Problems & Puzzles: Puzzles

Puzzle 815. Another prime pine Jan van Delden sent the following nice puzzle. It's a new version of the old (1999) Puzzle 77 from my pages: Let’s define our Christmas prime tree. See  Puzzle 77 for a different kind of tree. The tree has n rows of baubles. All baubles are decorated with different prime numbers.   As a non-minimal example (n=6):   2141 1493, 2789 257, 2729, 5381 71, 443, 7673, 13337 29, 113, 1187, 29363, 35993 5, 53, 281, 4409, 142067, 69143   Largest prime number: 142067 Our peak:                      2141 With sum:                     318957     We have:    29 = (5+53)/2, 113=(5+53+281)/3, 1187=(5+53+281+4409)/4, 29363=(5+53+281+4409+142067)/5,35993=(5+53+281+4409+142067+69143)/6   So the prime numbers on a row are equal to the moving average of the prime numbers on the row below. This moving average always starts at the prime on the far left side of the tree. In the example the number 5.   Try to answer the following questions using as large a tree as you possibly can!   Q1: Find a tree where the largest prime number is minimal.   Q2: Find a tree where the prime number at the peak is minimal.   Q3: Find a tree for which the sum of all primenumbers is minimized.   Q4: Is it possible to find trees where the ending digit D of all the primenumbers are equal?

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Contributopns came from Emmanuel Vantieghem, Vicente Felipe Izquierdo,

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Emmanuel wrote:

For n = 6, the tree with minimal maximal prime is :

677

653  701
 

587  719  797
 

347  827  983  1031
 

11  683  1787  1451  1223
 

3   19   2027   5099   107   83

 

There is another tree with smaller top :

 

593

467   719
 

191   743   1223
 

23   359   1847   2663
 

5   41   1031   6311   5927
 

3   7   113   4001   27431   4007
 

but it is possible that there exist a tree with a smaller top.

 

I did not try to minimize the sum of the primes (I think that the above solution could be it). Instead, I started the search for n = 7 and found among all the trees with 3 in the left lower corner the one with smallest maximal prime :

 

6569

6269  6869
 

5639  6899  8069
 

4259  7019  9419  6899
 

569  7949  12539  16619  11579
 

107  1031  22709  26309  32939  38219
 

3   211   2879   87743   40709   66089  69899
 

 

Finding the minimal top tree or the minimal sum tree would cost me at least a week of computing.

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Vicente wrote:

(n=6)

 

Q1:

677
653	701
587	719	797
347	827	983	1031
11	683	1787	1451	1223
3	19	2027	5099	107	83

 

Q2:

421
271	571
193	349	1171
19	367	661		3637
7	31	1063		1543		15541
3	11	79		4159		3463		85531

Q3: same tree in Q1

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