Help with a puzzle that is a bit too complex

[TheGreatGoatsbie]

Hi all, Bokke here,

I have made only two puzzles, which were reasonably well received, one was even featured on ctc! But I'm stuck with the next puzzle that is a bit too complex in its essence. I'm looking for some tips with regards to simplifying a rule set and adding a finishing rule that makes the break in a bit more accessible. I'm not too sure if this is the right place to ask for help.


Let me start at the beginning: I noticed that in all prime numbers between 0 and 100, the digits 1, 3 and 7 appear 9 times. This was my starting point.
As if now, I have the primary ruleset as follows: 
"Classic sudoku rules apply. Place all prime numbers between 0 and 100 on the diagonals. Double digit primes are read from top to bottom. Prime numbers containing one or more prime digits (pcn)may not touch each other orthogonally. The other prime numbers made up from non-prime numbers (npcn), can touch all other prime digits. Prime numbers made up out of two identical digits have ignored all the rules and lost each other. (The prime numbers between 0-100: 2/3/5/7/11/13/17/19/23/29/31/37/41/43/47/53/59/61/67/71/73/79/83/89/97 )"


This fits quite nicely since there are 16+4 pcn's making up a total of 36 digits. (13/17/23/29/31/37/43/47/53/59/67/71/73/79/83/97+ 2/3/5/7)
That allows the solver to deduce on which diagonals the prime contain, with a little bit of set theory (since the entire sets of 3's and 7's need to fit in there). I'm not sure what you would call these diagonals together,I have marked the boxes where the pcn's are forced to reside grey:

There is one problem with this setup, and that is the prime number 11. This number gets broken up, that is why I have named the puzzle "A Prime Example of the Loneliest Number".

This is the expected solution, which was quite a puzzle in itself:


I have tried several options for a break in, for example using x/v clues or kropki dots. But everytime the break in is either far too complex or far too easy.
I have marked the even digits, not as a prime clue, but because I think using some way to infer that those numbers are even, allows you to instantly infer that all other greyed out cells are odd.

I am open to suggestions of how to tackle the break in, and subsequently placing the 7s and 3s. After that one could deduce how the pairs are oriented.

Would you guys say that this is far too laborious of a concept to make a good puzzle with? Having the digits 1, 3 and 7 appear nine times is just so perfect!

For people who want to play with the set up a little, here's a link for editing the sudoku. Just click mode to change to editor.

Ps: I have never posted here so let me know if this post is either inappropriate, or misplaced!

[TheGreatGoatsbie]

I noticed that it might be more intuitive to read the prime numbers from left to right instead of downward. Here's the rotated puzzle

[Dandelo]

I just tried to understand the rules. Some points are not clear to me:

Is it allowed that two primes share a digit? E.g. the 3 in R4C4 is contained in instances of 13, 23, 31 and 37.

And if two pcn share a digit, do they touch orthogonally?

Do the rules apply to each instance of a prime?
The pcn 53 in R4C2/R5C1 touches the pcn 2 in R4C1.

[TheGreatGoatsbie]

I just tried to understand the rules. Some points are not clear to me:

Is it allowed that two primes share a digit? E.g. the 3 in R4C4 is contained in instances of 13, 23, 31 and 37.

And if two pcn share a digit, do they touch orthogonally?

Do the rules apply to each instance of a prime?
The pcn 53 in R4C2/R5C1 touches the pcn 2 in R4C1.

Good questions. First of all I should have clarified that the primes may not overlap, at all. While some prime numbers appear double in the grid, there is only one configuration that allows all prime numbers to appear, except 11.

But the last point is one I should have thought about. This is ofcourse the case for all single digit primes, of which 5 and 2 violate my current rule multiple times.

Let's say I want to place one set of all the primes in the grid on the diagonals. (left to right seems easier to me, but I'm not sure it is) For that set of primes, pcn's may never touch eachother(!) orthogonally, there are no overlapping primes. e.g. r2c2 - r3c3 - r4c4 is no valid instance of the primes 31 and 13, only one of them can count for this puzzle.

In other words, for every prime number, at least one of the pcn's has to appear in the grid that is not touching wor overlapping with another pcn, whereas at least one of every npcn has to appear in the grid.

So while I love the concept, I cannot really find a proper way to make it a non-laborious exercise. I think it is too much to keep track of the digits contained within the primes.

[Dandelo]

I think it's not necessary a problem if two or more pcn overlap. I don't believe you could avoid it. But just one of them may count for fulfilling the rule.

I think pcn/npcn are the right categories for the non-touching rule, but for the placement rule I suggest something else.

Maybe
"Mark 16 diagonally connected pairs of squares (plus 4 single squares?) that do not overlap. Then solve the Sudoku so that the marked pairs (plus ...?) contain each 2-digit (less than 100?) prime number."

Of course just if this is what you mean.

And another thought: If the rules are not ambiguous, it doesn't suit to the fun fact you want to describe. But the Sudoku rules (plus maybe some givens) will solve this problem.

[Dandelo]

I forgot: The non-touching rule then just applies to the pcn in the marked squares.

[TheGreatGoatsbie]

I think it's not necessary a problem if two or more pcn overlap. I don't believe you could avoid it. But just one of them may count for fulfilling the rule.

I think pcn/npcn are the right categories for the non-touching rule, but for the placement rule I suggest something else.

Maybe
"Mark 16 diagonally connected pairs of squares (plus 4 single squares?) that do not overlap. Then solve the Sudoku so that the marked pairs (plus ...?) contain each 2-digit (less than 100?) prime number."

Of course just if this is what you mean.

And another thought: If the rules are not ambiguous, it doesn't suit to the fun fact you want to describe. But the Sudoku rules (plus maybe some givens) will solve this problem.

Hmm, I like that way of approaching it, but the marking of the npcn's should also be included. So then it would be:
"Mark 21 pairs of diagonally connected squares and four single square that do not overlap. All primes under hundred have to appear once in these squares. Pairs of squares are read left to right as a two digit prime" 

I don't really follow the last thought, what do you mean exactly?

Thanks for thinking along!

[Dandelo]

It was meant as an idea how to avoid the problems. I haven't thought a lot about the puzzle itself.

I mean it might be impossible to find a ruleset which exactly fits to your idea, e.g. because of the multiple instances of a prime number. But if you weaken the ruleset, it probably allows multiple solutions. Then the puzzle could be made unique by some given clues.

Then it's not the wanted description of the configuration which you have in mind, but a solvable puzzle.