this page I show photographs of handicraft projects made using
magic squares, magic stars and related subjects as the theme.
is a rewarding experience to hold, for example, an actual 3
dimensional model of a magic star or cube, instead of simply viewing
a 2 dimensional version of it on paper.
Unfortunately, a photograph simply reverses the process, and we
are once again viewing a 2 dimensional version of the object.
||of order-5 (but lots of
||of order-6. Points also are
||Any pandiagonal magic square
may be considered as being on the surface of a torus instead of a plane.
||constructed with wooden blocks
||of order-3. Magic lines shown
||An 8-point star with 12 lines
of 3 numbers sum to 27.
||As above, but 3 missing numbers
inserted to now give 22 lines summing correctly.
||12 = 3 x 4, 56 = 7 x 8, 9 is
not equal to 0. (Cross-stitch)
||A magic square rainbow.
||A order-5 pandiagonal
associated magic square constructed with wooden dowels and metal washers.
block and dowel construction of an order 4 pantriagonal magic cube. The
numbers on each face of the cubelets represent a different magic cube.
||My order -13
Quadrant magic square has 14 patterns correct in all 4 quadrants.
|| In cross-stitch. The Lho-shu
order-3 square, order-3 cube # 3, and order-3 tesseract # 5 (Hendricks # 1).
Cross-stitched Magic Square
||This pandiagonal magic square required the use of
negative numbers so the constant could be made equal to 40. It was a
gift to my son for his 40th birthday and was presented to him in a
plastic desk stand.
On the back of the pattern are diagrams
showing how there are at least 428 different ways to form the number
40 using arrangements of 5 numbers.
See How many groups = 65? for more
information on this type of magic square.
The cross-stitched magic square is 4.5 inches by 4.5 inches, and
has 67 by 67 cross-stitches. (There are also red shadow back
stitches on the digits which do not show in the picture.)
Cross-stitched Magic Star
||This cross-stitched magic star is a reflected
version of index number 16 (of the 80 basic solutions).
The actual size of this project is 8 inches by 7 inches. The size
of each of the 12 cells is 3/4 inch.
Projects like this are a nice break from sitting in front of the
(And great to work on when sitting in front of the TV.)
The squashed shape is a result of lines such as 1 - 10 - 12 -3
having to be on a 45 degree angle when the fabric has square
A Pandiagonal Torus
||When explaining the system of broken diagonal pairs
in a pandiagonal magic square, the magic square is often explained
as being part of an infinite plane.
Sometimes the reader is advised to imagine the edges of the magic
square folding back and joining the opposite edge. In effect, the
magic square is on the surface of a torus instead of a plane.
Pictured here is a crude model of such a pandiagonal magic
square. Any of the 25 numbers may be considered the first number of
the first row. Then the square is built up by simply following the
lines around the torus. See
Unusual Magic Squares
for more information.
||This is a 3-D model of an order-4 pandiagonal magic
The wooden blocks are 3/4 inch, the dowels, 1/8 inch.
Here I do not use dowels to indicate the diagonals.
1 15 4 14
8 10 5 11
13 3 16 2
12 6 9 7
A Wooden Magic Cube
||An order-3 magic cube. Size is 4.5" x 4.5" x 4.5"
Magic sum is 42 on all the 9 rows, 9 columns, 9 pillars and 4
All of these lines are shown with dowels.
A Wooden 3-D Magic
||This is a model of the 3-D magic star discovered by
Aale de Winkel and myself in May, 1999.
It has 12 lines of 3
numbers all summing to 27, forming a star with 8 points.
It uses 14 of the numbers from 1 to 17 and is the minimal solution.
Numbers not used are the 3, 9 and 15.
The arrangement may be considered as 2 triangular based pyramids
pointing in opposite directions and intersecting at the midpoints of
It may also be considered as 8 numbers being placed at the corners
of an imaginary cube. Each of these
numbers is joined to two numbers
placed at corners of
the opposite face of the cube. In effect, the 5 numbers on
each face of the imaginary cube are connected in the form of an X.
A Wooden 3-D
Magic Star- 2
||Another view of the 3_D magic star. However, here
the line formed by the 3 unused numbers (3, 9, 15 which also sum to
27) are incorporated into the star figure.
The magic star, on it's
own has 12 lines summing to 27.
With the inclusion of this extra line of 3, 9 and 15, there are now
a total of 22 lines summing correctly and all numbers from 1 to 17
Except for the 1 line consisting of the 3, 9 and 15, I do not
show these extra lines with dowels. Two of them however, are formed
by the rubber bands holding the 3, 9 15 line in place.
An apology. I made the mistake of varnishing these models. When I
tried to illuminate them for pictures, I found the reflection
blocked out the numbers, so I had to photograph them with minimal
See 3-Dimensional Magic Stars
for more information.
A Number Pattern
This cross-stitch pattern shows on my 19" monitor (1024 x 768 pixals)
as almost actual size (14 squares/inch = 10 inches).
A Number Pattern-
||Colors didn't reproduce too good in this image, but
you get the idea.
dimensional magic square
||The idea for this novel magic square was sent to me
by Craig Knecht on June 5, 2001.
This is an order-5 pandiagonal
associated magic square.
The numbers in each cell are represented by metal washers. In this
picture, the model is suspended from the ceiling to illustrate the
balanced nature of all magic squares.
The 325 metal washers give this model a weight of almost 2
|Addendum June 30, 2012
On reviewing the above photo, I realize it is a bit confusing
because the front right peg, with it`s one washer is not clearly
Here I present the same magic square, but in the Dudeney standard
I also show a simple magic square with no extra features. These
examples show that all magic squares are balanced.
And finally I show a number square that is not magic, but is also
balanced. This illustrates the fact that some number squares are
balanced. Obviously not all are.
As mentioned above, Craig Knecht came up with the idea of
demonstrating how magic squares are balanced by using dowels and
washers to represent the integers in the cells of the square.
I apologize for still not getting it right. I should have used a
lower camera angle to better show that the base of the square is
indeed level when suspended from the ceiling.
Six magic cubes in One
This model is constructed using 64 3/4" wooden
blocks and connected by 1/8" hardwood dowels showing rows, columns,
pillars and triagonals. Numbers used are 1 to 384, which is 43
The six faces of each cubelet are painted six different
colors. The 64 faces of each color form a magic cube with constants ranging from
760 to 780. All 6 cubes have the same characteristics. They are:
The cube is pantriagonal, so each cube has 112 correct
lines. All 2x2 square arrays (including wrap-around) in each cube also add to
the constant, giving another 192 combinations per cube.
click to enlarge
White = 760, Blue = 764, Red = 768,
Pink = 772, Green = 776, Yellow = 780
This model was completed on July 28, 2002
Below is the listing for the cube with the white faces. The magic
constant is 760.
White S = 760
Top – 1 Bottom + 1 Bottom
Cross-stitch Quadrant Magic Square
|Here are pictures of a cross-stitch project I
started July 13, 200303 and completed on Dec. 1, 2003.
It is a
complement square to the one shown above.
For those who are interested in cross-stitch, it is on
count 16 Aida fabric and contains 18, 730 cross-stitches
(plus the back stitches for the text.
Quadrant magic squares are discussed here.
The complement of this square is on
Dimensions 2, 3, and 4
Click to enlarge
|This pattern is cross-stitched on 18-count (18
squares to the inch) Aida light mocha colored fabric.
The design was started about July 1, 2007, and the stitching was
finished (with interruptions) on Jan. 28, 2008
The magic square is
the only basic square of this order. The magic cube is index number
3. And the magic tesseract is index number 5. It is the first
tesseract John Hendricks constructed with his newly invented
tesseract pattern (in 1950), so by his numbering it is # 1.