Research archive
Project timeline
This page reconstructs the history of the research, from its first ideas and constructions to the proofs and the present form of the problem.
Any direct contribution of artificial intelligence to the substantive research is identified in the corresponding event, together with the model used and the division of work.
First encounter with the problem
Alexey Khalin introduces Vladimir Mynka to the problem of a 3×3 magic square of perfect squares. It does not yet lead to an independent investigation.
Independent research begins
A new wave of popular mathematics material brings Vladimir back to the problem, and independent work begins that month. He is then a first-year software engineering student and has run a mathematics community of about 5,000 subscribers for three years. The work starts without prior familiarity with the specialist literature.
A co-author joins
A co-author who wishes to remain anonymous joins the investigation. The first discussion already produces an initial theoretical basis, including the fact that the magic constant of a 3×3 square is three times its central entry. The result is later found to be well known.
Symmetries and the minimal square
A system of inequalities between the entries is formed, equivalence under rotations and reflections is separated out, vector-space properties are identified intuitively, and the notion of a minimal integral square is introduced.
Residues and Gaussian integers
The entries are studied through residues and quadratic residues. A combinatorial treatment of sums of two squares in the Gaussian integers, proposed by the co-author, recovers nearly all known arithmetic restrictions for a minimal 9/9 square: divisibility by 2, 3, and 5, and restrictions on prime divisors of the forms 4k+1 and 4k+3 in central, corner, and side entries. These conditions are later found to have already been published on Multimagie.
First article in the series
The mathemynka community publishes an introduction to the investigation and its dimensional model. The project writes the model m(E,x,y) explicitly for the first time, together with a first conclusion about the vector-space or module structure of magic squares.
Residues, the dir basis, and the first 5/9 cross
The second article applies residue theory to magic squares, introduces the basis that later develops into dir theory, and derives the first parametrization of the cross-shaped 5/9 case.
Combinatorial divisibility calculations
The third article formalizes combinatorial divisibility calculations for entries of a magic square for the first time within the series. This early version contains minor errors that are corrected as the theory develops.
Quadratic residues and the “knight's move”
The final article in the March series describes quadratic residues and the logic of the “knight's move”. A month later, this direction leads to a separate note on charming squares.
Magic, charming, and semimagic squares
After the first article, a reader observes that the product of two magic squares need not be magic, while a ternary product preserves the property. By summer, the observation develops into a theory of magic and charming squares and an algebra of semimagic squares.
First calculator and algorithms
Available literature is studied in parallel and compared with the independently obtained results. Vladimir develops the first calculator website, while the co-author develops the first research algorithms.
Article on norm theory
A new article gives a systematic account of norm theory and the Brahmagupta–Fibonacci identity as applied to sums of two squares. The theory was already classical, but now became part of the project's own exposition.