Problems & Puzzles: Puzzles

 Puzzle 1068 Follow up to Puzzle 1067 Redo Puzzle 1067 considering the Rules of reading and counting the primes, but now:Q1. For N=7, produce a solution with the maximal quantity of distinct primes considering the emirps Q2. Produce a solution for the largest N value you can work obtaining at least 4N+4 distinct primes.

During the week 18-24 Dec 2021, contributions came from Giorgos Kalogeropoulos

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Giorgos wrote:

Here is a 7x7 square that is an answer to Q1 and Q2:

1 6 6 1 1 6 1
6 0 0 9 5 8 9
6 2 9 9 2 2 1
1 5 9 9 5 0 9
1 0 5 2 5 5 1
1 0 5 1 0 5 1
1 1 9 1 1 9 9

and its rotation

6 6 1 1 6 1 1
1 5 0 1 5 0 1
1 5 5 2 5 0 1
6 0 5 6 6 5 1
1 2 2 6 6 2 9
6 8 5 6 0 0 9
1 9 1 1 9 9 1

This square is of the form 4N+4 having 32 distinct primes as column, rows and diagonals.
Also, those squares contain 12 emirps:  {1501501,1552501,1525501,1826501,1666211,9266221,1051051,1055251,1052551,9059951,1111661,1129991}.
But we only care about emirps that have at least one 6 or 9 (in order to be distinct).
So, the total number of distinct primes is 38.

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On Dec 29, 2021, Emmanuel Vantieghem wrote:

Here is the best 7x7 square I could find :

1   6   6   1   1   1   1
6   6   8   6   6   1   1
6   8   0   2   2   1   9
1   5   9   6   1   2   1
1   6   6   9   8   6   1
6   6   1   0   1   5   9
1   9   1   1   9   1   9
Row numbers : 1661111, 6686611, 6802219, 15961211669861, 6610159, 1911919,
Column numbers : 1661161, 6685669, 6809611, 1626901, 16218191112651, 1191199
Main diagonal : 1606859 ; 2nd diagonal : 1126661
All mentioned numbers are primes ; the emirps are underlined and give us 7 extra primes :
1661111, 1596121, 1669861, 1112651, 1621819, 1606859, 1126661
So, we read 23 primes in this square..

The tumbled square is :

6   1   6   1   1   6   1
6   5   1   0   1   9   9
1   9   8   6   9   9   1
1   2   1   9   6   5   1
6   1   2   2   0   8   9
1   1   9   9   8   9   9
1   1   1   1   9   9   1
Row numbers : 6161161, 6510199, 1986991, 1219651, 6122089, 1199899, 1111991
Column numbers : 6611611, 1592111, 6181291, 1069291, 1196089, 6995899, 1911991
Main diagonal : 6589091
2nd diagonal : 1999211
All mentioned numbers are primes ; the emirps are underlined and give us 4 extra primes : 1569121, 1929601, 9806911, 1129991
So, we read  20  primes in this square, different from those in the first square..

Total numbers of distinct primes in both squares is thus: 43.

But I think that it is not the maximum possible.

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