## Contents

 Introduction Order-5 Order-6 Order-7a Order-7b Order-8a Order-8b Order-9a Order-9b Order-9c Order-10a Order-10b Order-10c Order-11a Order-11b Order-11c Order-11d

See the next page for examples of the 14 magic star patterns of orders 12, 13 and 14.

The Corners of each square = 49

## Introduction

Magic stars are similar to Magic Squares in many ways. The order refers to the number of points in the pattern. A standard (normal or pure) magic star always contains 4 numbers in each line and consists of the series from 1 to 2n where n is the order of the star.

The magic sum (S) equals
(Sum of the series/number of points) plus 2 or
S = 4n + 2

Triangle points total 30, 32, 34, 36

### Basic & Equivalent Solutions

Each star has solutions that are apparently different but in fact are only rotations and/or reflections of the basic solution. For example, the order-10 star with its 10 degrees of rotational symmetry, each of which may be reflected, has 20 apparently different solutions. Only one of these is considered the basic solution.

Two characteristics determine the Basic Solution.

 The top point of the diagram has the lowest value of all the points. The valley to the right of the top point has a lower value then that of the valley to the left.

### Index Numbers

The basic solutions for a given order of magic star may be assembled in a numbered list.
Each solution consists of a list of the numbers appearing along each line in order as these order-6
examples:  # 1 (diagram below)       = 1   2   11  12   3   5   6   10   9    8   4   7
# 2 (the next in order)  = 1   2  11  12  4    3   7   8   10   5   6    9
The solutions are sorted by comparing the numbers from left to right and indexed accordingly.

Note that for order-6, the last 3 vacant cells in the search algorithm are all points. See 11c for
an explaination of required search time based on this fact.

## Example Magic Stars

### Order-5

Index # 6 of 12 basic solutions. Each of the 5 lines sum to 24.

This particular pattern (the only one of the 12) has numbers 1 to 5 at the points.

Order-5 is the smallest possible magic star. However, it is not a pure magic star because it cannot be formed with the 10 consecutive numbers from 1 to 10. The lowest possible magic sum (24) is formed with the numbers from 1 to 12, leaving out the 7 and the 11.

It is also possible to form 12 basic solutions with the constant 28, by leaving out the 2 and the 6.

### Order-6

Index # 38  of 80 basic solutions. Each of 6 lines sum to 26.

This particular pattern has the points also summing to the magic constant of 26. It is one of six solutions that have this property. The six complements of the above, have the valleys summing to 26. It is the only order that can have this property.

Order-6 is the smallest order that can form a pure magic star. It uses the numbers from 1 to 12.

It is the only order (at least to order-14) that can have more solutions then a higher order. (Order-7 has only 72 basic solutions.)

### Order-7a

Index # 28 of 72 basic solutions. The 7 lines each sum to 30.

Order-7 is the only one that has fewer solutions then a lower order (Order-6 has 80).

In this example, the points are all odd numbers.

This is the first of two patterns for order-7. Because 7 is a prime number, both Order-7 patterns are continuous. i.e. may be drawn without lifting pen from paper.

### Order-7b

This is index # 24 of a total of 72 basic solutions.

In this example, the points contain the numbers 1 to7.

Order-7 is the first order that has more then one
pattern. The number of patterns per order increases by one for each odd order.
Example; Order-5 and 6 have one pattern, 7 and 8 have two patterns, 9 and 10 have three patterns, etc

### Order-8a

Index # 36 of 112 basic solutions.
8 lines each sum to 34.

This is a non-continuous pattern, consisting of two super-imposed squares. In this example, the corners of each square also sum to the magic number 34.

### Order-8b

Index # 112 of 112 basic solutions. The 8 lines each sum to 34.

This is the first of two patterns for order-8. This one is continuous.

In this example, the points contain the eight highest numbers in the sequence.

### Order-9a

This is index # 3014 of 3014 basic solutions. It uses the 18 consecutive numbers from 1 to 18. Each of the nine lines sum to 38.

In this example, the points contain the numbers 10 to 18.

This pattern is continuous.

### Order-9b

Index # 1098 of 1676 basic solutions. Each line sums to 38.

This pattern is non-continuous. It consists of 3 super-imposed triangles, and in this case, the points of each triangle sum to the same value,31

### Order-9c

Index # 965 of 1676 basic solutions. S = 38.
This particular solution has all even numbers at the points.
Order-9c is a continuous pattern, i.e. it can be traced without lifting pen from paper. All orders have at least one continuous pattern, and if the order is a prime number, all the patterns will be continuous.
If the order number is composite, there will be the same number of non-continuous patterns as there are prime factor pairs.

### Order-10a

Index # 10 of 10882 basic solutions.

This pattern consists of two super-imposed pentagons with the corners of each summing to 62.

### Order-10b

Index # 10 of 115552 basic solutions. Uses the consecutive numbers from 1 to 20. The ten lines each sum to 20.

I am surprised there is such a large difference in the number of basic solutions between pattern B with 115552, and patterns A & C each with 10882.
However, all solutions matched up in complement pairs, and I could not find any equivalent solutions.

This is a continuous pattern.

### Order-10c

Index # 10 of 10882 basic solutions. Each line sums to 42.

This pattern is non-continuous. It consists of two 5-pointed stars (10 = 2 x 5).
The points of each pentagram sum to 54.

### Order-11a

Index # 500 of 53528 basic solutions. Each of the 11 lines sums to 46

Pattern A and pattern C have the same number of solutions, even though the point names are different (as does Order-8).
Patterns B and D both have the same number of solutions (75940) as expected, because they have the same point names.

### Order-11b

Index # 6 of 75940 basic solutions. Each line sums to 46.

There are 1,670,680 apparently different solutions for this pattern
(11 rotations times 2 reflections times 75940).

Search times vary widely for different patterns even though all use the same algorithm.
Order-11a = 62 days     Order-11c = 5.3 hours
(using a 200 Mhz Pentium Pro with 32 Megs of memory).

### Order-11c

Index # 26306 of 53528 basic solutions.

This is the first solution with a (the top number)
= 2

As explained in the introduction, the algorithm searches for suitable numbers for one line at a time. As the pattern nears completion, the only vacant cells are at points. The more consecutive points that remain vacant at the end, the longer the search time required.

For example: The last 5 empty cells are all points, but 11c has only 2 vacant cells that are points.

### Order-11d

Index # 41733 of 75940 basic solutions.

This is the first solution with a (the top number)
= 2

Because 11 is a prime number, all Order-11 patterns are continuous.

See the next page for examples of the 14 magic stars of orders 12, 13 and 14.

See my Geocities Page on Magic Stars for much more on Magic Stars