|Notes on Magic Squares and Cubes|
|by Walter Trump|
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 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.)
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.
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.
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.
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.
Strategy and source code of a special approximation|
Number of normal magic order-6 squares
Learn about the Monte Carlo backtracking algorithm.
Links to great sites about magic squares and cubes |
Each of these sites covers the subject very completely and contains many further links.