Problems & Puzzles: Puzzles

Puzzle 92A pile of prime-spheres

Days ago my friend Enoch Haga and me starting puzzling each other to construct a pile of balls (a tetrahedron) with the following properties: a) every ball contains a distinct prime, b) each prime-ball must be the sum of the prime numbers contained in the three balls from the immediate inferior level and in contact with the mentioned prime-ball.

Can you imagine a pile of balls? A new friend of these pages, Chuck Henry, kindly and quickly provided several beautiful photos generated by him that should help to visualize a pile of balls like the one we are talking about. Please click here 1 and here 2 (*).

Question:

Get the least solution for a pile of n levels, for n=5, 6 & 7.

_______
Notes
(*) These renderings are provided by Chuck Henry, Professor in the Sculpture Department at Virginia Commonwealth University in Richmond, VA. His work constructing the human form (perhaps the human fractal) from the images created inside clusters of reflective spheres and point light sources use Pi and The Golden Mean to organize the geometric structure. This work can be found at: read   http://saturn.vcu.edu/~chenry

Hints:

Here are (what we think) the minimal solutions for n (levels)= 2, 3 & 4:

***
n=2

19                 19=3+5+11

3, 5
11
***
n=3

71                 71=23+19+19

23, 29            23=13+7+3; 29 =7+17+5
19                 19=
3+5+11

13, 7, 17
3, 5
11

***
n=4

431

149, 151
131

83, 37, 53
29, 61
41

67, 13, 5, 17
3, 19, 31
7, 11
23

***

For n= 5 we have not found a "complete solution". This is the "best" I have found, 33/35= 94.28%, composites in bold & black.

22879

7213 7813
7853

2671 2131 2621
2411 3061
2381

997 577 797 457
1097 757 1367
557 937
887

199 349 89 109 79
449 139 599 269
509 19 499
29 419
439

***

For n=5 I have gotten a 35/35 solution but only after accepting (9) repeated (bold & black) primes:

5167

2039 1249
1879

673 593 193
773 463
643

191 241 101 31
241 251 61
281 151
211

47 107 37 17 7
37 97 47 7
107 107 7
67 37
107

***

Solution

At last, one complete solution for n=5, 35/35 primes without any repetition! Enoch Haga got it the Sunday 08/05/2000:

20431
23311 28751
4111 10061 17011
6571 3221 13411 971
31121 6101 13291 5861 15601

72493
37483 55823
13903 26693 31393
43793 22613 32563 22433

165799
78079 113909
80309 81869 86389

357787
240257 282167

880211

***

Enoch Haga found a smaller solution (13/05/2000) for n=5 35/35:

15991
13441 24071
13591 1151 541
11731 10211 2161 18911
19681 29501 14731 18461 1861

53503
28183 25763
35533 13523 21613
60913 54443 35353 39233

107449
77239 60899
150889 103319 96199

245587
331447 260417

837451

***

One more & smaller solution by Enoch Haga (11/3/2001) for n=5 is this:

15373 23173 8443 30253 8053
2063 4283 7283 6653
1213 10903 14923
6803 16673
21313

40609 35899 45979 44959
7559 22469 28859
18919 42499
44789

84067 104347 119797
48947 93827
106207

237361 317971
248981

804313

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