Magic Square Models



On this page I show photographs of handicraft projects made using magic squares, magic stars and related subjects as the theme.

It 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.


A Cross-stitched Magic Square

of order-5 (but lots of combinations).

A Cross-stitched Magic Star

of order-6. Points also are magic.

A Pandiagonal Torus

Any pandiagonal magic square may be considered as being on the surface of a torus instead of a plane.

Order-4 Magic Square

constructed with wooden blocks and dowels.

A Wooden Magic Cube

of order-3. Magic lines shown with dowels.

A Wooden 3-D Magic Star

An 8-point star with 12 lines of 3 numbers sum to 27.

A Wooden 3-D Magic Star- 2

As above, but 3 missing numbers inserted to now give 22 lines summing correctly.

A Number Pattern

12 = 3 x 4, 56 = 7 x 8, 9 is not equal to 0. (Cross-stitch)

A Number Pattern- 2

A magic square rainbow.

A 3 dimensional magic square

A order-5 pandiagonal associated magic square constructed with wooden dowels and metal washers.

Six magic cubes in One

A wooden block and dowel construction of an order 4 pantriagonal magic cube. The numbers on each face of the cubelets represent a different magic cube.

A Quadrant Magic Square

My order -13 Quadrant magic square has 14 patterns correct in all 4 quadrants.

Order-3, Dimensions 2, 3, and 4

 In cross-stitch. The Lho-shu order-3 square, order-3 cube # 3, and order-3 tesseract # 5 (Hendricks # 1).

rule-w.gif (2726 bytes)

A 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.)


A 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 computer.
(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 stitches.


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.


Order-4 Magic Square

This is a 3-D model of an order-4 pandiagonal magic square.

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"


1 17 24   23 3 16   18 22 2
15 19 8   7 14 21   20 9 13
26 6 10   12 25 5   4 11 27

Magic sum is 42 on all the 9 rows, 9 columns, 9 pillars and 4 triagonals.
All of these lines are shown with dowels.


A Wooden 3-D Magic Star

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 their sides.
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 are used.

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 lighting.

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- 2

Colors didn't reproduce too good in this image, but you get the idea.




A 3 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 pounds.

25 11 2 8 19
3 9 20 21 12
16 22 13 4 10
14 5 6 17


7 18 24 15 1

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 visible.

Here I present the same magic square, but in the Dudeney standard orientation.

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 times 6.

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

Magic constants:

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                   Top 1               Bottom + 1            Bottom
  1  355  217  187    367   37  151  205    109  247  325   79    283  121   67  289
   31  157  199     19  337  235  169    265  139   49  307    103  253  319   85
  301  271  133    313   91   97  259    163  193  379   25    229  175   13  343
  73  115  241     61  295  277  127    223  181    7  349    145  211  361   43

A 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 this page.

Order-3, 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.


This page was originally posted December 1999
It was last updated July 02, 2012
Harvey Heinz
Copyright 1998-2009 by Harvey D. Heinz