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.

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 solutions.)

Six is the only order that does not have at least one continuous pattern. The pattern consists of two super-imposed triangles.

This particular pattern has all even numbers at the points. Index # 28 is the complement of this one so has the points all odd numbers.

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).

This particular solution has the numbers 8 to 14 at the points. Its complement, # 24, has numbers 1 to 7 at the points.

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.

This 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.

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 pattern.

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 pattern is non-continuous. It consists of three super-imposed triangles (9 = 3 x 3). Nine has 1 prime factor pair so has one non-continuous pattern.

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.

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.

It uses the series of numbers from 1 to 20 (2 x 10) with each line summing to 42

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 pentagons.

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 here.

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.

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.

There are 1,670,680 apparently different solutions for this pattern.
(11 rotations times 2 reflections times 75,940.)

This pattern is continuous (as are all order-11).

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).

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.

This page was originally posted March 1998
It was last updated October 15, 2010
Harvey Heinz harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz