Multimagic Squares

@article{Derksen2007MultimagicS,
  title={Multimagic Squares},
  author={Harm Derksen and Christian Eggermont and Arno van den Essen},
  journal={The American Mathematical Monthly},
  year={2007},
  volume={114},
  pages={703 - 713}
}
In this paper we give the first method for constructing n-multimagic squares (and hypercubes) for any n. We give an explicit formula in the case of squares and an effective existence proof in the higher dimensional case. Finally we prove that n-multimagic squares do not exist for certain orders. 

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References

SHOWING 1-10 OF 25 REFERENCES
Unsolved problems on magic squares
  • G. Abe
  • Computer Science, Mathematics
    Discret. Math.
  • 1994
TLDR
This paper collects 23 unsolved problems or conjectures on magic squares, and some updated results related to these problems are mentioned.
Most-perfect Pandiagonal Magic Squares: Their Construction and Enumeration
Their construction and enumeration by Kathleen Ollerenshaw and David Brée. This book gives a method of construction and enumeration of all pandiagonal magic squares of a class known as
The Lost Squares of Dr. Franklin: Ben Franklin's Missing Squares and the Secret of the Magic Circle
  • P. C. Pasles
  • Mathematics, Computer Science
    Am. Math. Mon.
  • 2001
TLDR
Franklin's squares still appear today in popular mathematics books such as Theoni Pappas's Joy of Mathematics and Jan Gullberg's encyclopedic Mathematics: From the Birth of Numbers.
Mathematical Recreations and Essays
THIS edition differs from the third by containing chapters on the history of the mathematical tripos at Cambridge, Mersenne's numbers, and cryptography and ciphers, besides descriptions of some
MASTER’S THESIS
Web mediated communications revolutionized traditional social interactions. It is designed to facilitate information exchange between individuals and to enable people to connect with friends, family,
A magic cube
How Many Squares Are There, Mr. Franklin?: Constructing and Enumerating Franklin Squares
  • M. Ahmed
  • Mathematics, Computer Science
    Am. Math. Mon.
  • 2004
Story of the smallest trimagic square (January 2003), available at http://www.multimagie. com/English/Tri12story.htm
  • 2003
The Zen of Magic Squares, Circles, and Stars (Book)
...
1
2
3
...