---+---+---+
|8..|52.|...|
|...|...|9.1|
|...|...|3..|
+---+---+---+
|...|.4.|.5.|
|91.|...|...|
|.3.|...|...|
+---+---+---+ 15 clues     596 grid solutions.   source G Royle

i believe 29 more inserted, 44 in total

Code: Select all
+---+---+---+
|1..|...|...|
|...|1..|...|
|...|...|...|
+---+---+---+
|.1.|...|...|
|...|...|...|
|...|...|.1.|
+---+---+---+
|...|...|...|
|...|..1|...|
|...|...|..1|
+---+---+---+  6 clues 3 inserted

Code: Select all
+---+---+---+
|...|...|.1.|
|.2.|.5.|...|
|...|...|...|
+---+---+---+
|...|1..|...|
|...|...|2..|
|...|...|...|
+---+---+---+
|...|.7.|5.2|
|...|...|4..|
|1..|...|...|
+---+---+---+ 10 clues , 7 inserted

C
coloin

Posts: 1864
Joined: 05 May 2005

olimpia
Good question !

Here is a quick survey :
Code: Select all
*-----------*
|.1.|...|...|
|...|...|..1|
|...|.2.|...|
|---+---+---|
|...|...|...|
|1..|...|...|
|...|...|.1.|
|---+---+---|
|..2|...|...|
|...|..1|...|
|...|...|.2.|
*-----------*      8 clues

*-----------*
|.1.|...|...|
|...|...|..1|
|...|12.|...|
|---+---+---|
|...|.1.|...|
|1..|...|...|
|...|...|.1.|
|---+---+---|
|..2|...|1..|
|...|2.1|...|
|..1|...|.2.|
*-----------*      5 inserted

Code: Select all
*-----------*
|...|...|...|
|...|..2|...|
|1..|...|...|
|---+---+---|
|...|...|.2.|
|...|.1.|...|
|.2.|...|..1|
|---+---+---|
|..2|...|...|
|...|...|.12|
|...|1..|...|
*-----------*      10 clues

*-----------*
|2..|..1|...|
|...|..2|1..|
|1..|...|2..|
|---+---+---|
|..1|...|.2.|
|...|21.|...|
|.2.|...|..1|
|---+---+---|
|.12|...|...|
|...|...|.12|
|...|12.|...|
*-----------*      8 inserted

will give you more...

JPF
JPF
2017 Supporter

Posts: 3861
Joined: 06 December 2005
Location: Paris, France

### update again

Cool I really like the 6 clues/3 inserted. It improved on my 6/2 and looks way cool .
Updating the table again (and now I'm showing source in case anyone wants to go back and check):
Code: Select all
givens   qty solved  total   source
------   ----------  -----   ------
4         1           5
5         1           6
6         3           9      (Coloin)
7         3           10
8         5           13     (JPF)
9         5           14
10        8           18     (JPF)
...
13        16          29     (Olimpia)
...
15        29          44     (G. Royle)
16        47          63     (?)
17        64          81     (many)

Now I'm wondering where do we get a 1:1 payoff? That is at what point does "n" clues solve at least "n" positions
olimpia

Posts: 35
Joined: 14 November 2008
Location: USA

16 is also g. royle.
Some do, some teach, the rest look it up.

StrmCkr

Posts: 1088
Joined: 05 September 2006

Code: Select all
*-----------*
|..1|...|...|
|...|...|...|
|...|.2.|...|
|---+---+---|
|...|...|1..|
|...|..1|...|
|.2.|...|...|
|---+---+---|
|...|...|...|
|...|1..|...|
|...|...|.1.|
*-----------*     7 clues

*-----------*
|..1|...|...|
|...|.1.|...|
|...|.2.|..1|
|---+---+---|
|...|...|1..|
|...|..1|...|
|12.|...|...|
|---+---+---|
|.1.|...|...|
|...|1..|...|
|...|...|.1.|
*-----------*     4 inserted

Code: Select all
*-----------*
|...|1..|...|
|.2.|...|...|
|...|...|..2|
|---+---+---|
|...|...|...|
|2..|...|..1|
|1..|...|...|
|---+---+---|
|..1|...|...|
|...|...|.1.|
|...|..2|...|
*-----------*    9 clues

*-----------*
|...|12.|...|
|.2.|...|1..|
|.1.|...|..2|
|---+---+---|
|...|..1|...|
|2..|...|..1|
|1..|...|...|
|---+---+---|
|..1|...|...|
|..2|...|.1.|
|...|.12|...|
*-----------*   6 inserted

JPF
JPF
2017 Supporter

Posts: 3861
Joined: 06 December 2005
Location: Paris, France

Interesting question, but very hard for puzzles with more clues.

E.g. in 3 million random multisolution puzzles with 12 clues out of 5 different numbers we only found 1, which solved 9 numbers. It was better with only 4 numbers, but it needed hours to get one with 12+10, though i suspect, that the maximum is 12 or higher.

Here is an 11/9:
Code: Select all
+-------+-------+-------+
| . . . | 2 . . | . . . |
| 3 . . | . . . | . . 1 |
| . . 1 | . . . | . . . |
+-------+-------+-------+
| . . . | . . . | . . . |
| . . . | . . . | 2 . . |
| 1 . . | . . . | . . . |
+-------+-------+-------+
| . . . | . . 2 | . 1 . |
| . 2 . | . . 1 | . . . |
| . . . | . . . | . . 3 |
+-------+-------+-------+

11 clues and 9 solve

Code: Select all
+-------+-------+-------+
| . 4 . | . . . | . . 3 |
| . 1 . | . . . | . . . |
| . . . | . 3 . | . 2 . |
+-------+-------+-------+
| . . . | . . 1 | . . . |
| . . . | . . . | 2 . . |
| 2 . . | . . . | . . . |
+-------+-------+-------+
| . . . | 1 . . | . . . |
| 3 . . | 2 . . | . . . |
| . . . | . . . | . . 1 |
+-------+-------+-------+

12 clues and 11 solve
eleven

Posts: 2269
Joined: 10 February 2008

Here is a 14/20 :

Code: Select all
*-----------*
|...|1.2|...|
|3..|...|.4.|
|...|5..|...|
|---+---+---|
|.52|...|...|
|...|.6.|.3.|
|.1.|...|...|
|---+---+---|
|7..|.3.|...|
|8..|...|...|
|...|...|1..|
*-----------*     14 clues

*-----------*
|5..|1.2|...|
|32.|...|541|
|1..|5.3|...|
|---+---+---|
|.52|3..|.1.|
|...|.61|235|
|.13|.25|...|
|---+---+---|
|7.1|.3.|.5.|
|8.5|.1.|...|
|2..|.5.|1..|
*-----------*     20 solve

Here's a 12/13:
Code: Select all
*-----------*
|...|1..|...|
|.23|...|...|
|.4.|...|...|
|---+---+---|
|...|.4.|.2.|
|5..|...|...|
|1..|...|...|
|---+---+---|
|.6.|.2.|...|
|7..|...|1..|
|...|...|...|
*-----------*     12 clues

*-----------*
|.57|1..|...|
|.23|...|.1.|
|.41|...|...|
|---+---+---|
|3..|.4.|.21|
|5..|.1.|...|
|1..|...|...|
|---+---+---|
|46.|.21|...|
|73.|...|1.2|
|21.|...|...|
*-----------*    13 solve

This 15 clues is interesting :
Code: Select all
*-----------*
|1..|2..|...|
|...|...|3.4|
|...|...|5..|
|---+---+---|
|6..|...|.7.|
|...|...|.1.|
|.3.|.5.|...|
|---+---+---|
|...|1..|68.|
|.5.|...|...|
|..4|...|...|
*-----------*  15 clues ; 748 solutions

33 cells can be solved :
*-----------*
|195|234|768|
|..6|51.|3.4|
|4.3|...|5.1|
|---+---+---|
|61.|4.3|.75|
|54.|...|.13|
|.3.|.51|.46|
|---+---+---|
|3..|145|68.|
|.51|...|43.|
|.64|3..|15.|
*-----------*   33 solve

JPF
JPF
2017 Supporter

Posts: 3861
Joined: 06 December 2005
Location: Paris, France

Excellent results, JPF.

So to my surprise you need already 7 numbers for the 12/13 (at least my guess with 12+ was true).

Also great the 15/33, which i did not expect to be possible (more than the 15 clue with 596 solutions). With 7 in r8c4 and 8 in r5c5 this one gives a 17 clue also.
eleven

Posts: 2269
Joined: 10 February 2008

### 81 - 18 = 63 = 16 + 47

StrmCkr wrote:16 clues = 47 solved (2 solutions missing 18 clues)

this famous 16-clue puzzle was discovered by gfroyle --

gfroyle (2005.Jul.11) wrote:
Code: Select all
5 . 2 | . . . | 4 . .
. . . | 7 1 . | . . 3
. . . | . . . | . . .
-------+-------+------
. . . | . . 4 | 6 . .
. 7 . | 2 . . | . . .
. 1 . | . . . | . . .
-------+-------+------
6 . . | . . 2 | . . .
. . . | . 3 . | . 1 .
4 . . | . . . | . . .

is interesting, because it has only 16 clues.

It CANNOT be uniquely solvable, because it is missing both 8 and 9,
and so they can always be exchanged in any final solution.

But, this turns out to be the only problem --
the puzzle has exactly TWO solutions.

Pat

Posts: 3880
Joined: 18 July 2005

Thanks eleven

Here is an improvement :

14/22
Code: Select all
*-----------*
|...|..1|...|
|.2.|...|...|
|...|...|...|
|---+---+---|
|3.1|...|...|
|...|2..|..4|
|..5|...|...|
|---+---+---|
|.6.|42.|...|
|...|7..|5..|
|...|...|13.|
*-----------*     14 clues

*-----------*
|.3.|..1|...|
|.2.|3..|..1|
|.1.|..2|..3|
|---+---+---|
|3.1|...|...|
|...|25.|314|
|2.5|13.|...|
|---+---+---|
|163|425|...|
|..2|713|546|
|.5.|...|132|
*-----------*     22 solve

JPF
JPF
2017 Supporter

Posts: 3861
Joined: 06 December 2005
Location: Paris, France

### update

Here's a summary again (hopefuly no typo's):
Code: Select all
givens   qty solved  total   source
------   ----------  -----   ------
4         1           5
5         1           6
6         3           9      (Coloin)
7         4           11     (JPF)
8         5           13     (JPF)
9         6           15     (JPF)
10        8           18     (JPF)
11        9           20     (Eleven)
12        13          25     (JPF)
13        16          29     (Olimpia)
14        22          36     (JPF)
15        33          48     (JPF)
16        47          63     (G. Royle)
17        64          81     (many)

I charted the data and the curve is fairly smooth (obviously increases exponentially).

Its also interesting that many of these partial sudokus only put a few clue values in play. The 6/3 only has 1's (and solves just 1's). The puzzles with 7, 8, 9, and 10 clues only have two clue values (1 and 2). They are simple and cute, like a little baby
olimpia

Posts: 35
Joined: 14 November 2008
Location: USA

Here is a 13/17 :

Code: Select all
*-----------*
|...|1..|...|
|...|2..|...|
|.3.|...|.4.|
|---+---+---|
|...|51.|2..|
|.4.|...|...|
|...|...|...|
|---+---+---|
|...|.46|.3.|
|...|.3.|...|
|2..|...|...|
*-----------*    13 clues

*-----------*
|...|1..|.2.|
|...|2..|...|
|.32|6.5|.4.|
|---+---+---|
|3..|51.|2.4|
|.4.|32.|...|
|.2.|46.|...|
|---+---+---|
|...|.46|.32|
|...|.32|...|
|2.3|.51|4..|
*-----------*    17 solve

JPF
JPF
2017 Supporter

Posts: 3861
Joined: 06 December 2005
Location: Paris, France

Next