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

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

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

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.
JPF
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### Re: Minimal value of the sum of the digits of a minimal puzz

Scores:
Code: Select all
` Clues  Sum         Player                             Puzzle                                                         Distribution                                                                                                                                                         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             31  101         coloin    .....1...6..3.4.2...2..75413..1.......14.235..2.83..142.....43...3.....514.5....2           666641110             32  106         coloin    .....1......3.462...2...5413..1.......14.23..52.73..1425....43...3.....514.5...82           666651110             33                                                                                                                                          34                                                                                                                                          35  123         coloin    ....6.....7...1.23..2..3..1..4.523....54...1..2.138.45.4..251...5.6.4.3.2.631..54           666663110             36  129         coloin    ....6......7..1.23..2..36.1..4.523....54.6.1..2.13..45.4..251...5...4.3.2.6318.54           666664110             37                                                                                                                                          38                                                                                                                                          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 Aug 28, 2017 6:52 pm, edited 9 times in total.
JPF
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### Re: Minimal value of the sum of the digits of a minimal puzz

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

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`

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

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
coloin

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Joined: 05 May 2005

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

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

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`
coloin

Posts: 1877
Joined: 05 May 2005

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

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

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

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

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

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

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 ?
JPF
JPF
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### Re: Minimal value of the sum of the digits of a minimal puzz

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`

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

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`

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

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  = 61         ..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
Last edited by coloin on Fri Aug 25, 2017 12:30 am, edited 1 time in total.
coloin

Posts: 1877
Joined: 05 May 2005

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

Hi coloin,

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 ....