Here's one for n=7 (without the symmetry): 26 -26 43 24 -11 -3 -53 -51 20 17 -46 33 -57 22 61 -7 -55 18 -54 55 21 45 40 -62 63 51 -31 -47 -56 58 -2 12 -19 -29 41 32 -61 -14 -16 60 -45 -8 -12 -27 50 -41 -37 62 49 59 -43 -42 -21 10 0 -49 53 -32 38 -6 42 -60 -9 -25 44 29 -63 28 34 -20 -23 6 11 -50 56 35 3 37 -17 -24 -36 2 -48 -22 25 -40 -10 13 27 54 -44 14 31 57 36 -59 -4 -34 -1 47 -38 1 -5 -15 9 23 -52 19 48 16 -35 -13 8 4 -28 -58 -18 46 7 39 52 -30 30 5 15 -39 -33 On Fri, Dec 4, 2020 at 9:50 AM Tomas Rokicki <rokicki@gmail.com> wrote:
Here's one for n=6 (without the symmetry):
35 12 6 -14 -3 -36 -22 8 13 -35 44 -34 26 7 -26 -19 25 21 -24 32 -17 11 39 -40 33 20 -31 -5 -29 3 2 -9 -8 34 -25 36 -30 -7 -2 9 -33 4 0 -41 -11 17 -15 37 31 -4 15 -28 38 -6 -10 30 -32 -38 -13 19 40 10 41 -39 1 24 -18 5 22 -45 -37 -42 14 28 16 42 -44 23 45 -16 -1 27 -23 -20 -12 43 -27 29 -43 18 -21
On Fri, Dec 4, 2020 at 8:20 AM ed pegg <ed@mathpuzzle.com> wrote:
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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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