Magic hexagon

[bonanova]

The figure shows nineteen circles arranged hexagonally in fifteen rows of length 3, 4 or 5.

Place consecutive numbers in the circles so that .

1. At least one row in each of the three directions contains exactly one pair of numbers
   (e.g. 17 and 34) that differ by a factor of 2.
2. Exactly one of the six outer vertices contains a prime. (1 is not prime.)
3. Exactly one two of the six inner vertices (connected to the center) contains a prime.
4. The fifteen row sums are identical.

       

Edited December 7, 2014 by bonanova
Correcting Clue 3.

[k-man]

Can you clarify "consecutive numbers"? Does the puzzle require that all neighboring circles of a given circle had a number either 1 smaller or 1 greater than the number in the given circle? Or does it simply mean that the entire set of numbers used to fill in the puzzle must be a consecutive set?

[bonanova]

Entire set. The numbers are consecutive, and they're placed on the grid. Same for 'Magic squares, sort of' puzzle

[bonanova]

One of the numbers is 1.

[DejMar]

 

There is a published paper that demonstrates that there is a unique solution to the order-3 magic hexagon.

http://www.yau-awards.org/English/N/N92-Research%20into%20the%20Order%203%20Magic%20Hexagon.pdf

------ 3--17--18------ Condition 1 is satisfied with (4,8) in the fourth horizontal row,
----19-- 7-- 1--11---- and (2,4) in the second L-to-R-diagonal row.
--16-- 2-- 5-- 6-- 9-- Condition 2 is sastisfied {3,9,10,15,16,18} only 3 is prime.
----12-- 4-- 8--14---- Condition 3 fails {1,2,4,6,7,8} both 2 and 7 are prime.
------10--13--15------Condition 4 is satisfied - all fifteen rows sum to 38.

Edited December 6, 2014 by DejMar

[gavinksong]

There is a published paper that demonstrates that there is a unique solution to the order-3 magic hexagon.

http://www.yau-awards.org/English/N/N92-Research%20into%20the%20Order%203%20Magic%20Hexagon.pdf

------ 3--17--18------ Condition 1 is satisfied with (4,8) in the fourth horizontal row,

----19-- 7-- 1--11---- and (2,4) in the second L-to-R-diagonal row.

--16-- 2-- 5-- 6-- 9-- Condition 2 is sastisfied {3,9,10,15,16,18} only 3 is prime.

----12-- 4-- 8--14---- Condition 3 fails {1,2,4,6,7,8} both 2 and 7 are prime.

------10--13--15------Condition 4 is satisfied - all fifteen rows sum to 38.

Edit: I'm dumb. Edited December 6, 2014 by gavinksong

[gavinksong]

Are we allowed to use negative numbers? And if we are, how does primality work?

[DejMar]

Are we allowed to use negative numbers? And if we are, how does primality work?

Primality only applies to positive integers (i.e., natural numbers) by definition. Yet, in a non-formal definition of a prime number, the negative of a positive prime number can be associated as being the same prime number. Thus, with the second definition permitting negative values, 2 and -2, for example, are the same prime, while by the first definition negative numbers are not prime.

[bonanova]

DejMar has it.

 

The puzzle has no solution for the clues as given, which contain an error in clue 3.

My bad; there is one even prime number.

 

Nice solve.