Notes on Magic Squares and Cubes  
by Walter Trump

Red Square nebula There are more magic squares of order 10 than elementary particles in our universe and in trillions of other universes (if they exist).

Number of magic squares  

NEW 2024-05-31:  Hidetoshi Mino succeeded now in calculating the correct number of magic 6x6-squares.

The results of historical and computer enumeration of magic squares

perfect cube shown on the cover of the book 'Mathematics for elementary teachers' Magic Cubes as extensions of magic squares to 3 dimensions.

Perfect magic cubes

NEW 2024-06-20:  Holger Becker discovered that a perfect magic cube of order 7 was already published in 1833 by Ferdinand Julius Brede alias de Fibre.

Read about the fascinating features and the exciting history of these magic objects.

MS of subtraction "quadrata subtractionis" published by Adam Kochański (Poland) in 1686.

Magic Squares of Subtraction  

The old paper "Considerationes quaedam circa Quadrata et Cubos Magicos"
was rediscovered by Henryk Fukś (Canada) and translated from Latin to English.

Trimagic Squares Trimagic 12x12-squares are known since 2002-06-08.

Trimagic Squares of order 12  

Here some new squares with special properties are presented.
For example: a trimagic square for the 60th anniversary of LEGO.

area magic square Graphic interpretation of magic squares

Area Magic Squares  

Each area size is equal to the associated entry of the magic square.
Area magic squares were first suggested by William Walkington in December 2016.

bimagic square Bimagic squares are still magic when all entries are squared.

All bimagic squares of order 8  

There are exactly 192·136,244 = 26,158,848 bimagic 8x8-squares.

tromino square Ultramagic squares are both pandiagonal and centrally symmetrical.

Ultramagic squares of order 7

Properties, transformations and beautiful examples of these squares.
Read how to determine all 20,190,684 different squares.

Dürer's famous magic square of 1514 2 + 8 + 9 + 15   is a magic (trimagic) series of order 4.

Magic series

For orders m≥1000 the amount of series is very huge. Nevertheless exact enumerations could be done.
Read all about these building blocks of magic objects.

angel How many magic squares of higher orders are there?

Estimates of the number of magic squares (cubes, hypercubes)  

Magic series and probability considerations enable us to estimate the numbers of magic objects.

Monte Carlo Casino Strategy and source code of a special approximation

Number of classical magic order-6 squares

Learn about the Monte Carlo backtracking algorithm.

Harvey's cube made of wood Links to great sites about magic squares and cubes

Each of these sites covers the subject very completely and contains many further links.

Walter Trump, Nürnberg, Germany, ms(at), © 2003-05-05 (last modified: 2024-04-08)