A - B - C / \ / \ D E F G / \ / \ H - I - J - K - L \ / \ / M N O P \ / \ / Q - R - S | Place the numbers 1-19 in the hexagon, such that each of the 12 lines of three numbers have the same sum. What are the lowest and highest possible common sums? (Or is this possible?) |
Source: Original. Based on a puzzle from Sudipta Das.
Here is Al Zimmermann's summary:
Here are the attainable common sums and the number of ways (disregarding rotations and reflections) in which they can be attained: Sum Number of Ways --------------------- 22 4 23 2 24 6 25 6 26 4 27 11 28 5 29 8 30 0 31 8 32 5 33 11 34 4 35 6 36 6 37 2 38 4 These are the four ways for achieving a sum of 22: 1---9--12 1--13---8 / \ / \ / \ / \ 18 19 8 6 18 19 12 10 / \ / \ / \ / \ 3--17---2--16---4 3--17---2--16---4 \ / \ / \ / \ / 14 15 13 11 14 15 9 7 \ / \ / \ / \ / 5--10---7 5---6--11 1--14---7 1--18---3 / \ / \ / \ / \ 18 19 13 6 19 17 15 9 / \ / \ / \ / \ 3--17---2--11---9 2--16---4---8--10 \ / \ / \ / \ / 15 16 12 5 14 12 13 7 \ / \ / \ / \ / 4--10---8 6--11---5 The four solutions for a sum of 38 can be derived by replacing each number n in the above four solutions by the number 20-n.