Big Magic Stars
All of the normal magic stars (from orders 6 to 14) on these pages have been found by a computer search. Unlike magic squares, with their regular pattern of an orthogonal array, magic stars have a different pattern for each order and, in fact, multiple patterns within each order. To find solutions for magic stars with these different patterns would require a separate algorithm for each.
I have developed a search algorithm which requires only modest program alterations to adapt to each different pattern. Using an exhaustive search routine has the added advantage that the solutions may be placed in an ordered list.
However, with increased order size, comes increasing work to adapt the program, and increasing search time to find solutions. For this reason, I have not yet tackled orders greater then 14.
On June 26, 2001, Simon Whitechapel sent me an e-mail with a link to a Web site he published. It contains 6 magic stars, one each of orders 15 to 20. They are all of pattern A and all were found by him using a Pascal program.
I show 6 new solutions on this page. All are obtained from his
solutions by complimenting and then normalizing .
I then show the other possible patterns for each order. They appear in blank form, and await the discovery of solutions.
Six Solutions - Orders 15 - 20
Other Patterns - Order-15
Other Patterns - Order-16
Other Patterns - Order-17
The 7 order-17 patterns of normal magic stars all use the numbers from
1 to 34.
Other Patterns - Order-18
The 7 order-18 patterns of normal magic stars all use the numbers from
1 to 36.
Types of patterns: A = two 9 sided figures (enneagons), B = 3 hexagons, C = two 9a stars, E = 6 triangles, G= two 9c stars. D & F = continuous.
Other Patterns - Order-19
Other Patterns - Order-20
Conclusion and a challenge
From the forgoing, it is obvious that there are multiple patterns per order and the number of patterns increase by one for each new odd order.
Notice that pattern 'A' has 4 intersections per line, 'B'
has 6, 'C' has 8, 'D' has 10, and 'E' has 12, 'F' has 14, 'G' has 16, and
'H' has 18 intersections per line. For a normal magic star, the numbers
are always placed on the two outside intersections of the lines.
The challenge! I welcome, and will acknowledge, all solutions submitted for these patterns.
This is the last solution on the file. Anyone wish to draw a diagram for this huge star?
Data for a 50-v magic star (magic total = 202).
Good work, Simon!