Magic Squaresproof atlas

About

Research on 3×3 magic squares

The project studies the arithmetic and algebraic structure of 3×3 magic squares. Its central open question is whether a magic square can consist of nine positive, pairwise distinct perfect squares; the partial k/9 problems, congrua, and the algebra of magic matrices are treated as connected parts of the same investigation.

1. Problem statement

1.1. Magic square

A magic square of order n is an n×n array of numbers in which the entries in every row, every column, and the two diagonals joining opposite corners have the same sum M, called the magic constant. Distinctness of the entries is not part of this definition; it is imposed separately when needed.

∑j=1naij=M (1≤i≤n),∑i=1naij=M (1≤j≤n),∑i=1naii=∑i=1nai,n+1−i=M.\sum_{j=1}^{n}a_{ij}=M\ (1\le i\le n),\qquad \sum_{i=1}^{n}a_{ij}=M\ (1\le j\le n),\qquad \sum_{i=1}^{n}a_{ii}=\sum_{i=1}^{n}a_{i,n+1-i}=M.

A magic square of order 3 has nine entries. Explicitly, the condition is the equality of eight sums: three rows, three columns, and two diagonals.

(abcdefghj),a+b+c=d+e+f=g+h+j=M,a+d+g=b+e+h=c+f+j=M,a+e+j=c+e+g=M.\begin{gathered} \begin{pmatrix}a&b&c\\d&e&f\\g&h&j\end{pmatrix},\\[3pt] a+b+c=d+e+f=g+h+j=M,\\ a+d+g=b+e+h=c+f+j=M,\\ a+e+j=c+e+g=M. \end{gathered}

1.2. A magic square of squares and the strong 9/9 problem

In this project, a magic square of squares means an integral magic square of order 3 whose nine entries are positive, pairwise distinct perfect squares. Thus every entry has the form q² for a positive integer q, and no two entries are equal.

aij=qij 2,qij∈Z>0,(i,j)≠(k,ℓ) ⟹ qij 2≠qkℓ 2.a_{ij}=q_{ij}^{\,2},\qquad q_{ij}\in\mathbb Z_{>0},\qquad (i,j)\ne(k,\ell)\ \Longrightarrow\ q_{ij}^{\,2}\ne q_{k\ell}^{\,2}.

The strong 9/9 problem is to construct such a square or prove that none exists. It remains open.

1.3. The weak 7/9 problem

In this problem, a k/9 square is an integral magic square of order 3 with pairwise distinct entries and at least k positive perfect-square entries. The remaining entries are not required either to be squares or to be nonsquares. Consequently, an 8/9 or 9/9 solution would automatically solve the 7/9 problem.

One class of 7/9 squares is known; whether any others exist is unknown.

A representative of the known class is the Bremner–Sallows square:

B=(5652289237322324252137⋅2633151⋅147152722052).\mathcal B=\begin{pmatrix} 565^2&289^2&373^2\\ 23^2&425^2&137\cdot2633\\ 151\cdot1471&527^2&205^2 \end{pmatrix}.

Let Rᵢ denote the row sums, Cᵢ the column sums, and D₁ and D₂ the two diagonal sums. The exact magic-square certificate is then the equality below. The seven displayed roots are pairwise distinct, while the other two entries are not squares:

R1=R2=R3=C1=C2=C3=D1=D2=541875,R_1=R_2=R_3=C_1=C_2=C_3=D_1=D_2=541875,6002<137⋅2633=360721<6012,4712<151⋅1471=222121<4722.600^2<137\cdot2633=360721<601^2,\qquad 471^2<151\cdot1471=222121<472^2.

Rotations, reflections, and multiplication of every entry by the same positive perfect square produce only trivially equivalent variants and do not count as a new solution.

There are exact logical implications between the two formulations. Any 9/9 square would be a new 7/9 solution. A proof that the known 7/9 class is unique would also rule out 8/9 and 9/9 squares, whereas a proof of nonexistence for 9/9 alone would not settle uniqueness for 7/9.

2. Scope of the research

The 9/9 and 7/9 problems provide the main objective, but they do not exhaust the project. We study which sets of entries can simultaneously be perfect squares, construct parametric families of such squares, investigate divisibility restrictions and representations by sums of squares, and study ordinary matrix multiplication of magic and semimagic squares.

The notation k/9 means that the selected k entries are guaranteed to be squares; some of the remaining entries may also happen to be squares. Two placements of selected entries are regarded as the same when one is obtained from the other by rotating or reflecting the whole square.

3. Main results

The principal results of the project are listed below. Each result is accompanied by the precise scope of the proved statement, its limitations, and links to the detailed derivation.

3.1. Complete coverage of the 5/9 positional types

For each of the 23 essentially different placements of five square entries, we construct an explicit nondegenerate parametric family of integral magic squares. Thus no 5/9 positional type remains without a construction. Here nondegenerate means that admissible parameter values exist for which all nine entries are positive and pairwise distinct.

The subset of rational solutions covered by each displayed formula is stated separately. For the ABCDG pattern, one polynomial formula likewise covers only a subfamily, but a separate conic-bundle algorithm enumerates every rational solution and has an explicit inverse. Completeness of a formula and completeness of an effective algorithm must therefore be distinguished.

3.2. Progress on 6/9

All 16 positional 6/9 types have exact systems of equations, dedicated articles, infinite rational families, and positive specializations of exact type 6/9. For the two parallel types, a complete tfmn classification of all nondegenerate rational solutions is proved. The fourteen nonparallel types are reduced through chosen genus-one fibrations to elliptic K3 models; explicit non-torsion sections give infinite families. For most of these K3 surfaces, the exact geometric rank and global completeness of the chosen rational chart remain undetermined.

Bremner represented all sixteen 6/9 configurations as intersections of three quadrics in ℙ⁵ and identified on each associated surface an elliptic fibration that yields an infinite one-parameter family. For one configuration he proved that the surface is a smooth K3, found the singular fibers with passport 4I₄+4I₂, and computed rank 2 over the function field and rank 20 for the Néron–Severi lattice. The present atlas records explicit models and proved characteristics for all sixteen patterns; for the two parallel patterns it additionally gives a complete tfmn classification of nondegenerate rational solutions.

3.3. Magic, charming, and semimagic squares

The product of two 3×3 magic squares is generally no longer magic, but it remains semimagic: all row and column sums are equal. Such products form a distinguished three-dimensional class that we call charming squares. Magic and charming squares alternate under multiplication: magic times magic is charming, magic times charming is magic, and charming times charming is charming.

After the common scalar component is separated, these two classes give the even and odd parts of the five-dimensional associative algebra of semimagic matrices. Explicit formulas for the determinant, adjugate, and characteristic polynomial have also been obtained for this algebra.

Moreover, over the rational numbers every 3×3 semimagic square has an explicit decomposition as the sum of a magic square and a product of two magic squares:

S(E,x,y,z,w)=M(E,x,y)+M(0,1,0)M(0,z,−w).S(E,x,y,z,w)=M(E,x,y)+M(0,1,0)M(0,z,-w).

The second term is the charming square C(0,z,w). Thus magic and charming squares do more than span the five-dimensional space by linear combinations: the decomposition of every element is explicit.

3.4. Congrua and the F4+ surface

Three squares in arithmetic progression determine a right triangle whose area equals the common difference. This connects part of the magic-square problem with the classical congruent number problem. If F(a,b)=ab(a²−b²), then values of F that differ by a rational square represent the same class in this problem.

F(a,b)=ρ2F(a,d)F(a,b)=\rho^2F(a,d)

The project introduces the complete F4+ model for pairs of Pythagorean areas differing by a rational square. Its auxiliary rational elliptic surface has exact rank 2, while the two Pythagorean parametrizations define two independent points on the universal congruent-number curve over the function field of F4+. Consequently, that curve has rank at least 2. The construction gives a unified way to obtain identities between congrua and related families of magic squares, including some 6/9 families.