Problems & Puzzles: Puzzles

Puzzle 305. A new version of Puzzle 209

11
13-17
03-07-05

In the shown triangular arrangement of R=3 rows of the first R.(R+1)/2 odd primes, all the sum of three contiguous primes produce a distinct prime

11+13+17 = 41
13+03+07 = 23
13+17+07 = 37
17+07+05 = 29

Q1. Get solutions for R=4 to 10, by a systematic approach and describe it.

Contributions came from Jacques Tramu & Anurag Sahay.

***

Jacques wrote:

Positions of primes :

1
2 3
4 5 6
7 8 9 10

My systematic approach is to use brute force search
and try all primes for positions 1, 2,3, 4 , ...
But the search ( 55!  possibilities for 10 rows) can be  heavily cut
if you remark there is some symetry.
For example  2 == 3 , 1 == 10 , 1 == 7
Dont try 11 in position 2 if you have tried it in position 3.
Dont try 17 in position 10 if you have tried it in position 1, etc .

The results (up to 11 rows) :

3
7  13
5  17  11 <== 3

3
5  11
17   7  23
13  29  31  19 <== 4

3
5  11
17   7  19
13  23  41  43
29  47  37  53  31 <== 5

3
5  11
7  31  19
23  29  47  61
13  37  73  59  43
17  53  41  67  71  79 <== 6

3
5  11
7  31  17
37  23 109  67
29  41  19 101  89
13  97  53  59 103  47
43  71  83  61 107  73  79 <== 7

3
5  11
7  31  17
37  23  53  79
101  43  13  97 107
29  67  41  83  89 151
19  61  71 139 109 113 157
73  47  59 103 131 127 149 137 <== 8

3
5  11
7  31  17
13  41  59  37
23  47  79  53  61
29  19  73 131 109 101
151 179  71 113 139 191 157
89 149 103 163 167 193 173 137
107  43 181  83  67  97 197 199 127 <== 9

263
257 241
251 211 239
233 199 197 223
229 181 191 103 173
227 137 149 163 167 139
193 67 73 151 53 43 17
23 11 101 107 79 41 13 59
47 37 19 61 71  7  5 179 109
3 29 83 131 31 89 97 127 113 157 <======== 10

331
317 293
313 277 311
307 263 271 281
283 239 199 269 223
257 229 193 251 127 227
241 101 191 233 109 137 139
211 89 157 19 167 103 181 179
197 163 107 17  5  7 53 79 131
61 73 47 43 37 11 29 67 83 97
149 173 151 13 71 23  3 41 31 59 113 <==== 11

***

Anurag wrote:

My approach does not use any smart problem-specific logic. It is partly stochastic with heuristic guidance.

R=3

13
11-07
03-05-17

R=4:

19
7 17
11 13 31
3 5 23 29

R=5

19
53 17
41 13 29
43 47 11 7
37 23 3 5 31

R=6:
71
61 79
73 17 31
03 07 19 53
37 13 11 59 67
41 29 05 43 47 23

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