Magic Star Definition

Contents

The pattern for Order-7a is shown below to help illustrate the definition.
Each of the 20 patterns from Order-5 to Order-12d may be seen at magic star examples.

Magic Star - Defined

As a prerequisite to a comprehensive study of Magic Stars. the first requirement is a formal definition. As I have not been able to find one in the literature, this is the definition I propose.

Definition
A pure magic star is a set of integers 1, 2, 3, ..., 2n which are placed at the 2n exterior points of intersection of the lines which form a regular polygram, such that the sum of the four integers found in any of the n lines is given by:   S = 4n+2   where S is called the magic sum, and n is the order of the star.

NOTES:

1. This definition is for a pure magic star.
2. A pattern of lines forming a star. The number of points defines the order.
3. Each line contains four cells to hold numbers.
4. These cells are located at the intersection of two lines forming a point (peak) or valley.
5. The numbers in the cells are the consecutive integers from 1 to 2n, (n is the order of the magic star).
6. The four numbers in each line must sum to the magic constant (S) for that order.
7. The cells are named and listed in the order illustrated in figure a, i.e. running along each line in order.
8. For the standard position:
   a. The top point has the lowest value of all the points.
   b. The top right valley has a lower value then the top left valley.

Star Definition Illustrated

NOTES regarding the above points

1. Point 1 is because there can exist impure magic stars, such as order-5, prime number, series not starting at one, etc.
2. Points 3 and 4 are needed with this pattern because it is possible to have more then 4 numbers on a line, or the four numbers could be placed in different locations.
3. Point 8.a) eliminates rotations, 8.b) eliminates horizontal reflections.
4. The numbers used (for the order-7) are consecutive from 1 to 14.
5. For listing and comparison purposes, the cell values are noted in alphabetical order as shown in the above figure a. This also shows the order of the lines.
6. The first line is always the one moving from the top point down to the right. Other lines continue as shown here. If a line is completed before the pattern is completed (a non-continuous pattern such as order-6), the next line starts with the next vacant point to the right of the top point (a).

Traditionally, cell designations have been shown row by row from top down and reading from left to right.. I have chosen to show them starting from the top but reading along each line in order

1. Because it seems more logical to read each line of the pattern as a unit rather then row by row where each number would be from a different line (except for horizontal lines). Also the rows become confusing as the complexity of the pattern increases.
2. Because it is more practical in a search algorithm to solve each line in order.

To support my definition

1. I have seen no definition after searching recreational mathematics literature.
2. I have seen no mention of the fact there are more then one pattern for each magic star (except for the smallest pure one, Order-6).
3. I have seen no lists of solutions published except for Orders 6, 7 and 8. But orders 7 and 8 have two lists each. No one has mentioned this.

I apologize to the hobbyists out there who have compiled lists of solutions using the row by row method, but as I started investigating the higher orders found that method simply to awkward and open to error. However, for orders 6 and 7, I have converted several of these lists to my system and found they compare perfectly so everyone seems to be coming up with the same results. My hope is that this definition will become a standard so that we can all compare our findings to the mutual benefit of the subject.

Trenkler Stars

Marián Trenkler has published a paper on magic stars in which he has defined magic stars with four numbers per line as type S or type T.

The stars I am mainly concerned with on this site (and have defined above), are Type S.

Type T stars differ in that the numbers that here appear in the valleys are located at intersections in the interior of the star.

His paper also defines;
Weakly-magic stars as those constructed from number sets that are not consecutive.
Almost-magic stars as those using consecutive numbers but have n – 2 lines summing to 4n + 2, and the others sum to 4n + 1 and 4n + 3.
An order-5 magic star with 10 numbers from 1 to 12 is a weakly-magic star. An order-5 magic star using numbers 1 to 10 is an almost-magic star.

For more details and examples go to Trenkler.htm (on this site)

Value of Magic Sum

In magic star patterns, each number appears in two lines so the formula to calculate the magic constant is:

S = Sum of the series / number of points * 2

For the 5-point magic star the sum of 1 to 12 = 78 less 7 and 11 = 60 so 60/5 * 2 = 24.
For the 6-point magic star the sum of 1 to 12 = 78 so 78/6 * 2 = 26
For the 7-point star the sum of 1 to 14 = 105 so 105/7 * 2 = 30, etc...

In general, when n is the order, S = 4n + 2.

Multiple Patterns per Order

What do I mean by pattern?

I have been using the word pattern ambiguously to refer to an arrangement of numbers that compose a magic star, and also to the arrangement of lines that together form the diagram of a star. It is this second definition I wish to discuss at this time.

The order-6 magic star can be formed with only one pattern of lines. However, from order-7 upwards, there are at least 2 patterns possible for each order.  My magic star definition limits the position of the cells (numbers) to four per line and at the peaks and valleys only, or there would be many more patterns possible.

Even though the patterns change completely from order to order, there is a surprisingly similar correspondence of peaks. Or, because every fourth designation as we trace through the pattern is a peak, maybe not surprising. The list below illustrates this similarity. Note that one point of the two patterns of order-8 differs, yet the number of solutions is the same. With all the other orders, if the points are similar, the number of solutions is the same, if they differ the solution total differs also.When we get to Order-11, though, there are two solutions (a & c) that differ from the pair with similar point names. They also differ from each other, but the number of solutions is the same (just as order-8). Presumably the same will apply to Order-12, but so far I have a complete solution list only for 12b.

Points, patterns & total solutions

 5    a  d  g  i  j     not a pure magic star     12   Continuous    11   seconds
 6    a  d  g  j  k  l                            80   2 triangles   7    seconds
 7a   a  d  g  j  l  m  n                         72   Continuous    1.4  minutes
 7b   a  d  g  j  l  m  n                         72   Continuous    1.1  minutes
 8a   a  d  g  j  m  n  o  p                     112   2 squares     1.1  minutes
 8b   a  d  g  I  m  n  o  p                     112   Continuous    4.7  minutes
 9a   a  d  g  j  m  o  p  q  r                 3014   Continuous    2.21 hours
 9b   a  d  g  j  l  n  p  q  r                 1676   3 triangles   18   minutes
 9c   a  d  g  j  l  n  p  q  r                 1676   Continuous    19.3 minutes
10a   a  d  g  j  m  p  q  r  s  t             10882   2 pentagons   24   hours
10b   a  d  g  j  m  o  q  r  s  t            115552   Continuous    6.5  hours
10c   a  d  g  j  m  p  q  r  s  t             10882   2 pentagrams  24   hours
11a   a  d  g  j  m  p  r  s  t  u  v          53528   Continuous    62   days
11b   a  d  g  j  l  n  p  r  t  u  v          75940   Continuous    33.6 hours
11c   a  d  g  i  k  m  o  q  s  u  v          53528   Continuous     5.3 hours
11d   a  d  g  j  l  n  p  r  t  u  v          75940   Continuous    20.2 hours
12a   a  d  g  j  m  p  s  t  u  v  w  x     >800000   2 hexagons       to assign the 
12b   a  d  g  j  m  o  q  s  u  v  w  x      826112   3 squares     39.5 days (+ 46 days
12c   a  d  g  j  l  n  p  r  t  v  w  x     >800000   4 triangles      complement pairs)  
12d   a  d  g  j  m  o  q  s  u  v  w  x      826112?  Continuous    
13a   a  d  g  j  m  p  s  u  v  w  x  y  z        >3,000,000  Continuous      
13b   a  d  g  j  m  p  s  u  v  w  x  y  z        >3,000,000  Continuous      
13c   a  d  g  j  m  p  r  t  v  w  x  y  z        >3,000,000  Continuous      
13d   a  d  g  j  m  o  q  s  u  w  x  y  z        >3,000,000  Continuous      
13e   a  d  g  j  l  n  p  r  t  v  x  y  z        >3,000,000  Continuous      
14a   a  d  g  i  k  m  o  q  s  u  w  y  aa  bb   >7,000,000  Continuous      
14b   a  d  g  j  m  o  q  s  u  w  y  z  aa  bb   >7,000,000  Continuous      
14c   a  d  g  j  m  p  s  v  w  x  y  z  aa  bb   >7,000,000  2 heptagons      
14d   a  d  g  j  m  p  s  v  w  x  y  z  aa  bb   >7,000,000  2 7-point star-b pattern   
14e   a  d  g  j  m  p  s  v  w  x  y  z  aa  bb   >7,000,000  2 7-point star-a pattern   

The orders are in pairs, with an odd and the next even order having the same number of patterns.
If two patterns have the same point names, the number of basic solutions is the same and the value and the value of cell a changes at the same solution number (except for order-9).

I have included patterns for Order-13 and Order-14 as a point of interest. However, it will probably be sometime in the future before these ten complete lists of solutions will be found. At this time, personal computers are simply not fast enough to explore the large number of permutations.

The last column (above) indicates the length of time required to search for all the basic solutions for each pattern. The computer used is a 200 Mhz Pentium II with 32 Megs of RAM. The program is written in QuickBasic, then compiled and run in a DOS window.
The number of consecutive letters as the final cell names is an indication of relative running time. For example, 11a has five consecutive letters at the end and took 62 days to run; 11c has only two and took 5.3 hours.

Pattern naming convention. Originally I had rather arbitrarily assigned names a, b, c, etc to the various patterns of an order of magic star. In January, 2001, Aale de Winkel suggested a systematical way of applying these labels.
Imagine the points of a star diagram as being points on a circle. Then each point in turn is connected by a line to another point, by moving around the circle clockwise. If we step once and connect to the second point, the pattern is called 'A'. Stepping twice, and connecting to the third point, produces pattern 'B'. etc.
Another way to look at this subject:
'A' has 4 intersections per line, 'B' has 6, 'C' has 8, 'D' has 10, and 'E' (required for orders 13 and 14) has 12 intersections per line.

By Feb. 16, 2001, all relevant pages have been revised to show the new pattern names.

Number of Basic Solutions

With increasing order, and a corresponding increase in the length of the series, the complexity of the pattern goes up as is to be expected. We would expect the number of basic solutions to also go up. And so they do (I’ve tested to order-12), except for the case of order-7 which has less solutions then order-6.

Another exception to the increasing number of solutions is order-11, where all 4 patterns have less solutions then Order-10b. Is it coincidence that in both cases, it is the first occurrence of two patterns with the same number of solutions? (In the case of Order-11, this is a second pair.)

The number of solutions for each order is always an even number because each basic solution has a complement, which, after normalizing, is also a basic solution. Another way to say this is that all basic solutions come in pairs that are complements of each other.

Orders 6, 8 and 10 do not have solutions with all low numbers at the points and therefor also do not have solutions with all high numbers at the points (because of complements). All other orders do have such solutions, including order-12, thus ending speculation that this is a feature of even orders.

Order-5 has been included in this study for comparison purposes although it is not a pure magic star.
With this one exception this study is limited to pure magic stars only.
Lists are of basic solutions only. Each order has 2n-1 disguised versions. So order-14 with over 7 million basic solutions has over 196 million apparently different solutions.
Because of the large file sizes for order-9 and higher, I include condensed lists only, showing some solutions at the start and end, and when the value of cell a changes.

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Harvey Heinz   harveyheinz@shaw.ca
This page last updated March 11, 2006
Copyright © 1998, 1999, 2000 by Harvey D. Heinz