Magic Star Definitions
The pattern for Order-7a is shown below to help illustrate
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.
NOTES regarding the above points
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
Another example of the need for a standard is on my primestars page.
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.
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;
For more details and examples go to Trenkler.htm (on this site)
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
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.
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 (explanation for high count here) 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 NC 12b a d g j m o q s u v w x 826112 3 squares 39.5 days (+ 46 days to assign 12c a d g j l n p r t v w x >800000 4 triangles NC the complement pairs) 12d a d g j m o q s u v w x 826112? Continuous NC 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.
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.
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.
By Feb. 16, 2001, all relevant pages have been revised to show the new pattern names.
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 (including order-12) do have such solutions, 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.