Problems & Puzzles: Puzzles
Puzzle 209. Triangles of primes
Here you asked to find a K-triangular 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+17+07 = 29
Example for K=4
1. Can you provide a formula to calculate the quantity of embedded equilateral triangles in an K-triangular 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?
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.
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
Q3 becomes irrelevant after Q2.