H. E. Dudeney Features
The following features of the order-6 magic star were reported by H. E. Dudeney in Modern Puzzles, 1926. This book has been included in itís entirety in 536 Puzzles & Curious Problems Charles Scribnerís Sons, NY 1967 where this information appears in condensed form on pages 349-350.
Although Dudeney thought incorrectly that there were only 74 solutions, I have confirmed that these claims apply to all 80 solutions (shown on the previous page).
We must be full of admiration and amazement for this man who did so much for recreational mathematics, of which his work with magic squares and magic stars was but a small part. Also, letís remember, he did it all without the help of computers, or even desk calculators.
Because the interior cells of each triangle is common to both triangles, the total for all the cells in each main triangle is 78 - triangle point total.
Because the total of the six points is always even, there can be no solution with consecutive numbers at the points.
If the sum of the three points of a large triangle is subtracted from 26, this difference, plus the value of one of the 3 points will be equal to the sum of the two valleys opposite. i.e. 26 - (J + K + L) = J + F + H.
Diamonds & Point Pairs
The three diamonds A + C + K + H, J + E + G + I and D + F + L + B each sum to 26.
One or two pairs of opposite points always equals 13. Three pairs of points never do.
Opposite small triangles
Opposite small triangles will always sum to the same value,
i.e. A + B + I = E + F + K.
Another Dudeney claim was that all the solutions could be classified in three groups depending which of several pairs sum to 13. He referred to the first, and biggest, group as regular, and the other two groups as irregular. I simply call them class I, II and III.
This claim also checks out.
The patterns below will often require rotation and/or reflection in order to match the basic solutions.