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to this page of unusual magic
stars. New material will be added here as it becomes available and
time permits. |
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Six in Six
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Two order-6 magic stars with some cells common to
both.
The blue inner star is basic solution #1 with S = 26.
The 3 peaks on each triangle sum to 19.
The green outer star is not a pure magic star because the
numbers are not consecutive. S = 48.
The 3 peaks on each triangle sum to 53.
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Eight in Eight (a)
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2 super-imposed order-8 pattern B stars. The small
star is a pure
magic star so S = 34. It is solution # 30.
The large star has S = 68.
The valley cells are common to both stars although they are in
the interior of the large star (not the valleys).
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Eight in Eight (b)
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An order-8B star super-imposed over an order-8A
star.
The type A star is index #26 so S = 34.
The type B star is not normal because numbers are not
consecutive. S=76
The valley cells of the type A star are the peak cells of
the type B star.
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Five-in-a-row (8)
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An order-8A type star but contains 12 rows of 5
numbers. Numbers 1-25. S=65. This type of star is now known as an
Isomorphic magic star. [1]
This pattern has a direct relationship to an order-5 pandiagonal
magic square as shown on the right.
[1] See more on isomorphic
stars here. |
| 1 |
19 |
23 |
15 |
7 |
| 25 |
12 |
6 |
4 |
18 |
| 9 |
3 |
20 |
22 |
11 |
| 17 |
21 |
14 |
8 |
5 |
| 13 |
10 |
2 |
16 |
24 |
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Five-in-a-row (6)
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This magic hexagon commemorates the year 1993.
It consists of the 13 prime years of the 20th century and
the first six prime years of the 21st century. Each line
of
five primes sum to the constant 9875.It was designed by Alan Wm.
Johnson. Jr. and appeared
in Recreational & Educational Computing Newsletter,
Dec. 1992, vol. 7, No. 8.
See the REC homepage at
http://members.aol.com/DrMWEcker/REC.html
Sorry. This link no longer works (2010).
See my other material from REC
.(Use Back button to return)
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Five-in-a-row (6b)
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This star contains the consecutive numbers
1 to 19 in 9 rows of 5 numbers.Each row sums to 46. |
Magic Star -
magic triangles
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This magic hexagon uses the numbers 1 to 13 to form
6 lines summing to 28. Each of the six small triangles of 3
numbers, as well as the two large triangle points, sum to 21.
Because the number seven is missing, this is not a pure
magic star.
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This magic hexagon uses 12 prime numbers to form 6
lines summing to 308. Each of the six small triangles of 3
numbers, as well as the two large triangle points, sum to 231.
Both of these stars
from Harry Langman, Ph.D., Play
Mathematics, Hafner Publishing Co. 1962.
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Hexagram - magic
triangles
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This star contains the consecutive numbers 1 to 12
arranged so that all six small triangles sum to 17.The complement
of this star (each number
subtracted from 13) has each triangle summing to 22.
There are two other such arrangements,
with the triangles summing to 18 and 19.
Each of these has a complement, summing
respectively, to 21 and 20.
Note that the star, as a whole, is not magic.
From Harry Langman, Ph.D.,
Play Mathematics, Hafner Publishing Co. 1962.
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Multiplying
magic hexagram
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When the four numbers in each line are multiplied
together,
the product, when divided by 13, leaves a remainder of one.
This pattern was designed by
David M. Collison (1937-1991).
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Five-in-one
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This pattern was constructed by a Mr. Morton about
1915. It appears as fig. 678 on page 348 of W. S. Andrews, Magic
Squares and Cubes, Dover Publ. 1960 The innermost star (heavy
red) is an upside-down Pattern 7A, and uses the consecutive numbers
from 1 to 14 so is a pure magic star. It is an equivalent to basic
solution number 71.
The blue star (S = 120) also has 4 numbers per line. It is also
an upside down Pattern 7A.
The Violet star (S = 144) has 6 numbers per line and is an upside
down pattern 7B.
The light red star (S = 340) has 10 numbers per line and is also
a pattern 7B.
The Green outside star (S = 520) also has 10 numbers per line and
is a pattern 7A.
The author used the numbers from 1 to 101 for this construction.
Six of these numbers were not used and three of them were used
twice.
Can you find these missing and
duplicate numbers? |
Three-in-one
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On examining the patterns of some higher order magic
stars, we see that the inner part of the pattern is actually another
pattern for the same order. By adding 2n to each number of a
basic star of that order we obtain a new magic star. To the left I
show an order-10 pattern C star. The outside 20 numbers form the
basic solution 5304 (the first one with a = 2) and the magic sum 42.
The inside part of the pattern is order-10 pattern A. Here it
forms a magic star with the consecutive numbers from 21 to 40. These
were obtained by adding 20 to each number of the basic solution
(also # 5304) of pattern A. This star has the magic sum of 122.
The entire pattern forms a magic star with 8 numbers per line and
the magic constant 164.
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The order-11 star is pattern C with pattern A
inside. The overall pattern uses consecutive numbers from 1 to 44.
For order-11 we add 22 to each number of the inside star.Magic
sums are S = 46, S = 134, S = 180
Index numbers are 11-C # 27224 and 11-A # 26306.
A similar star can be constructed using 12C with 12A inside.
Similar patterns could be made using 9A with 9C inside, 11B with
11D inside and 12B with 12D inside. In these 3 cases however, the
inside star would be very small in comparison with the outside one.
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Garrick's 6-in-a-row
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This pattern was sent to me on Mar. 5, 2002 by
Garrick Wells of Charlotte, North Carolina, U. S. A.. He reported
that it took him 5 1/2 weeks to design it.
It is an Order-12, pattern B star, but not standard because it
contains more then 4 numbers per line.
It uses the numbers from 1 to 36 with each of the 12 lines summing
to 111.
In addition, the valley numbers form the corners of two
overlapping hexigons of 6 numbers each that sum to the same 111.
These are 19, 17, 13, 22, 16, 24 and
23, 14, 15, 18, 21, 20.
Notice that the 12 lowest numbers appear in the point (peak)
positions, the 12 middle numbers appear in the valleys, and the 12
highest numbers appear in the interior of the star.
Of course if the star is complemented by subtracting each number
from 37, the positions of the smallest and largest numbers would be
reversed. |
Magic Stars - Magic
Squares
I have a page showing magic stars mapped to magic squares.
My unusual squares page has an example of a
star embedded in a square
and a fancy order-8 star
mapped to an order-4
square.
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