Problems & Puzzles: Puzzles Puzzle 209. Triangles of primes Here you asked to find a Ktriangular arrange of the first K*(K+1)/2 odd primes, such that the sum of the three primes in the vertex of every equilateral triangle embedded in the triangular arrange, add up to a prime number. Example for K=3 05 05+17+07 = 29
Example for K=4 19 Questions: 1. Can you provide a formula to calculate the quantity of embedded equilateral triangles in an Ktriangular array? 2. Can you find one solution for every 4<K<=10? 3. Do you devise a systematic approach in order to get the solutions asked in 2?
Solution: Q1. Correct solutions were sent by: J. Wharf, J.C. Rosa, J. Arioni, J. L. Pe, J. Heleen, J. vanDelden, Rudolph Knjzek, J. K Andersen & J. Wiesenbauer. The formulas sent are in a variety of forms (A(k) is the asked quantity of embedded equilateral triangles):
Come of them also noticed that this question was also solved
as a sequence in the OIS::
ID Number: A002717 (Formerly M3827 and N1569) Sequence:
0,1,5,13,27,48,78,118,170,235,315,411,525,658,812,988,1188,
1413,1665,1945,2255,2596,2970,3378,3822,4303,4823,5383,5985,
6630,7320,8056,8840,9673,10557,11493,12483,13528,14630,
15790,17010,18291,19635,21043,22517 Name: Floor(n(n+2)(2n+1)/8). Comments:
Number of triangles in triangular matchstick arrangement of side n.
or in in the "Book of Numbers" by John H. Conway and
Richard K. Guy on p.83.
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Q2. Solutions for K=5 were sent by Wharf, Rosa,
Andersen & Wiesenbauer.
But only Wharf and Andersen discovered the
reasons why there are no
solutions for K=>6. Both more or less say the same:
Both also
found all the solutions for K=4 & K=5
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Q3 becomes irrelevant after Q2.
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