Problems & Puzzles: Puzzles

Puzzle 839. Follow up to Puzzle 835. Emmanuel Vantieghem sent the following nice puzzle: Do there exist  NxN  squares of positive integers such that all  2N(N-1)  sums of adjacent elements give the set of the consecutive primes from  3 till the 1+2N(N-1)-th prime? The first I could find was N=3: 1 2 3 6 11 8 17 20 21 which gives the first twelve odd primes:     3 = 1 + 2 ; 5 = 2+3  17 = 6 +11 ; 19 = 11+8  37 = 17+20; 41 = 20+21 and   7 = 1 + 6 ; 23 = 6+17  13 =2 +11 ;31 = 11+20  11 =3 + 8 ; 29 = 8 +21 This is my biggest example. N=7:   1,    2,     3,    4,     7,  6,    11  18,   29,   38,  33,   46, 37,  90   5,    68,   21, 116,  15, 136, 91  24,   89,  218, 23,  176, 57, 106  79,  192, 119, 228, 173, 50, 267 198, 161, 248, 191,  38, 129, 164 185, 236,  21,  242,  71, 260, 119 Q. Find the smallest solution for N=3 to 10. "Smallest" means using the smallest larger integer (in bold&red in the Emmanuel's solutions)

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On March 1, 2017, Dmitry Kamenetsky wrote:

Here are my best answers for Puzzle 839:

 

3x3
score: 21
17 20 21 
6 11 8 
1 2 3 

4x4
score: 49
12 35 38 45 
11 26 3 8 
48 41 2 11 
49 30 1 6 

5x5
score: 91
30 37 4 55 76 
1 10 19 82 91 
2 3 34 75 88 
15 4 39 64 63 
64 85 22 49 34 

6x6
score: 200
6 1 2 27 200 33 
91 190 3 20 51 106 
10 9 8 23 80 177 
57 4 33 104 9 64 
172 105 118 63 44 15 
97 166 45 134 105 68 

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