6/9 Magic-Square Patterns: The Sixteen Positional Types

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1. What 6/9 means

Let S be a set of six positions in an ordinary 3×3 magic square. A 6/9 configuration with pattern S is a magic square in which every entry indexed by S is a rational or integral square. The remaining three entries may also happen to be squares: the number 6 is a proved lower bound, not a requirement that exactly six entries be square.

We use the general coordinate form

\begin{pmatrix} A&B&C\\D&E&F\\G&H&J \end{pmatrix} = \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}.

For every selected position P, introduce a square root qₚ and require P=qₚ². This gives six equations that are linear in E,x,y and quadratic in the roots.

2. Why there are always three conditions

Elimination theorem

For every six-cell pattern S, the matrix of the selected linear forms has rank 3. Eliminating E,x,y therefore leaves exactly three independent homogeneous quadratic relations among the six roots. Over ℚ, these three relations are not only necessary but also sufficient to recover a unique triple E,x,y.

Proof

The row associated with an entry P corresponds to a point (αₚ,βₚ) in the 3×3 coefficient grid, since P=E+αₚx+βₚy. A line contains at most three distinct points of this grid. Hence six selected points cannot be collinear and affinely span the plane; the rows (1,αₚ,βₚ) have rank 3.

\dim\ker L_S^{\,T}=6-\operatorname{rank}L_S=3.

Let R₁,R₂,R₃ be a basis of the left kernel of Lₛ. The equation Lₛ(E,x,y)ᵀ=qₛ⁽²⁾ immediately implies Rᵢ(qₛ⁽²⁾)=0. Conversely, in a finite-dimensional vector space over ℚ, the orthogonal complement of the left kernel equals the image of Lₛ. Thus the three equations place qₛ⁽²⁾ in the image of Lₛ, while full rank makes E,x,y unique.

q_S^{[2]}\in\operatorname{im}L_S \quad\Longleftrightarrow\quad R_1(q_S^{[2]})=R_2(q_S^{[2]})=R_3(q_S^{[2]})=0.

3. Why there are exactly sixteen types

The complement of a six-cell pattern contains three entries. Taking complements commutes with every rotation and reflection of the square, so the 6/9 orbits correspond bijectively to the orbits of three-element subsets of the nine entries.

Apply Burnside's lemma. The identity fixes all C(9,3)=84 triples. A ±90° rotation has cycles of lengths 1,4,4 and fixes no triple. A 180° rotation fixes the center and has four pairs of opposite entries; an invariant triple must contain the center and one such pair, giving 4 possibilities. Every reflection has three fixed entries and three exchanged pairs. An invariant triple either contains all three fixed entries or one fixed entry together with one pair, giving 1+3·3=10 possibilities.

N_{6/9}=N_{3/9} =\frac{84+0+4+0+4\cdot10}{8} =16.

4. All positional types

The atlas chooses one canonical representative from each orbit. It records the original six-square system, three preferred independent relations, and the strongest result proved so far. Color denotes the type of quadric, while shades of one color distinguish separate conditions of the same type.

Applying the eight elements of D₄ to each three-cell complement never produces the complement in another row: the rows are distinguished by center membership, the numbers of corners and edge cells, and their adjacency or opposition. Hence the table contains sixteen distinct orbits, and the Burnside count proves that the list is exhaustive.

For concise descriptions of color signatures, we introduce geometric names. They refer to the selected triple of quadrics, not to the shape of the six-cell pattern:

red–red–redtriangularred–red–yellowrectangular

Among the preferred bases in the atlas, 3 types are triangular and 8 are rectangular. In particular, both parametrized tfmn classes, ABEFGJ and ABDFHJ, are rectangular.

Proof atlas

All 16 orbits and triples of quadrics for 6/9

Every card contains the original system and three preferred independent relations. Shades of one color distinguish separate conditions of the same mathematical type.

progression of squaresGaussian normx²+2y² normStatus shows the strongest proved result, not isolated examples.

ABCDEF

complement: GHJ

red(DEF)yellow(ACDE)blue(BCDE)

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E-x-y&=d^2\\E&=e^2\\E+x+y&=f^2\end{aligned}\right.

d^2+f^2=2e^2

a^2+d^2=c^2+e^2

b^2+2c^2=d^2+2e^2

One progression and two quadrics on the shared C,D,E block.

Canonical system

ABCDEG

complement: FHJ

red(CEG)yellow(ACDE)yellow(ABEG)

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E-x-y&=d^2\\E&=e^2\\E+y&=g^2\end{aligned}\right.

c^2+g^2=2e^2

a^2+d^2=c^2+e^2

a^2+b^2=e^2+g^2

The CEG progression and two independent Gaussian norms.

Canonical system

ABCDEH →

complement: FGJ

red(CDH)red(BEH)yellow(ACEH)

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E-x-y&=d^2\\E&=e^2\\E+x-y&=h^2\end{aligned}\right.

d^2+h^2=2c^2

b^2+h^2=2e^2

a^2+c^2=e^2+h^2

An intersecting red-red-yellow type with shared entry H.

RectangularK3: 2I₄+8I₂; 2≤rank≤4

ABCDEJ

complement: FGH

red(BDJ)red(AEJ)yellow(ACDE)

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E-x-y&=d^2\\E&=e^2\\E-x&=j^2\end{aligned}\right.

b^2+d^2=2j^2

a^2+j^2=2e^2

a^2+d^2=c^2+e^2

Two intersecting progressions and a yellow compatibility relation.

RectangularK3: 2I₄+8I₂; 1≤rank≤4

ABCDFG

complement: EHJ

red(BFG)yellow(BCDG)blue(ACFG)

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E-x-y&=d^2\\E+x+y&=f^2\\E+y&=g^2\end{aligned}\right.

b^2+f^2=2g^2

b^2+c^2=d^2+g^2

2a^2+g^2=c^2+2f^2

A progression, a Gaussian norm, and an x²+2y² norm without the center.

Canonical system

ABCDFH

complement: EGJ

red(AFH)red(CDH)yellow(BDFH)

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E-x-y&=d^2\\E+x+y&=f^2\\E+x-y&=h^2\end{aligned}\right.

f^2+h^2=2a^2

d^2+h^2=2c^2

b^2+h^2=d^2+f^2

An intersecting red-red-yellow type with shared entry H.

RectangularK3: 2I₄+8I₂; 1≤rank≤4

ABCDGJ

complement: EFH

red(BDJ)yellow(ACGJ)yellow(BCDG)

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E-x-y&=d^2\\E+y&=g^2\\E-x&=j^2\end{aligned}\right.

b^2+d^2=2j^2

a^2+j^2=c^2+g^2

b^2+c^2=d^2+g^2

One progression and two independent Gaussian norms.

Canonical system

ABCDHJ

complement: EFG

red(BDJ)red(CDH)yellow(ABHJ)

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E-x-y&=d^2\\E+x-y&=h^2\\E-x&=j^2\end{aligned}\right.

b^2+d^2=2j^2

d^2+h^2=2c^2

a^2+j^2=b^2+h^2

An intersecting pair of progressions with shared entry D.

RectangularK3: 2I₄+8I₂; 1≤rank≤4

ABCEGH

complement: DFJ

red(CEG)red(BEH)yellow(ACEH)

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E&=e^2\\E+y&=g^2\\E+x-y&=h^2\end{aligned}\right.

c^2+g^2=2e^2

b^2+h^2=2e^2

a^2+c^2=e^2+h^2

The same K3 quartic as ABCEGJ, with a different cell interpretation.

RectangularK3: 2I₄+8I₂; 2≤rank≤4

ABCEGJ

complement: DFH

red(CEG)red(AEJ)yellow(BEGJ)

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E&=e^2\\E+y&=g^2\\E-x&=j^2\end{aligned}\right.

c^2+g^2=2e^2

a^2+j^2=2e^2

b^2+e^2=g^2+j^2

The same K3 quartic as ABCEGH, with a different cell interpretation.

RectangularK3: 2I₄+8I₂; 2≤rank≤4

ABCGHJ

complement: DEF

yellow(ACGJ)yellow(ABHJ)blue(ACHJ)

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E+y&=g^2\\E+x-y&=h^2\\E-x&=j^2\end{aligned}\right.

a^2+j^2=c^2+g^2

a^2+j^2=b^2+h^2

a^2+2c^2=2h^2+j^2

The unique pattern without a red progression: two Gaussian and one blue norm.

Canonical system

ABDEFH

complement: CGJ

red(AFH)red(DEF)red(BEH)

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-x-y&=d^2\\E&=e^2\\E+x+y&=f^2\\E+x-y&=h^2\end{aligned}\right.

f^2+h^2=2a^2

d^2+f^2=2e^2

b^2+h^2=2e^2

Three red conditions with two shared centers.

TriangularK3: 4I₄+4I₂; 1≤rank≤2

ABDEFJ

complement: CGH

red(BDJ)red(DEF)red(AEJ)

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-x-y&=d^2\\E&=e^2\\E+x+y&=f^2\\E-x&=j^2\end{aligned}\right.

b^2+d^2=2j^2

d^2+f^2=2e^2

a^2+j^2=2e^2

A different topology of three red conditions.

TriangularK3: 4I₄+4I₂; 1≤rank≤2

ABDFHJ →

complement: CEG

red(AFH)red(BDJ)yellow(BDFH)

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-x-y&=d^2\\E+x+y&=f^2\\E+x-y&=h^2\\E-x&=j^2\end{aligned}\right.

f^2+h^2=2a^2

b^2+d^2=2j^2

b^2+h^2=d^2+f^2

A parallel red-red-yellow type without the central entry.

Rectangulartfmn parametrization

ABEFGH

complement: CDJ

red(AFH)red(BFG)red(BEH)

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E&=e^2\\E+x+y&=f^2\\E+y&=g^2\\E+x-y&=h^2\end{aligned}\right.

f^2+h^2=2a^2

b^2+f^2=2g^2

b^2+h^2=2e^2

A triangle of three pairwise square means.

TriangularK3: 4I₄+4I₂; 1≤rank≤2

ABEFGJ →

complement: CDH

red(BFG)red(AEJ)yellow(ABFJ)

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E&=e^2\\E+x+y&=f^2\\E+y&=g^2\\E-x&=j^2\end{aligned}\right.

b^2+f^2=2g^2

a^2+j^2=2e^2

a^2+b^2=f^2+j^2

A parallel red-red-yellow type containing the center.

Rectangulartfmn parametrization

5. The eight progressions in a magic square

The general form contains exactly eight triples of positions whose values identically form an arithmetic progression. This is a statement about entry values and does not require the positions to lie on one geometric line.

\begin{array}{llll} A+J=2E, & F+H=2A, & B+D=2J, & B+H=2E,\\ B+F=2G, & D+H=2C, & C+G=2E, & D+F=2E. \end{array}

These identities correspond to AEJ, AFH, BDJ, BEH, BFG, CDH, CEG, and DEF. If the three entries are squares, they form an arithmetic progression of squares. In particular, AFH and BDJ are the less obvious progressions whose consecutive positions are separated by a knight's move.

The list is complete by the coefficient-grid model: an arithmetic progression corresponds to a line through its three points, and the 3×3 grid has exactly three horizontal lines, three vertical lines, and two diagonals containing three points.

Each such progression supplies one of the pattern's three quadratic conditions. The mere presence of two or three progressions does not yet make tfmn applicable: two progressions must be disjoint and have the same direction and difference in the general square.

6. The two parallel classes

There are three disjoint progressions in the x direction, AEJ, BFG, and CDH, and three in the y direction, AFH, BDJ, and CEG. Choosing two progressions in one direction gives six patterns:

\begin{aligned} x:\;&ABEFGJ,\ ACDEHJ,\ BCDFGH,\\ y:\;&ABDFHJ,\ ACEFGH,\ BCDEGJ. \end{aligned}

Four of them contain the center E and are equivalent to ABEFGJ. Two omit the center and are equivalent to ABDFHJ. Rotations and reflections preserve the center, so these two orbits are distinct; the symmetries within each group show that no further orbit occurs here.

ABCDEFGHJ

ABEFGJ: AEJ and BFG

ABCDEFGHJ

ABDFHJ: AFH and BDJ

The exact scope of tfmn

Among the sixteen positional types, exactly ABEFGJ and ABDFHJ are unions of two disjoint parallel progressions. In each, parametrizing the two progressions reduces the comparison of their differences to equality of their tf values. These are the two well-developed tfmn classes of 6/9.

7. What happens in the other fourteen types

The remaining patterns may contain intersecting progressions, a single progression, or no progression at all. In every case the full problem still consists of three independent quadrics, but the natural coordinates and the way those quadrics are solved change. Four- and five-cell subpatterns provide the familiar progression and norm relations, yet all three conditions must be made compatible at once.

Consequently, examples of 6/9 squares in other positional types do not by themselves constitute a parametrization. Each type requires a generating rational surface or family, a proof that E,x,y are recovered correctly, and a separate determination of its coverage. This classification is the next layer of the 6/9 investigation.

Next: fmn and tfmn →Open the 6/9 constructor Open the standalone atlas Theory of 4/9 and 5/9 patterns