Problems & Puzzles: Puzzles

Puzzle 323. Primes in a Sudoku solution

I will suppose that you already know+ the lately very popular numbers puzzle named Sudoku . If not, please see a very complete description of it in this Wikipedia article.

But, in short, the very basic definition of a Sudoku puzzle goes like this:

'The aim of the puzzle is to enter a number from 1 through 9 in each cell of a grid, most frequently a 9×9 grid made up of 3×3 subgrids (called "regions"), starting with various numbers given in some cells (the "givens"). Each row, column and region must contain only one instance of each number'.

I will only recall just one pertinent theoretical result about the number of Sudoku solutions:

'...the number of valid Sudoku solution grids for the standard 9×9 grid was calculated by Bertram Felgenhauer in 2005 to be 6,670,903,752,021,072,936,960 [3], ... This number is equivalent to 9! × 72^2 × 2^7 × 27,704,267,971, the last factor of which is prime...'

So, now you know that there are enough distinct valid solution for the purpose of our puzzle.

Now I will give you only two examples of Sudoku solved grids that I have produced for the puzzle of this week (hiding how I have obtained them):

Solution Min:

 1 2 3 5 6 4 8 9 7 4 5 6 8 9 7 2 3 1 7 8 9 2 3 1 5 6 4 9 7 5 1 2 3 6 4 8 2 3 1 4 8 6 9 7 5 6 4 8 7 5 9 3 1 2 5 6 4 9 7 8 1 2 3 8 9 7 3 1 2 4 5 6 3 1 2 6 4 5 7 8 9

Solution Max:

 1 2 3 5 9 4 6 7 8 5 4 8 7 1 6 3 9 2 7 6 9 2 8 3 4 1 5 3 9 6 1 2 5 8 4 7 2 1 7 4 6 8 9 5 3 4 8 5 9 3 7 1 2 6 6 7 4 3 5 9 2 8 1 9 3 1 8 7 2 5 6 4 8 5 2 6 4 1 7 3 9

What in particular these Sudoku solutions show us?

Well, the so called 'Solution Min' has only 2 primes of three digits, considering all the possible 3 digits numbers (8x2x9=144) in the nine regions++:

R4: 839
R7: 691

In its turn, the so called 'Solution Max' has 45 primes (only 33 of them are distinct primes):

R1: 769, 157, 389, 149, 743, 967, 751, 983, 941, 347
R2: 283, 463, 617
R3: 293, 197
R5: 937, 149*, 263, 587, 167, 521, 739, 941* 761, 569
R6: 953, 157*, 359, 751*
R7: 139
R8: 359*, 641, 953*, 683, 173
R9: 281, 739*, 257, 863, 149*, 269, 761*, 937*, 941*, 167*
The repeated primes are marked with an asterisk (*)

But I do not pretend that my solutions are the true minimal or maximal solutions, just a good starting point encouraging you to improve them.

Questions:

1. Obtain a Sudoku solution having zero three digit primes (among the 144 three digits possible numbers in the 9 regions)

2.1 Obtain a Sudoku solution having more than 45 three digits primes (among the 144 three digit possible numbers in the 9 regions).

2.2 Obtain a Sudoku solution having more than 33 three digits distinct-primes (among the 144 three digit numbers in the 9 regions).

____________
(+) I know the Sudoku game after one workmate, Ismael Flores, show me a puzzle from a magazine some 4-5 weeks ago. Thanks Ismael for this new toy for me.
(++) You are not able to form three digits primes from numbers from two distinct regions.

Contributions cane from Adam Stinchcombe and Anurag Sahay:

I have generated a solution to part #1 (zero primes):

345 678 912
786 129 453
291 534 867
534 867 291
678 912 345
129 453 786
453 786 129
867 291 534
912 345 678

***

Anurag found an answer to Q3:

40 primes, all distinct primes.

1 2 3 9 6 8 5 4 7
5 4 8 3 1 7 2 6 9
7 6 9 5 2 4 1 8 3
4 3 1 6 7 2 9 5 8
6 5 2 4 8 9 7 3 1
9 8 7 1 3 5 4 2 6
2 7 6 8 5 1 3 9 4
8 1 4 2 9 3 6 7 5
3 9 5 7 4 6 8 1 2

...

Later he sent a solution to Q2:

I found a solution with 47 primes (19 of them distinct primes):

623 941 587
419 875 236
758 362 194

362 194 758
941 587 623
875 236 419

236 419 875
194 758 362
587 623 941

***

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