The New Sudoku Players' Forum

by **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

by **JPF** » Mon Feb 20, 2017 10:20 am

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

by **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

by **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

JPF

by **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

by **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

by **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

by **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

by **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.

by **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

by **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.

is the function sum(n) strictly increasing?
does the "minimal" distribution always have a missing digit?
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

by **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

JPF

by **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

JPF

by **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 = 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

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

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