Problems & Puzzles: Collection 20th

Coll.20th-007. Symmetrical (Dihedral) prime tiling of the plane.

On March 14 2018, Jan van Delden wrote:

Consider the following hexagon:

In the center-hexagon (cyan) the consecutive primes 7 .. 23 are given such that all sums of 3 neighbouring tiles add up to a prime number.
Here these sums have the additional property that they form 6 consecutive prime numbers, from 31=7+11+13 until 59=23+17+19.

In new triangles primes may be added if they form prime numbers, when added to existing triangles using similar constraints.

In the large hexahedron (yellow-blue-cyan) strips of the following form can be found (and their rotations):

In the example above only type A is used.
Letís extend this procedure to the star (blue-cyan).

For new primes in the blue triangles we use all strips op type A and type B.

In order to compute p the following constraints are imposed (using type A):

  p+11+7 is prime

  p+11+13 is prime

But p and q are situated in a strip of type B, so the next constraint is also imposed:

  p+11+13+23+q is prime

Similarly for the other blue-triangles (and all new strips).

So a total of 6x2 constraints of type A and 6 constraints of type B are added.
The result is that for the star-shape (cyan-blue) all constraints of type A and B are imposed.

We call a solution minimal if the sum of the primes involved is minimal. Please note that in these solutions might not be unique.

Q1:         The given solution is not minimal. Find the minimal solution.
Q2:         Find a minimal solution for the star (blue-cyan).
                Is there a solution using consecutive primes?

Q3:         Find a minimal solution for the large hexagon (yellow-blue-cyan).
                Is there a solution using consecutive primes?
                Try to impose type A,B and C constraints if possible.

Q4:         Is it possible to find solutions to Q2/Q3 where all resulting prime sums are consecutive?


 

Contributions came from Emmanuel Vantieghem,

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Emmanuel wrote on July 13, 2018:

Here are my results about the puzzle Coll.20th-007.

Q1. This is a minimal solution 

Q2. This is a minimal solution : all primes from  5  to  43  are in use

Q3. This might be minimal, but I have serious doubt.
It would take too much time to work through all possibilities.

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On July 17, 2008, Jan van Delden wrote:

Emmanuelís solutions to Q1 en Q2 are fine.

 

Q2:

 

A solution to Q2 with 24 distinct prime sums (sum=348 , minimal given 24 distinct prime sums).

Q3:

 

Emmanuelís solution is fine, however it is not minimal.

I found (sum = 1198):

 

 

It uses a subset of 26 consecutive primes starting with 5, 71 and 97 are not present.

 

Prime number 2 and 3 canít be part of the solution (simple modular proof).

The sum of the first 24 primes starting with 5 equals 1156.
So there is little slack to improve upon this solution.

The sum of the first 24 primes starting with 7 equals 1254, so 5 should be present in the minimal solution.

If one starts with 5 and omits 7, the sum of the first 24 primes equals: 1252.
If we repeat this analysis one can show that the minimal solution should at least contain the primes 5..61.

 

In the displayed figures the colors indicate the way the algorithm works.

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