THE MAGIC HEXAGON
Why not solve it the easy way!
By G. P. Robert
The magic hexagon puzzle consists of arranging the numbers 1 to 19 in a
hexagonal pattern (see below) in such a way that each row in any direction adds up
Unlike magic squares, which can be easily constructed if you know the trick,
the solution to the Hexagon does not appear to possess any discernable regular
pattern. The number of possible arrangements is a formidable
121,645,100,408,832,000. However, since any arrangement can be rotated six times
through 60 degrees, and each of these positions has a mirror image, the actual
number of distinctly different arrangements is reduced to a mere
Inspection of the hexagon reveals one or two relationships which might help in
reducing the prodigous task of finding solutions. For example, since the sum of the
lines (A+B+C)+(D+E+F+G)+(H+I+J+K+L)=3x38 and (A+D+H)+(C+G+L)=2x38, then
it follows that the remaining triangle of numbers (B+E+F+I+J+K) must equal 38.
Addition to this triangle of the line (M+N+O+P) gives the large central triangle
(B+E+F+I+J+K+M+N+O+P)=2x38. Within this larger triangle the lines (B+E+I+M) and
(B+F+K+P) together equal 2x38. B, which occurs in both lines, must therefore equal
the small residual triangle (J+N+O). Similar relationships will, of course, exist in each
of the 12 possible orientations of the Hexagon.
Use can be made of some of these relationships to shorten the search for a
solution by trial and error. The listed program, which will run on a Nascom 1 or 2
under Nas-Sys, and could be adapted to run on any Z80 based microcomputer,
systematically fills the hexagon from available numbers stored in a table at the
bottom of the screen. At each position where a check can be carried out, an
appropriate line or triangle of numbers is added up. If the total is 38 the program
proceeds to the next position. If not, then a number is selected and the test is
repeated. If the end of the table is reached without a satisfactory number being found
the program retreats to the previous position and the search is continued.