Minimal value of the sum of the digits of a minimal puzzle

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Minimal value of the sum of the digits of a minimal puzzle

Postby JPF » Mon Feb 20, 2017 10:19 am

Based on an interesting question raised by Mike here, I suggest this new challenge:

What is the minimum value of the sum of the digits of a minimal puzzle?

As a benchmark, Jason calculated here the minimum for the known 17s : 58
I confirm this number.
There are 30 puzzles with sum = 58.
Here is an exemple :
Code: Select all
 . . . | 4 2 . | 1 . .
 . 3 . | . . . | . . 5
 . . . | . . 1 | . . .
-------+-------+-------
 7 2 4 | . . . | . . .
 . . . | 3 . . | . 6 .
 . . 1 | . . . | . . .
-------+-------+-------
 8 . . | 5 1 . | . . .
 . . . | . . . | 2 3 .
 . . . | . . . | . . .
all have a digits-distribution 4 3 3 2 2 1 1 1 0

I propose to set up records for different numbers of clues.
The next post is supposed to list the records.

JPF
Last edited by JPF on Wed Feb 22, 2017 7:07 pm, edited 1 time in total.
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Re: Minimal value of the sum of the digits of a minimal puzz

Postby JPF » Mon Feb 20, 2017 10:20 am

Scores:
Code: Select all
Clues Sum    Player    Puzzle                                                                           Dist.     
                                                                                                                   
   17  58   JasonLion ...42.1...3......5.....1...724.........3...6...1......8..51..........23.......... 433221110 
   18  59         JPF ..7....36.2..1.............3..1...4.....5.1......2....85....2..1..4........3....1 533221110 
   19  60      coloin 1.25...........3....41......1....6.5...812........4..........2..3...1.......37..1 633221110 
   20  62         JPF ..........51........3.1..42.....35....2......7....6.1.2..14........2.1..1...8...3 643221110 
   21  64         JPF .4.......1.......2.2....3.1...1.624...............5...8...1....2.3.7.1.5.1.23.... 653221110 
   22  66      coloin .....3......12...4..3....2.7.2....1...85..2...1.......45..1....2....61..1..2..3.. 663221110 
   23  68      coloin ...512.3..1.6.....4723.........2..1...1.482..........3.3......1.2...14.........2. 664311110 
   24  71      coloin ...513.2..126....347.2.........32.1...1.48...3.........2......1.....14........23. 665311110 
   25  74      coloin ...513.2..126....347.2.........32.1...1.483............2.3....1.3...14........2.. 666311110 
   26  78      coloin ...513.24.126....3.7........3...2.1......8...4..1...32.234....1.4..21......3..... 666411110 
   27  82      coloin ...543.21..2.613...7.2...4...3...18..2..14.3.4.......2.34..2.1....1....3......... 666511110 
   28  86      coloin ...51......2.634..3742...1...3...184.......3.14......2.31..2...2..4..3..4..1..2.. 666611110
   29  91         JPF 1....35.....2.1.4...4..7..32..3..4..5.31.42....1.2..3.4.....126..2......31.8....4 666621110
   30  96         JPF .....1......3.4.2...2...5413..1...7.8.14.23...2.63..142.....43...3.....514.5....2 666631110

                                                                                                                   
                                                                                                                   
   39 149 m_b_metcalf .............71.23.72..3681..4.523.6..54.6.1.62.137.45.4..251...5...4.3.26.31..54 666665310 
   40 159 m_b_metcalf ..........51.78236.782.3541..5.241.3..43....5.1...6.2...76.2.14.4..3.6..1.6.47352 666655420 


JPF
Last edited by JPF on Sat May 06, 2017 7:41 pm, edited 8 times in total.
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Re: Minimal value of the sum of the digits of a minimal puzz

Postby JPF » Wed Feb 22, 2017 6:11 pm

Here is a first proposal for 18 clues:

Code: Select all
 . . . | . . . | . . .
 . . . | . 1 . | . 2 5
 . . . | . 3 4 | . . .
-------+-------+-------
 . . . | . . . | 3 1 .
 . . 2 | 5 . 8 | . . .
 . . . | . . . | . . .
-------+-------+-------
 . 2 . | . . . | . . .
 . . . | 2 6 . | . . 1
 3 4 . | . . 1 | 7 . .   Sum of values of 18 clues = 60 - distribution : 443221110


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Re: Minimal value of the sum of the digits of a minimal puzz

Postby JPF » Thu Feb 23, 2017 1:56 pm

Code: Select all
 . . 7 | . . . | . 3 6
 . 2 . | . 1 . | . . .
 . . . | . . . | . . .
-------+-------+-------
 3 . . | 1 . . | . 4 .
 . . . | . 5 . | 1 . .
 . . . | . 2 . | . . .
-------+-------+-------
 8 5 . | . . . | 2 . .
 1 . . | 4 . . | . . .
 . . . | 3 . . | . . 1   Sum of values of 18 clues = 59 - distribution : 533221110


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Re: Minimal value of the sum of the digits of a minimal puzz

Postby coloin » Wed Mar 01, 2017 12:26 am

Code: Select all
+---+---+---+
|...|...|.2.|
|.21|...|.4.|
|3.8|1..|...|
+---+---+---+
|...|.35|1..|
|...|.2.|..7|
|...|.41|..2|
+---+---+---+
|1.5|...|.3.|
|2..|3..|...|
|6.3|21.|...|
+---+---+---+    minimal, Sum of values of 24 clues = 72 - distribution : 665221110

I looked at some of the 39 clue puzzles and it seems that 666xxxxxx is not precluded generally, but a 25 clue 666221110 didnt turn up after a brief search
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Re: Minimal value of the sum of the digits of a minimal puzz

Postby JPF » Wed Mar 01, 2017 6:23 pm

Nice 24 clues coloin!

Code: Select all
 . . 3 | . . 8 | . . .
 . . . | . . . | 1 . .
 . . . | . . 1 | 2 4 .
-------+-------+-------
 4 2 . | . 1 . | . . .
 . . . | 3 . . | 5 . .
 . . . | . . . | . . 1
-------+-------+-------
 . . . | 6 . . | . . .
 2 . . | . . 5 | . . .
 1 7 . | . . . | . 2 3     Sum of values of 19 clues = 61 - distribution : 543221110

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Re: Minimal value of the sum of the digits of a minimal puzz

Postby coloin » Thu Mar 02, 2017 12:51 am

Code: Select all
+---+---+---+
|...|...|...|
|...|12.|.34|
|...|...|.21|
+---+---+---+
|45.|3..|...|
|23.|..8|1..|
|1..|...|3..|
+---+---+---+
|7.2|.3.|.1.|
|3.6|5.1|2..|
|...|..2|...|
+---+---+---+  Sum of values of 25 clues = 75 - distribution : 666221110
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Re: Minimal value of the sum of the digits of a minimal puzz

Postby m_b_metcalf » Fri Mar 03, 2017 3:44 pm

coloin wrote:I looked at some of the 39 clue puzzles and it seems that 666xxxxxx is not precluded generally, ...

From the full, optimised set of 2650 puzzles the minimum obtained was:
Code: Select all
 . . . . . . . . .
 . . . . 7 1 . 2 3
 . 7 2 . . 3 6 8 1
 . . 4 . 5 2 3 . 6
 . . 5 4 . 6 . 1 .
 6 2 . 1 3 7 . 4 5    39 clues
 . 4 . . 2 5 1 . .
 . 5 . . . 4 . 3 .
 2 6 . 3 1 . . 5 4    Sum of values of clues = 149 
                      666665310


Regards,

Mike
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Re: Minimal value of the sum of the digits of a minimal puzz

Postby Mathimagics » Fri Mar 03, 2017 9:26 pm

For 17 clues, assuming that puzzles with 5 x 1's don't exist (proving that might be an interesting exercise), then one way to beat 58 would be 44222111 = 57.
Last edited by Mathimagics on Sat Mar 04, 2017 6:06 pm, edited 1 time in total.
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Re: Minimal value of the sum of the digits of a minimal puzz

Postby m_b_metcalf » Sat Mar 04, 2017 9:21 am

Code: Select all
 . . . . . . . . .
 . 5 1 . 7 8 2 3 6
 . 7 8 2 . 3 5 4 1
 . . 5 . 2 4 1 . 3
 . . 4 3 . . . . 5
 . 1 . . . 6 . 2 .
 . . 7 6 . 2 . 1 4
 . 4 . . 3 . 6 . .
 1 . 6 . 4 7 3 5 2  Sum of values of clues =  159 
                    No. of givens =  40
                    666655420


Both puzzles have the same sum.

Regards,

Mike
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Re: Minimal value of the sum of the digits of a minimal puzz

Postby JPF » Sat Mar 04, 2017 10:29 am

Based on those first results, questions can be raised.

Let sum(n) be the minimal value of the sum of the digits of a minimal puzzle with n clues.

  1. is the function sum(n) strictly increasing?
  2. does the "minimal" distribution always have a missing digit?
  3. what about the function r(n) = sum(n)/n ; seems to stay between 3 and 4 with a minimum at around 3 for n=24,25 ?
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Re: Minimal value of the sum of the digits of a minimal puzz

Postby JPF » Mon Mar 13, 2017 11:25 am

Code: Select all
 . . . | . . . | . . .
 . 5 1 | . . . | . . .
 . . 3 | . 1 . | . 4 2
-------+-------+-------
 . . . | . . 3 | 5 . .
 . . 2 | . . . | . . .
 7 . . | . . 6 | . 1 .
-------+-------+-------
 2 . . | 1 4 . | . . .
 . . . | . 2 . | 1 . .
 1 . . | . 8 . | . . 3     Sum of values of 20 clues = 62 - distribution : 643221110

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Re: Minimal value of the sum of the digits of a minimal puzz

Postby JPF » Thu Apr 06, 2017 9:51 pm

Code: Select all
 . 4 . | . . . | . . .
 1 . . | . . . | . . 2
 . 2 . | . . . | 3 . 1
-------+-------+-------
 . . . | 1 . 6 | 2 4 .
 . . . | . . . | . . .
 . . . | . . 5 | . . .
-------+-------+-------
 8 . . | . 1 . | . . .
 2 . 3 | . 7 . | 1 . 5
 . 1 . | 2 3 . | . . .   Sum of values of 21 clues = 64 - distribution : 653221110

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Re: Minimal value of the sum of the digits of a minimal puzz

Postby coloin » Fri Apr 07, 2017 9:37 pm

With a single maximal 3-rookery we can get all ...
Code: Select all
.27..6......72..65..6....274.2.6..7.6.31.72...7...2..651.67...226...97..7..2..6..   37      999221110  non minimal 
                                                                                                                   
.27..6......7....5..6.....74......7...31..2.......2...51..7.........9.........6..   18      533221110  = 59         
..7..6.......2...5..6....2.4..........31.7....7...2...51......22....97........6..   19      543221110  = 60         
..7..6.......2...5..6....2.4.2........31.72...7.......51.......2....97.....2..6..   20      643221110  = 62         
.....6......72...5..6....2.4......7...31......7...2...51..7...22....97..7.....6..   21      653221110  = 64         
.....6......72...5..6....2.4.2....7...31..2...7.......51..7....2....97..7..2..6..   22      663221110  = 66         
.....6......72...5.......274.2....7...31.72.......2..651.......26...97..7.....6..   23      664221110  = 69         
............72..65.......274...6....6.31.7....7...2...51.6....226...97..7.....6..   24      665221110  = 72         
............72..65.......274.2.6..7.6.31.72.......2...51.6.....26...97..7.....6..   25      666221110  = 75     

But we can actually do better than this ....
C
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Re: Minimal value of the sum of the digits of a minimal puzz

Postby JPF » Sat Apr 08, 2017 10:47 am

Hi coloin,

Thanks for your post, but:
I don't see how the sum of the digits of this puzzle :
Code: Select all
.27..6......7....5..6.....74......7...31..2.......2...51..7.........9.........6..
equals 59.
Actually, I do, but I'm too lazy to do it ;)
see the summary here.
Edit :
Your 19 clues-puzzle has a distribution 543221110 ; then the minimum sum of the digits is 61.
Here is the puzzle equivalent :
Code: Select all
..2..3.......1...4..3....1.7..........85.2....2...1...45......11....62........3..

What do you mean by single maximal 3-rookery?

you wrote:But we can actually do better than this ....

Go ahead!, that's the purpose of this thread.

JPF
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