Problems & Puzzles: Puzzles Puzzle 92. A 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. _______ Hints: Here are (what we think) the minimal solutions for n (levels)= 2, 3 & 4: *** 19 19=3+5+11 3, 5 71
71=23+19+19 431 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 *** For n=5 I have gotten a 35/35 solution but only after accepting (9)
repeated (bold & black) primes: *** 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 72493 165799 357787 880211 *** Enoch Haga found a smaller solution (13/05/2000) for n=5 35/35: 15991 53503 107449 245587 837451 *** One more & smaller solution by Enoch Haga (11/3/2001) for n=5 is this: 15373 23173 8443 30253 8053 40609 35899 45979 44959 84067 104347 119797 237361 317971 804313
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