Problems & Puzzles: Puzzles

Puzzle 1124 Pentactys   On Feb 21, 2023, Giorgos Kalogeropoulos sent the following nice follow-up puzzle to the Puzzle 1123, related to a figure now named "Pentactys":   We can find 24 primes in Triangle "a" by concatenating the numbers forwards and backwards along the lines:   2, 3, 5, 7, 11, 13, 31, 37, 41, 43, 47, 59, 67, 73, 89, 127, 149, 631, 1063, 1213, 4813, 10631, 13841, 14813   Also, there are 40 primes in Triangle "b" :  2,3,5,7,11,13,17,37,41,47,61,71,73,79,83,97,101,113,131,137,139,173,479,617,839,911,1013,1193,2137, 3911,5119,6173,10139,11131,11311,12113,14713,51193,83911,1211311   Note that the 2-digit numbers don't change when we concatenate backwards. e.g. 14 stays 14 and doesn't become 41   Q1. Can you rearrange the numbers from 0 to 14 in order to produce more than 40 primes? Q2. What is the maximum number of primes that can be produced? Q3. How many configurations (disregarding rotations and reflections) produce this maximum?

---

From Feb 25 to Jan 3, sent contributions Emmanuel Vantieghem, Gennady Gusev and Oscar Volpatti

***

Emmanuel wrote:

I found two triangles with 42 primes :
               0
           14     4
       13      6      12
    10     1      5       2
 3     7       9       8     11
Giving the primes
   2, 3, 5, 7, 11, 13, 17, 19, 37, 59, 61, 71, 73, 79, 89, 97,
   101, 103, 107, 113, 131, 211, 251, 379, 461, 613, 617, 811,
   1013, 1259, 1319, 9811, 11897, 12211, 12613, 13103, 31013,
   79811, 118973, 379811, 412211, 1413103

and
                  1
             6       7
        13     10      9
      5      3       2      8
  4     14     12      0      11
Giving the primes
  2, 3, 5, 7, 11, 13, 17, 23, 29, 53, 61, 67, 71, 79, 89, 97,
  103, 107, 109, 179, 313, 613, 811, 823, 971, 1013,
  1229, 1361, 1613, 4513, 7103, 8971, 9811, 11897, 12011,
  13109, 51361, 79811, 143107, 451361, 1412011, 41412011

I do not know if  42  is the maximum nor do I know if there are other 42-prime solutions.

 

P.S. Giorgos' triangle A has 25 primes (139 is missing)  and triangle B  has 41 primes (211 missing).  

 

***

Gennady wrote:

I found 1 triangle: 8;  11,  3;  0,  2,  4;  9,  5, 10,  7; 14,  1,  6, 13, 12; that gives 42 primes:

2, 3, 5, 7, 11, 13, 19, 23, 43, 47, 59, 61, 83, 107, 113, 127, 137, 149, 211, 251, 311, 347, 523, 613, 743, 811,

1013, 1109, 1213, 1361, 1523, 1613, 3251, 5107, 9011, 10211, 12743, 21013, 71059, 95107, 141613, 149011.

 

***

Oscar wrote:

Checking all possible digit configurations was an easy task for a tetractys; I think that exhaustive search is a feasible approach for a pentactys too, and I guess that many puzzlers actually completed such task.
Unfortunately, my poor code would need about ten days to terminate on my humble laptop, so I stopped after checking all configurations with digit "0" on top vertex (about 20% of all distinct configurations).
I found 15 nice configurations, each producing 42 distinct primes; I'll use the two-line notation borrowed from pux

zzle 1123.

0;  5, 4;  13, 6, 12;  10, 1, 14, 2;  3, 7, 9, 8, 11.
2, 3, 5, 7, 11, 13, 17, 19, 37, 61, 71, 73, 79, 89, 97, 101, 103, 107, 113, 131, 149, 211, 379, 461, 613, 617, 811, 1013, 1319, 2141, 9811, 11897, 12149, 12211, 12613, 13103, 31013, 79811, 118973, 379811, 412211, 513103.

0;  6, 4;  13, 1, 12;  14, 10, 7, 2;  5, 8, 3, 9, 11.
2, 3, 5, 7, 11, 13, 17, 29, 37, 41, 61, 71, 73, 79, 83, 97, 101, 103, 107, 113, 127, 131, 179, 211, 613, 617, 839, 911, 971, 1013, 1193, 3911, 5839, 8101, 12113, 12211, 13103, 14107, 31013, 51413, 83911, 412211.

0;  6, 13;  8, 1, 10;  3, 4, 7, 5;  11, 2, 12, 9, 14.
2, 3, 5, 7, 11, 13, 17, 23, 41, 43, 47, 59, 61, 71, 79, 83, 97, 101, 107, 113, 113, 127, 131, 149, 179, 211, 241, 311, 347, 613, 617, 683, 743, 971, 1013, 2129, 5743, 8311, 12211, 24113, 68311, 112129, 912211.

0;  6, 13;  8, 1, 10;  3, 4, 7, 9;  11, 2, 12, 5, 14.
2, 3, 5, 7, 11, 13, 17, 23, 41, 43, 47, 59, 61, 71, 79, 83, 97, 101, 107, 109, 113, 127, 131, 149, 211, 241, 311, 347, 479, 571, 613, 617, 683, 743, 1013, 8311, 9743, 12211, 13109, 24113, 68311, 1491013.

0;  7, 4;  9, 1, 6;  8, 10, 13, 12;  11, 5, 3, 2, 14.
2, 3, 5, 7, 11, 13, 17, 19, 23, 41, 47, 53, 61, 71, 79, 89, 97, 101, 103, 109, 113, 131, 313, 613, 619, 811, 1013, 1153, 1213, 1423, 2131, 3109, 3511, 5101, 6133, 9103, 9811, 11897, 21317, 79811, 81013, 1423511.

0;  9, 5;  12, 7, 10;  2, 4, 1, 13;  11, 3, 8, 6, 14.
2, 3, 5, 7, 11, 13, 17, 23, 41, 43, 47, 59, 61, 71, 79, 83, 97, 101, 107, 113, 127, 131, 179, 211, 241, 311, 347, 613, 617, 683, 743, 971, 1013, 2129, 5743, 8311, 12211, 14683, 24113, 68311, 112129, 912211.

0;  9, 14;  10, 7, 12;  13, 1, 4, 2;  5, 6, 8, 3, 11.
2, 3, 5, 7, 11, 13, 17, 23, 41, 43, 47, 61, 71, 79, 83, 97, 101, 107, 109, 113, 127, 131, 149, 211, 241, 311, 347, 479, 613, 617, 683, 743, 1013, 1471, 5683, 8311, 9743, 12211, 13109, 24113, 68311, 513109.

0;  11, 5;  3, 2, 14;  8, 4, 12, 13;  6, 1, 7, 9, 10.
2, 3, 5, 7, 11, 13, 17, 23, 41, 43, 47, 61, 71, 79, 83, 97, 109, 113, 127, 139, 179, 211, 241, 311, 347, 617, 683, 743, 971, 1013, 1097, 1213, 1423, 2129, 8311, 12211, 41213, 51413, 68311, 112129, 841213, 912211.

0;  12, 4;  13, 1, 6;  14, 10, 7, 2;  5, 8, 3, 9, 11.
2, 3, 5, 7, 11, 13, 17, 29, 37, 41, 61, 67, 71, 73, 79, 83, 97, 101, 103, 107, 113, 131, 179, 211, 673, 839, 911, 971, 1013, 1193, 1213, 1217, 3911, 5839, 6113, 6211, 8101, 13103, 14107, 31013, 51413, 83911.

0;  12, 13;  5, 10, 6;  8, 9, 7, 1;  2, 11, 3, 4, 14.
2, 3, 5, 7, 11, 13, 17, 37, 41, 43, 47, 59, 61, 67, 71, 73, 79, 89, 97, 107, 109, 113, 179, 211, 311, 593, 613, 673, 811, 911, 971, 1013, 1213, 1361, 1613, 2113, 8971, 12107, 13109, 141613, 144311, 1191013.

0;  12, 13;  5, 10, 6;  11, 9, 7, 1;  8, 2, 3, 4, 14.
2, 3, 5, 7, 11, 13, 17, 23, 29, 37, 41, 43, 47, 59, 61, 67, 71, 73, 79, 97, 107, 109, 179, 211, 593, 613, 673, 811, 823, 911, 971, 1013, 1213, 1361, 1613, 11971, 12107, 12511, 13109, 17911, 141613, 291013.

0;  12, 13;  14, 10, 6;  8, 9, 7, 1;  2, 11, 3, 4, 5.
2, 3, 5, 7, 11, 13, 17, 37, 41, 43, 47, 61, 67, 71, 73, 79, 89, 97, 107, 109, 113, 149, 179, 211, 311, 613, 673, 811, 911, 971, 1013, 1213, 1361, 1493, 1613, 2113, 8971, 12107, 13109, 51613, 54311, 1191013.

0;  14, 4;  13, 6, 12;  10, 1, 5, 2;  3, 7, 9, 8, 11.
2, 3, 5, 7, 11, 13, 17, 19, 37, 59, 61, 71, 73, 79, 89, 97, 101, 103, 107, 113, 131, 211, 251, 379, 461, 613, 617, 811, 1013, 1259, 1319, 9811, 11897, 12211, 12613, 13103, 31013, 79811, 118973, 379811, 412211, 1413103.

0;  14, 11;  5, 9, 8;  3, 7, 10, 12;  2, 4, 1, 13, 6.
2, 3, 5, 7, 11, 13, 17, 23, 37, 41, 43, 47, 53, 59, 71, 73, 79, 89, 97, 101, 107, 109, 113, 131, 149, 241, 479, 571, 613, 811, 911, 1013, 1213, 1453, 6131, 8101, 12107, 13109, 24113, 47911, 612811, 1491013.

0;  14, 12;  5, 9, 8;  6, 7, 10, 3;  2, 4, 1, 13, 11.
2, 3, 5, 7, 11, 13, 17, 41, 47, 59, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 149, 241, 311, 313, 479, 571, 1013, 1283, 1297, 7103, 8101, 8311, 11131, 11311, 13109, 24113, 67103, 128311, 411311, 1491013.

***