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  61         JPF ..3..8.........1.......124.42..1.......3..5..........1...6.....2....5...17.....23 543221110 
   20  62         JPF ..........51........3.1..42.....35....2......7....6.1.2..14........2.1..1...8...3 643221110 
   21                                                                                                             
   22                                                                                                             
   23                                                                                                             
   24  72      coloin .......2..21....4.3.81.........351......2...7....41..21.5....3.2..3.....6.321.... 665221110 
   25  75      coloin ............12..34.......2145.3.....23...81..1.....3..7.2.3..1.3.65.12.......2... 666221110 
   26                                                                                                             
                                                                                                                   
   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 Mon Mar 13, 2017 11:56 am, edited 2 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


JPF
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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

JPF
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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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