The Magic Hexagon

arithmetic/magic.hexagram New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

The Magic Hexagon

Twelve circles have been placed at the vertices and intersections of a
hexagon.  How can the numbers 1 to 12 be placed, one in each circle, so
that the sums of the numbers in every row, and also the sum of the six
vertices, are equal?

-- Garth Williams <williams@engmlab.uct.ac.za>

[ Editor: the figure is a six-pointed star (two overlapping
triangles): six points and six intersections, with six rows of four
circles. ]

Show solution

Here is one way:

              1         3
                   9
             12        11
        2                    5
              6         4
                  10
              7         8

-- Dennis Yelle <dennis@netcom.com>

[ Editor: the points of the star are 1, 3, 5, 8, 7, 2, and the rows
are 1+9+11+5, 5+4+10+7, 7+6+12+1, 3+11+4+8, 8+10+6+2 and 2+12+9+3;
each of these, and the six points, sum to 26. ]