Problems & Puzzles: Puzzles

Puzzle 673 Sudoku & primes, again My friend Jaime Ayala proposes a new puzzle related to Sudoku solutions & prime numbers (See puzzle 323): Let's suppose we have a "standard sudoku solution", SS Let's associate to each one of the nine integers in SS a distinct prime number from a set of nine consecutive primes in order to get the "transformed sudoku solution", TS Let's obtain the concatenation of these prime numbers in TS, by row (left to right) and by column (up to down), obtaining a total of 18 integers. How many primes are in these 18 integers? I have computed the following valid example: If the set of integers 1 to 9 is associated to the set of nine consecutive primes from 11 to 41, this is a valid couple of SS & TS: SS   ST 5 1 2 9 8 6 3 7 4   23 11 13 41 37 29 17 31 19 3 4 7 2 1 5 6 8 9 17 19 31 13 11 23 29 37 41 6 8 9 3 4 7 2 5 1 29 37 41 17 19 31 13 23 11 8 5 6 7 2 9 1 4 3 37 23 29 31 13 41 11 19 17 1 2 3 4 5 8 7 9 6 11 13 17 19 23 37 31 41 29 7 9 4 1 6 3 5 2 8 31 41 19 11 29 17 23 13 37 9 7 1 8 3 2 4 6 5 41 31 11 37 17 13 19 29 23 2 3 5 6 9 4 8 1 7 13 17 23 29 41 19 37 11 31 4 6 8 5 7 1 9 3 2 19 29 37 23 31 11 41 17 13 BTW, in this valid couple of SS & TS we have zero primes in the 18 integers obtained by the asked concatenation! Q. Get a SS with 18 primes in ST, or the best you can. _____ You are able to use the set of 9 consecutive primes 11 to 41, or another set if you prefer.

---

Contributions came from Giovanni Resta, Emmanuel Vantieghem. Both found a  16 primes distinct example.

Giovanni claims that, based in his exhaustive approach (delivered on request), 16 is the maximum possible for consecutive sets of primes below 1000.

***

Giovanni wrote:

For Problem 673, I searched all the 9-primes consecutive ranges
below 1000 and the maximal number of primes in a Sudoku
scheme I found is 16. This was obtained for the 9-uples
starting at 11, 17, 19, 29, 31, 37, 41, 47, 59.

For example, for the numbers 11,13,17, 19,23,29, 31,37,41 I obtained:

6 1 5 7 2 3 4 9 8 =  29 11 23 31 13 17 19 41 37
2 9 4 1 6 8 7 5 3 =  13 41 19 11 29 37 31 23 17
7 3 8 5 9 4 2 6 1 =  31 17 37 23 41 19 13 29 11
4 2 6 8 3 1 9 7 5 =  19 13 29 37 17 11 41 31 23
8 7 3 4 5 9 1 2 6 =  37 31 17 19 23 41 11 13 29
9 5 1 2 7 6 8 3 4 =  41 23 11 13 31 29 37 17 19
5 4 9 6 1 2 3 8 7 =  23 19 41 29 11 13 17 37 31
1 6 2 3 8 7 5 4 9 =  11 29 13 17 37 31 23 19 41
3 8 7 9 4 5 6 1 2 =  17 37 31 41 19 23 29 11 13

All the 9 horizontal numbers are primes, plus
7 of the vertical ones, namely:

291331193741231117
114117133123192937
231937291711411331
311123371913291741
132941172331113719
173719114129133123
193113411137172329

***

Emmanuel wrote:

This is (up to now : my PC continues search to better ones) my best solution: 16 primes.
 

  11  41  19  37  13  17  31  23  29
  37  29  17  31  41  23  19  13  11
  13  31  23  29  11  19  37  17  41
  41  37  29  17  19  31  23  11  13
  23  11  13  41  37  29  17  19  31
  17  19  31  11  23  13  41  29  37
  29  13  37  23  17  41  11  31  19
  31  23  41  19  29  11  13  37  17
  19  17  11  13  31  37  29  41  23
                             *    *

All rows give primes and all columns except those marked with * give primes !

***