Problems & Puzzles: Puzzles

 Puzzle 807. Symmetrical compositions of twin primes Natalia Makarova sent the following nice puzzle: We consider the consecutive twin primes (p1, p1+2), … (p2, p2+2), …, (p3, p3+2), … , (pn, pn+2), where n > 2 and p1 < p2 2 find the composition with a minimal value of p1. Examples: n=3 (5, 7), (11, 13), (17, 19) Symmetrical composition: [5, 11, 17] 5+17=2*11 n=4 (29, 31), (41, 43), (59, 61), (71, 73) Symmetrical composition: [29, 41, 59, 71] 29+71=41+59 I found the following solutions: n=5 [155861, 155891, 156059, 156227, 156257] n=6 [59, 71, 101, 107, 137, 149] n=7 [227927459, 227927597, 227927639, 227927699, 227927759, 227927801, 227927939] n=8 [41387, 41411, 41519, 41609, 41759, 41849, 41957, 41981] n=9 [54793185527, 54793185659, 54793185989, 54793186169, 54793186559, 54793186949, 54793187129, 54793187459, 54793187591] n=10 [34623805211, 34623805421, 34623805787, 34623806249, 34623806771, 34623807017, 34623807539, 34623808001, 34623808367, 34623808577] You can record solution briefly in the following form: n=10 34623805211: 0, 210, 576, 1038, 1560, 1806, 2328, 2790, 3156, 3366 Solutions for n = 9 are required to solve the puzzle for n=3. I have another solutions for n = 9: 354584248349: 0, 132, 372, 678, 900, 1122, 1428, 1668, 1800 388743941039: 0, 42, 240, 282, 450, 618, 660, 858, 900 403147629431: 0, 126, 420, 750, 768, 786, 1116, 1410, 1536 463060598321: 0, 390, 906, 1116, 1218, 1320, 1530, 2046, 2436 584591273177: 0, 372, 744, 1122, 1152, 1182, 1560, 1932, 2304 But I have not got a magic square of order 3. Questions: Q1. Find the minimal solutions for n > 10  Q2. Find more solutions for n = 9.

Natalia Makarova wrote on Nov. 6. 2015:

I have found three solutions for n=9

1110317288231: 0, 450, 648, 756, 1038, 1320, 1428, 1626, 2076

2007253835681: 0, 6, 420, 1896, 1938, 1980, 3456, 3870, 3876

2188700058659: 0, 792, 1038, 1428, 1590, 1752, 2142, 2388, 3180

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She wrote again on Nov 12, 2015:

Jaroslaw Wroblewski found 4 solutions for n = 9, which is obtained from a magic square of order 3

204860134660098317297: 0, 42, 60, 84, 102, 120, 144, 162, 204 422229725797687239077: 0, 42, 84, 120, 162, 204, 240, 282, 324
5646440666838544810187: 0, 42, 84, 210, 252, 294, 420, 462, 504
6082062789438398013047: 0, 12, 24, 240, 252, 264, 480, 492, 504

But it is possible that a minimal solution has not been found.

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