Let k be three times a Triangular number. k=3 T(n), 3, 9, 18, 30, 45, ... Arrange numbers -k to k in a hexagon so that all hex lines have sum zero. From Tomas Rokicki, here are all the solutions for hex size 3. {{{-9,1,8},{3,4,-5,-2},{6,-7,0,7,-6},{2,5,-4,-3},{-8,-1,9}},{{-9,1,8},{6,5,-4,-7},{3,-8,-3,9,-1},{2,4,0,-6},{-5,-2,7}},{{-9,3,6},{7,-8,5,-4},{2,1,0,-1,-2},{4,-5,8,-7},{-6,-3,9}},{{-9,4,5},{7,3,-6,-4},{2,-8,-2,9,-1},{1,6,0,-7},{-3,-5,8}},{{-9,5,4},{6,2,-7,-1},{3,-8,0,8,-3},{1,7,-2,-6},{-4,-5,9}},{{-8,1,7},{6,3,-4,-5},{2,-9,0,9,-2},{5,4,-3,-6},{-7,-1,8}},{{-8,2,6},{9,-5,3,-7},{-1,-4,0,4,1},{7,-3,5,-9},{-6,-2,8}},{{-8,3,5},{7,-9,6,-4},{1,2,0,-2,-1},{4,-6,9,-7},{-5,-3,8}},{{-8,3,5},{7,4,-7,-4},{1,2,0,-2,-1},{-9,-6,9,6},{8,-3,-5}},{{-8,3,5},{9,-7,4,-6},{-1,-2,0,2,1},{6,-4,7,-9},{-5,-3,8}},{{-7,1,6},{9,-4,3,-8},{-2,-5,0,5,2},{8,-3,4,-9},{-6,-1,7}},{{-7,2,5},{3,6,-8,-1},{4,-9,0,9,-4},{1,8,-6,-3},{-5,-2,7}},{{-7,2,5},{6,8,-5,-9},{1,-2,-6,3,4},{-8,-1,9,0},{7,-3,-4}},{{-7,3,4},{9,-8,5,-6},{-2,-1,0,1,2},{6,-5,8,-9},{-4,-3,7}},{{-6,7,-1},{9,-7,2,-4},{-3,-8,6,0,5},{8,-2,3,-9},{-5,1,4}},{{-5,2,3},{9,-8,6,-7},{-4,-1,0,1,4},{7,-6,8,-9},{-3,-2,5}},{{-5,2,3},{9,-7,6,-8},{-4,-2,0,1,5},{7,-6,8,-9},{-3,-1,4}},{{-5,3,2},{9,-8,6,-7},{-4,-2,1,0,5},{7,-6,8,-9},{-3,-1,4}},{{-5,8,-3},{9,-2,-6,-1},{-4,-7,0,7,4},{1,6,2,-9},{3,-8,5}}} Here's a hex size 4 solution. {{-12,-11,15,8},{-7,-10,-3,4,16},{13,-2,17,-9,-1,-18},{6,5,-14,0,14,-5,-6},{18,1,9,-17,2,-13},{-16,-4,3,10,7},{-8,-15,11,12}}, Here's a hex size 5 solution. {{-27,15,2,-13,23}, {22,11,-17,28,-19,-25}, {4,7,-18,30,-21,24,-26}, {9,1,-6,-29,-10,12,3,20}, {-8,-14,16,-5,0,5,-16,14,8}, {-20,-3,-12,10,29,6,-1,-9}, {26,-24,21,-30,18,-7,-4}, {25,19,-28,17,-11,-22}, {-23,13,-2,-15,27}} At Wikipedia's entry on magic hexagons there is an order 8 solution by Louis K. Hoelbling. Above the size 2 hexagon, do solutions always exist? --Ed Pegg Jr