Big Magic Stars
Introduction
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.
See these solutions on his site at http://www.geocities.com/aladgyma/articles/scimaths/polmagic.htm
I show 6 new solutions on this page. All are obtained from his
solutions by complimenting and then normalizing .
See my http://www.geocities.com/~harveyh/magicstar.htm
page for more information on these steps.
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
|
Order-15 pattern A |
Order-16 pattern A |
|
Order-17 pattern A |
Order-18 pattern A |
Order-19 pattern A |
Order-20 pattern A |
Other Patterns - Order 15
 |
 |
 |
 |
 |
Order- 15 uses numbers from 1 to 30. Types of patterns: All orders have at least 1 continuous pattern. |
 |
 |
 |
 |
 |
Order- 16 uses numbers from 1 to 32. Types of patterns: |
 |
 |
 |
 |
 |
 |
The 7 order-17 patterns of normal magic stars all use the numbers
from 1 to 34.
Each of the 17 lines sum to the magic constant of
70.
Because 17 is a prime number, all 7 patterns are continuous (they may be
drawn without lifting pen from paper).
Other Patterns - Order 18
 |
 |
 |
 |
 |
 |
The 7 order-18 patterns of normal magic stars all use the numbers
from 1 to 36.
Each of the 18 lines sum to the magic constant of 74.
|
Types of patterns: A = 2 - 9 sided figures
(enneagons), B = 3 hexagons, C = 2 - 9a stars, E = 6 triangles, G= 2 - 9c
stars, |
 |
 |
 |
 |
 |
 |
 |
The 8 order-19 patterns of normal magic stars all use the numbers from 1 to 38. Each of the 19 lines sum to the magic constant of 78. Because 19 is a prime number, all 8 patterns are continuous.
|
Normal magic stars ( a review) All normal magic stars |
Other Patterns - Order 20
 |
 |
 |
 |
 |
 |
 |
The 8 order-20 patterns of normal magic stars all use the numbers from 1 to 40. Each of the 20 lines sum to the magic constant of 82. Each pattern has 40 aspects due to rotations and
reflections. |
Description of order-20 patterns. A = 2 decagons |