Puzzles related to Patterns Game patterns

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Re: Puzzles related to Puzzle Game patterns

Postby m_b_metcalf » Tue Oct 04, 2016 4:51 pm

Code: Select all
 . . . . . 4 . . 6
 . . 2 8 . . 7 . .
 . 8 . . 7 . . 3 .
 . 9 . . . . . . 2
 . . 5 . . . 6 . .
 2 . . . . 5 . 8 .
 . 5 . . 1 . . 6 .
 . . 8 . . 7 9 . .
 9 . . 6 . . . . . nice x-sudoku, pattern 277
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Re: Puzzles related to Puzzle Game patterns

Postby m_b_metcalf » Tue Nov 22, 2016 10:39 pm

Code: Select all
 7 . . . 5 . . . .
 . 1 . 3 . . . 7 .
 . . 4 . . . 5 . .
 . 9 . . . 6 . . .
 5 . . . 9 . . . 1
 . . . 8 . . . 9 .
 . . 2 . . . 3 . .
 . 3 . . . 9 . 8 .
 . . . . 7 . . . 6  nice X-sudoku, pattern from game 278
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Re: Puzzles related to Puzzle Game patterns

Postby m_b_metcalf » Wed Feb 01, 2017 3:32 pm

Game 279:
Code: Select all
 . 1 2 . 9 . . . .
 3 7 . 2 . . . . .
 9 . . . . . 2 . .
 . 3 . . . 6 . 4 .
 6 . . . 4 . 3 . .
 . . . 7 . . . . 9
 . . 5 . 6 . 8 . .
 . . . 3 . . . . 5
 . . . . . 5 . 7 .  x-sudoku, easy

 . 3 1 . 2 . . . .
 4 9 . 1 . . . . .
 7 . . . . . 9 . .
 . 1 . . . 6 . 9 .
 9 . . . 5 . 2 . .
 . . . 2 . . . . 8
 . . 7 . 1 . 3 . .
 . . . 9 . . . . 7
 . . . . . 5 . 4 .  x-sudoku, fairly hard
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Re: Puzzles related to Patterns Game patterns

Postby m_b_metcalf » Fri Feb 17, 2017 10:09 am

Here is a valid puzzle for Patterns Game 280, the sum of whose clue values is 91. Challenge: can anyone find a valid puzzle with a lower sum?
Code: Select all
 1 . . 2 . . . . 3
 . 4 . . 1 . . 5 .
 . . 6 . . . 1 . .
 3 . . . . 1 . 6 .
 . 1 . . 4 . 5 . .
 . . . 6 . . . . .
 . . 4 . 6 . . . .
 . 5 . 7 . . . 3 2
 7 . . . . . . 8 .   ED=7.8/1.2/1.2. Sum of values of clues = 91


Regards,

Mike

P.S. Is it possible that a moderator change the title of this thread from 'Puzzles related to Puzzle Game patterns' to 'Puzzles related to Patterns Game patterns'? Thanks
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Re: Puzzles related to Patterns Game patterns

Postby JPF » Fri Feb 17, 2017 10:57 am

In your puzzle, swap digits 2 and 6:
Code: Select all
 1 . . | 6 . . | . . 3
 . 4 . | . 1 . | . 5 .
 . . 2 | . . . | 1 . .
-------+-------+-------
 3 . . | . . 1 | . 2 .
 . 1 . | . 4 . | 5 . .
 . . . | 2 . . | . . .
-------+-------+-------
 . . 4 | . 2 . | . . .
 . 5 . | 7 . . | . 3 6
 7 . . | . . . | . 8 .  sum = 83

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Re: Puzzles related to Patterns Game patterns

Postby blue » Fri Feb 17, 2017 11:32 am

Code: Select all
2 . . 4 . . . . 1
. 1 . . 3 . . 2 .
. . 4 . . . 3 . .
7 . . . . 6 . 5 .
. 5 . . 4 . 2 . .
. . . 1 . . . . .
. . 2 . 1 . . . .
. 3 . 2 . . . 1 8
1 . . . . . . 6 .   ED=1.5/1.2/1.2. Sum of values of clues = 74
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Re: Puzzles related to Patterns Game patterns

Postby JPF » Fri Feb 17, 2017 5:48 pm

74 seems to be tough to beat:
Code: Select all
 1 . . | 4 . . | . . 2
 . 2 . | . 3 . | . 1 .
 . . 8 | . . . | 3 . .
-------+-------+-------
 2 . . | . . 1 | . 6 .
 . 5 . | . 4 . | 1 . .
 . . . | 3 . . | . . .
-------+-------+-------
 . . 1 | . 2 . | . . .
 . 3 . | 5 . . | . 2 4
 4 . . | . . . | . 7 .   ED=9.0/1.2/1.2. Sum of values of clues = 74


or is there an explanation why the lowest bound must be 74 ?

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Re: Puzzles related to Patterns Game patterns

Postby blue » Fri Feb 17, 2017 6:02 pm

JPF wrote:or is there an explanation why the lowest bound must be 74 ?

None that I know of.

Nice puzzle ! No 9, and one each of 6,7,8.
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Re: Puzzles related to Patterns Game patterns

Postby m_b_metcalf » Fri Feb 17, 2017 7:41 pm

The best I got so far was 76:

Code: Select all
 3 . . 5 . . . . 1
 . 2 . . 4 . . 3 .
 . . 1 . . . 2 . .
 2 . . . . 3 . 1 .
 . 4 . . 2 . 6 . .
 . . . 1 . . . . .
 . . 6 . 7 . . . .
 . 1 . 3 . . . 2 5
 8 . . . . . . 4 .  ED=1.5/1.2/1.2  Sum of values of clues =  76 


I, too, was wondering about a minimum (for this pattern). To this end, I constructed an Ur-puzzle putting in minimum values by hand such that the basic Sudoku constraints are fulfilled, as well on the constraints on minimality (number of occurrences of individual values, etc.). If I got it right, it's this:
Code: Select all
 1 . . 2 . . . . 3
 . 2 . . 3 . . 1 .
 . . 3 . . . 2 . .
 2 . . . . 1 . 3 .
 . 1 . . 2 . 4 . .
 . . . 3 . . . . .
 . . 1 . 4 . . . .
 . 3 . 1 . . . 2 4
 4 . . . . . . 5 .   Sum of values of clues = 57, 355536 solutions.
 

However, to fulfill also the constraint that a minimum of eight individual values are required, I replaced three 4s by 6, 7 and 8 giving a sum of 66, the absolute theoretical minimum. So, there is still some way to go from 74 (except I can't find any with a smaller sum)! Please check this reasoning.

Incidentally, I looked at the patterns game results and the 17s file, finding:

Code: Select all
 . . . . . . . 1 .
 . . . . . 1 . 2 3
 . . 4 . 5 . . . .
 . . . 2 . . . 4 .
 . . 5 . 1 . 6 . .   Patterns Game minimum
 . 1 . . . 3 . . .
 . . . . 7 . 4 . .
 3 2 . 8 . . . . .
 . 5 . . . . . . .   ED=7.9/1.2/1.2, jpf. Sum of values of clues =  67

and
Code: Select all
 . . . . 1 . 4 . .
 . 2 . . . . . . .
 . . . . 3 . . . .
 1 . 4 . . . . . .
 . . . 3 . . . 2 .   17s minimum
 6 . . . . . . 5 .
 . . . 2 . . 7 3 .
 4 . . . . . 1 . .
 . 8 . 5 . . . . .   No. of givens =  17. Sum of values of clues =  61 


Regards,

Mike
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Re: Puzzles related to Patterns Game patterns

Postby blue » Sat Feb 18, 2017 10:27 pm

m_b_metcalf wrote:I, too, was wondering about a minimum (for this pattern). To this end, I constructed an Ur-puzzle putting in minimum values by hand such that the basic Sudoku constraints are fulfilled, as well on the constraints on minimality (number of occurrences of individual values, etc.). If I got it right, it's this:
Code: Select all
 1 . . 2 . . . . 3
 . 2 . . 3 . . 1 .
 . . 3 . . . 2 . .
 2 . . . . 1 . 3 .
 . 1 . . 2 . 4 . .
 . . . 3 . . . . .
 . . 1 . 4 . . . .
 . 3 . 1 . . . 2 4
 4 . . . . . . 5 .   Sum of values of clues = 57, 355536 solutions.
 

However, to fulfill also the constraint that a minimum of eight individual values are required, I replaced three 4s by 6, 7 and 8 giving a sum of 66, the absolute theoretical minimum. So, there is still some way to go from 74 (except I can't find any with a smaller sum)! Please check this reasoning.


Hi Mike,

Here's one with 8 digits and a sum of 61. It has multiple solutions.

Code: Select all
6 . . 2 . . . . 1
. 2 . . 7 . . 8 .
. . 1 . . . 2 . .
1 . . . . 3 . 2 .
. 5 . . 2 . 1 . .
. . . 1 . . . . .
. . 2 . 1 . . . .
. 1 . 3 . . . 4 2
3 . . . . . . 1 .


Edit: This pattern of 1's, couldn't be a part of a minimal puzzle though. 1r9c8 would always be redundant.

Cheers,
Blue.
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Re: Puzzles related to Patterns Game patterns

Postby m_b_metcalf » Sun Feb 19, 2017 9:30 am

blue wrote:Edit: This pattern of 1's, couldn't be a part of a minimal puzzle though. 1r9c8 would always be redundant.

As would 2r3c7.
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Re: Puzzles related to Patterns Game patterns

Postby dobrichev » Sun Feb 19, 2017 1:56 pm

# puzzles per digits distribution for PG 280, set of 52,309,057 puzzles (+ 1 invalid)
Hidden Text: Show
Code: Select all
333332222   11338874
333333221   6839072
333333311   273822
333333320   173571
433322222   9124913
433332221   13463370
433333211   2206145
433333220   698253
433333310   58160
443222222   786510
443322221   3440331
443332211   1847159
443332220   389687
443333111   74465
443333210   145158
444222221   64603
444322211   176122
444322220   28187
444332111   43584
444332210   42408
444333110   3016
444422111   1442
444422210   866
444431111   168
444432110   670
444441110   2
533222222   152565
533322221   450582
533332211   184236
533332220   39846
533333111   5831
533333210   11856
542222222   5552
543222221   72250
543322211   99847
543322220   16076
543332111   16478
543332210   16656
543333110   831
544222211   2954
544222220   379
544322111   3030
544322210   2069
544331111   183
544332110   676
544421111   18
544422110   22
544431110   2
552222221   72
553222211   364
553222220   46
553322111   222
553322210   165
553331111   9
553332110   30
554222111   10
554222210   6
554321111   8
554322110   9
554331110   1
632222222   224
633222221   1667
633322211   1890
633322220   262
633332111   284
633332210   232
633333110   12
642222221   98
643222211   414
643222220   49
643322111   249
643322210   169
643331111   12
643332110   34
644222111   10
644222210   4
644322110   6
644331110   1
652222211   1
653322110   1

Code: Select all
2..1....3
.4..2..1.
..3...5..
5....1.6.
.2..4.1..
...2.....
..7.1....
.1.4...28
6......3. Sum of values of clues = 74


The above is at the bottom with 653322110 distribution.
This with 644331110 distribution has the same sum.

More generally, the trailing zero is a must (I can't figure out how the rest of the digits could compensate + 1 * 9).
We know why a secondary zero at right is forbidden.
What was the maximal number occurrences of the same digit still not causing redundancy (regardless of the pattern)?
The maximal number of the consequent ones at the right end is somehow limited by the number of different clues required to resolve parts of the solution grid (bands, rookeries) and I never heard for investigations in this direction.

It is unlikely a 17-clue puzzle to hold the minimal sum. Most likely the 18-19 or even 20 clues area is of interest.
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Re: Puzzles related to Patterns Game patterns

Postby m_b_metcalf » Sun Feb 19, 2017 3:05 pm

dobrichev wrote:What was the maximal number occurrences of the same digit still not causing redundancy (regardless of the pattern)?

The rule is:

a) no digit may occur more than 6 times;

b) if there are 5 or 6 occurrences, they must not be such that a horizontal band with 3 crosses a vertical band with 3.

For 23 clues, the minimum is thus 6*(1 + 2 + 3 ) + 1*(4 + 5 + 6 + 7 + 8) = 66, as already stated. For 17 clues, we have 6*1 + 5*2 + 1*(3 + 4 + 5 + 6 + 7 + 8) = 49 (which seems unlikely).

I modified blue's sum=61 puzzle to conform to these rules, obtaining:

Code: Select all
 6 . . 2 . . . . 3
 . 3 . . 7 . . 8 .
 . . 4 . . . 2 . .
 1 . . . . 3 . 2 .
 . 5 . . 2 . 1 . .
 . . . 1 . . . . .
 . . 2 . 1 . . . .  from blue's
 . 1 . 3 . . . 4 2
 9 . . . . . . 1 .
Sum of values of clues =   73, 94164 solutions

I found this too:
Code: Select all
 1 . . 3 . . . . 2
 . 2 . . 8 . . 1 .
 . . 3 . . . 5 . .
 3 . . . . 4 . 6 .
 . 5 . . 2 . 1 . .
 . . . 1 . . . . .
 . . 5 . 3 . . . .
 . 1 . 2 . . . 4 7
 4 . . . . . . 2 .
Sum of values of clues =   75

but, given your result, anything below 74 looks extremely unlikely.

Regards,

Mike
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Re: Puzzles related to Patterns Game patterns

Postby dobrichev » Sun Feb 19, 2017 3:48 pm

Thanks, Mike.

666111110 distribution is the theoretical maximum.
6xxxxxxxx distribution exists in ~1/1000 of the puzzles (PG 280)
66xxxxxxx distribution doesn't exist or is extremely rare
65xxxxxxx puzzles exist, ~1/25,000,000
655xxxxxx none
654xxxxxx none
653xxxxxx rare, ~1/50,000,000

BTW this is the distribution by min(sum).
Code: Select all
minimal sum of givens, #puzzles
74   2
75   7
76   9
77   36
78   56
79   859
80   187
81   2369
82   2138
83   19736
84   4626
85   60059
86   22362
87   155287
88   173345
89   221897
90   650127
91   1919409
93   573345
94   4138584
95   2206145
96   224
97   13468922
98   152565
99   786510
100   173571
101   9398735
102   6839072
105   11338874

The symmetrical twin of each of the puzzles isn't processed.

On the other end, max(min(sum)) of 105 isn't rare, ~1/5 of the puzzles.

There are gaps at 92, 103, 104. Maybe for a reason?
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Re: Puzzles related to Patterns Game patterns

Postby JasonLion » Sun Feb 19, 2017 7:06 pm

Permuting the digits of the 17s optimally gives quite a number of 58s.

For example
Code: Select all
. . . 4 2 . 1 . .
. 3 . . . . . . 5
. . . . . 1 . . .
7 2 4 . . . . . .
. . . 3 . . . 6 .
. . 1 . . . . . .
8 . . 5 1 . . . .
. . . . . . 2 3 .
. . . . . . . . .    No. of givens =  17. Sum of values of clues =  58


All of the 17 clue sum of 58 puzzles have the distribution:
433221110
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