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

The results of historical and computer enumeration of magic squares

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

All bimagic squares of order 8   (NEW 2014-05-10)

There are exactly 192·136,244 = 26,158,848 bimagic 8x8-squares.
We found them all!     Co-author: Francis Gaspalou
(Important update 2014-08: Dwane Campbell could solve the hot spot problem.)

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   (Important update 2013: formulae proved and new algorithm found by Dirk Kinnaes)

Read all about these building blocks of magic objects.
Don't be afraid of exact numbers with more than 100 digits.

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.

perfect cube shown on the cover of the book 'Mathematics for elementary teachers' How many magic lines can a magic cube of small order possess?

Perfect magic cubes

The successful search for the smallest perfect magic cube in 2003.

Monte Carlo Casino Strategy and source code of a special approximation

Number of normal 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, © 2003-05-05 (last modified: 2014-08-19)