About the 3×3 Magic Square of Squares Project

1. Problem statement

Every magic square of order 3 is uniquely determined by three coordinates E, x, and y; its magic sum is 3E:

\mathcal M(E,x,y)=\begin{pmatrix} E+x & E-x+y & E-y\\ E-x-y & E & E+x+y\\ E+y & E+x-y & E-x \end{pmatrix}

The full problem asks for integers E, x, and y for which all nine entries are positive, pairwise distinct squares of integers. The existence of such a 9/9 square remains neither proved nor disproved.

E,x,y\in\mathbb Z,\qquad \mathcal M(E,x,y)_P=q_P^2>0\quad(P\in\Omega),\qquad q_P^2\ne q_Q^2\ (P\ne Q)

The project's current task is to describe solutions for a subset of cells S whose entries are guaranteed to be squares. Cells outside S are unrestricted and may also happen to be squares.

\mathcal M(E,x,y)_P=q_P^2\quad(P\in S),\qquad |S|\in\{4,5\}

2. Relation to congrua

The basic red case occurs when three square-valued entries form an arithmetic progression. Their common nonzero difference d is a congruum:

r^2,\ s^2,\ t^2,\qquad r^2+t^2=2s^2,\qquad d=s^2-r^2=t^2-s^2

This progression gives a right triangle with sides t−r, t+r, and 2s and area d. Thus the search for red triples is connected with the classical congruent number problem. The project uses this correspondence as one lemma, not as a solution of the full magic-square problem.

3. Object of classification

The group D4 acts on the nine cells by rotations and reflections. The project classifies four- and five-cell masks S modulo this action. There are exactly 23 orbits at each level, and taking complements gives a bijection between the 4/9 and 5/9 levels.

After roots qₚ are introduced, the original system is linear in E, x, and y. Eliminating these three coordinates gives one quadric when |S|=4 and two independent quadrics when |S|=5. They are necessary and sufficient over Q; integral coordinates are obtained by clearing the common denominator through homogeneous scaling.

q_S^{[2]}\in\operatorname{im}L_S\quad\Longleftrightarrow\quad R_i(q_S^{[2]})=0\quad(1\le i\le |S|-3)

4. Scientific novelty

The project's novelty does not lie in the general form of a magic square, the notion of a congruum, or the statement of the open 9/9 problem. It lies in treating all partial square masks at the 4/9 and 5/9 levels within one system:

a complete list of D4 orbits at both levels and correction of the project's earlier catalog from 22 to 23 orbits;
a uniform derivation of the necessary-and-sufficient quadratic systems directly from the coordinates E, x, and y;
explicit parametrizations for every orbit, each with a separate proof of its coverage domain and exceptional locus;
a canonical colored basis of quadrics that ties each mask color to a specific equation rather than decorative markup.

5. Contribution of the current version

All 23 orbits at the 4/9 level have a complete rational parametrization with an algorithm for obtaining integral coordinates.
For 15 of the 23 orbits at the 5/9 level, the employed parametrization is proved complete. For each of the other eight, the widest subset whose coverage is proved is stated explicitly, together with the exceptional locus.
Every family formula is accompanied by a derivation from its quadrics, reconstruction of E, x, and y, and an explicit coverage status; identity verification is not presented as a coverage proof.
Exact polynomial certificates are kept in the proof core and distinguished from formulas whose proofs have not yet been formalized.

6. Scope and limitations

The 4/9 and 5/9 classification does not prove the existence of a 9/9 square.
The notation k/9 guarantees that the selected k entries are squares but does not assert that the remaining entries are nonsquares.
Completeness of the mask list, sufficiency of the quadrics, and completeness of a particular parametrization are distinct claims and are proved separately.
No claim of historical priority over the entire literature is made before a dedicated bibliographic review is completed.