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 99 grid made up of 33 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 99 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:

Adam wrote, for Q1:

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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