This page contains examples of magic stars from order-5
to order-11. Because there are multiple patterns for each order greater
then order-6, there are a total of 16 different magic star examples.
The Magic Star Examples-2 page shows the 14 patterns for orders 12, 13 and
||Order-5......#2 of 12
||Order-6......# 31 of 80
|Order-7a......# 47 of 72
||Order-7b....# 72 of 72
||Order-8a....# 70 of 112
|Order-8b......# 45 of 112
||Order-9a....# 1,615 of 3,014
||Order-9b....# 289 of 1,676
|Order-9c......# 1,676 of
||Order-10a..# 10,882 of
||Order-10b..# 57,777 of
|Order-10c....# 10,882 of
||Order-11a..# 27,224 of
||Order-11b..# 41,733 of
|Order-11c....# 26,306 of
||Order-11d..# 41,733 of
||Magic Star Examples-2..Orders
Example magic Stars
|Index # 2 of 12 basic solutions. Each of the five
lines sum to 24.
This solution has the points also summing to 24.
Because 5 is a prime number, this is a continuous pattern.
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.
|Index # 31 of 80 basic solutions. Each line sums to
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 (except for order-10a) that has more
solutions then a higher order. (Order-7 has only 72 basic
Six is the only order that does not have at least one continuous
pattern. The pattern consists of two super-imposed triangles.
|Index # 47 of 72 basic solutions. Each line sums to
This particular pattern has all even numbers at the points.
Index # 28 is the complement of this one so has the points all odd
There are 2 star designs for this order, both having the same
number of solutions. Order-7 uses the consecutive numbers from 1 to
14 (2 x 7).
Order-7 has less basic solutions then the lower order 6, the only
such case (at least to Order-12).
|Index # 72 of 72 basic solutions. Each of the seven
lines sum to 30.
This particular solution has the numbers 8 to 14
at the points. Its complement, # 24, has numbers 1 to 7 at the
Each solution contains the consecutive series of integers from 1
to 14. The sum of this series is 105. There are 7 lines with each
number appearing in 2 lines, so the magic sum is (2 * 105) / 7 = 30.
Because 7 is a prime number, both designs of this order are
continuous, i.e. do not contain two or more distinct closed loops.
|Index # 70 of 112 basic solutions.
Order-8a consists of two super-imposed squares 8 = 2 x 4).
particular solution has the corners of each square also summing to
the constant 34 and is one of eighteen such solutions. In fact, all
solutions have the corners of the two squares summing to the same
value, which can range from 19 to 49.
Order-8 uses the consecutive numbers from 1 to 16 (2 x 8). Each
of the 8 lines sum to 34
All basic solutions have the lowest point value at the top. The
valley to the right of the top point has a lower value then the
value to the left of it. Each solution is written as a string of
numbers moving along the lines in order, starting from the top
point. The solutions are then sorted in ascending order, starting
with the first number, to determine the index number.
|Index # 45 of 112 basic solutions. Each line sums to
The points consist of the numbers 1 to 9 with no number 7, for
a point total of 38. This is the smallest point total possible. The
points of solution number 44 also sum to 38 but it uses the integers
1 to 10, with no 8.
All orders, except order-6 have more then one star pattern. The
number of patterns increases by one with each increasing odd order.
Also all orders (except six) have at least one continuous pattern
and, if the order is not a prime number, at least one non-continuous
|Index # 1,615 of 3,014 basic solutions. Each of the
9 lines sum to 38. It uses the series of numbers from 1 to 18 (2 x
This particular solution has the all even numbers at the
points. Its complement, # 995, has all odd numbers at the points.
They are 1 of 24 such pairs of solutions for Order-9a. By
coincidence, there are also 24 pairs of solutions with all low
numbers and all high numbers at the points.
The pattern is continuous. It is one of three star patterns for
Order-9, two of which are continuous.
|This is index # 289 of 1,676 basic solutions.
pattern is non-continuous. It consists of three super-imposed
triangles (9 = 3 x 3). Nine has 1 prime factor pair so has one
This particular solution has the points of each of the three main
triangles summing to 26 and is one of 2 such solutions. There are
twelve solutions with the three triangles summing to the same value,
which ranges from 23 to 34.
|Index # 1,676 of 1,676 basic solutions. Each line
sums to 38.
Becasue it is at the end of the list, this solution
has the nine high number of the series at the points. It's
complement, the other member of complement pair # 524, is solution
number 671. Naturally, it has the nine low 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 (except order-6), 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.
|Index # 10,882 of 10,882 basic solutions.
the series of numbers from 1 to 20 (2 x 10) with each line summing
This pattern is non-continuous. It consists of two super-imposed
pentagons (10 = 2 x 5). In each case, the five points of each
pentagon sum to the same value, and the ten cells in each pentagon
will sum to the same value because all valleys are common to both
|Index # 57,777 of 115,552 basic solutions. This
pattern is continuous.
Number 57,777 is the first basic solution
with a = 2. The last such solution is 84,776, i.e. exactly 27,000
basic solutions with a = 2.
There is a surprising difference between the number of solutions
for Order-10b and the number for patterns a & c. Have I overlooked
something? I could find no equivalent solutions and there are
exactly 57776 complement pairs (115552/2). (This number was later
confirmed by others)
Number of complement pairs = 57,776.
Last solution with a equal to1 = 57,776!
Order-10b has more solutions then order-11 just as order-6 has
more then order-7.
An explanation for the high count is
|Index # 10,882 of 10,882 basic solutions. Each line
sums to 42.
This pattern is non-continuous. It consists of two
5-pointed stars (10 = 2 x 5). In each case, the five points of each
star sum to the same value , and the ten cells in each star will sum
to the same value because all valleys are common to both stars.
|Index # 27,224 of 53,528 basic solutions. Each line
sums to 46.
Order-11 uses the consecutive numbers from 1 to 22
(2 x 11).
This is the first of four patterns for Order-11. All are
continuous because 11 is a prime number.
|Index # 41,733 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 75,940.)
|Index # 26,306 of 53,528 basic solutions. Each of
the 11 lines sums to 46.
This pattern is continuous (as are all
Search times vary widely for different patterns even though all
use the same algorithm.
Order-11a = 62 days Order-11d = 5.3 hours
(using a 200 Mhz Pentium Pro with 32 Megs of memory).
|Index # 41,733 of 75,940 basic solutions. Each line
sums to 46.
This is the first solution (of this pattern)
starting with the number 2 (position a).
This pattern is continuous because 11 is a prime number.
Go to Magic Star Examples-2 to see the
16 order 12, 13 and 14 examples.