## Magic Square of Squares

Goal:

A magic square is a square of distinct (usually positive) integers such that the sum of each row, each column and each diagonal is the same. A magic square of squares is a magic square such that each of its entries is a square. It is an open problem to decide if a 3x3 magic square of squares exists. For the 4x4 case, Euler gave a solution, which in modern algebra terms stems from the multiplicativity of the norm of quaternions.

Many variations are possible. One can, for instance, ask for bimagic squares. Those are magic squares that are also magic when one squares their entries. Another variation are magic cubes.

Christian Boyer has created an interesting website http://www.multimagie.com/.

One goal is to study known constructions and examples, and to use them to make a nice collection of interesting magic squares and cubes and to illustrate them in an attractive way (this requires some effort especially for cubes).

Another goal is to create "congruence magic squares of squares" using modular arithmetic.

Supervisors: Gabor Wiese, N.N.

Difficulty level: Any.

Tools:

• Any programming language or any Computer Algebra System such as sagemath
• Image processing software

Literature:

• Christian Boyer: Some Notes on the Magic Squares of Squares Problem, The Mathematical Intelligencer, 2005
• O. M. Cain: Gaussian Integers, Rings, Finite Fields and the Magic Square of Squares, arXiv:1908.03236 