Problems & Puzzles: Puzzles

Puzzle 835. Goldbach squares. Dmitry Kamenetsy sent the following nice puzzle: A Goldbach square of order N is a NxN square filled with odd primes such that the sum of any two adjacent cells is exactly one of the even numbers from 6 to 4+4N(N-1) inclusive. Every even number in this range must occur exactly once. For example, here is a Goldbach square of order 3:   7 5 3  17 11 3  3 7 19  The sums across rows are:   7+5 = 12 5+3 = 8 17+11 = 28 11+3 = 14 3+7 = 10 7+19 = 26   The sums down columns are: 7+17 = 24 17+3 = 20 5+11 = 16 11+7 = 18 3+3 = 6 3+19 = 22 Notice that every even number from 6 to 28 appears exactly once. Dmitry has calculated a solution for a square 10x10, that I will publish next week Q1: What is the largest Goldbach square you can find? Q2: Do Goldbach squares exist for every N>=2?

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Contributions came from Emmanuel Vantieghem.

Emmanuel wrote:

I send you my biggest solution up to now : N=7
 

  3   3   5   5   7   7  11

 13  17  17  19  19  37  17

 19  23  29  29  31  31  47

 47  47  43  53  53  61  41

 29  71  61  59 101  47  83

 97  73  13  97  61  89  83

 19  79  67  67  61  79  59

There are also solutions for  N = 2, 4, 5, 6.

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As I promised, here is the Dmitry's N=10 solution that he sent some weeks ago:

89 11 101 71 271 13 5 89 59 197 

107 227 23 251 67 19 3 211 29 47 

13 83 47 79 43 313 3 139 127 131 

3 223 11 163 7 13 181 181 37 61 

89 79 73 151 3 173 73 131 227 97 

199 149 179 97 41 7 127 233 71 47 

97 17 11 31 5 7 109 53 37 31 

257 89 251 101 131 19 37 241 67 7 

79 109 31 191 29 11 43 19 163 53 

127 163 181 89 179 23 59 5 47 19 

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