Order-6 magic square

Why the sum is 111

The thirty-six numbers from 1 to 36 total 36 × 37 / 2 = 666. Six rows of equal sum share that total, so each row is 666 / 6 = 111. In general the magic constant of an order-n square is M = n(n²+1)/2: 15 at order 3, 34 at order 4, 65 at order 5, 111 at order 6, 505 at order 10.

A complete example

Here is a square produced by the method, with each cell's class in small type. The six rows, the six columns and both diagonals are 111.

An order-6 magic square of the protocol, with its labelling

36R	2B	3B	34J	5B	31R
7B	29R	9B	10B	26R	30J
13B	14B	22R	21R	23J	18B
24V	20B	16R	15R	17J	19V
25B	11R	28V	27V	8R	12J
6R	35V	33B	4J	32V	1R

It is not associative: two cells opposite through the centre do not always sum to 37. No singly even magic square is, and that is an old theorem, not a weakness of the method.

The pointwise construction

Each cell receives one of four classes, and its value follows from its coordinates (r for the row, c for the column, numbered 0 to 5):

B = 6r + c + 1  ·  V = 6r + (5−c) + 1  ·  R = 37 − B  ·  J = 37 − V

B reads the row left to right, V reads it backwards, and R and J are their complements to 37. Cell (2, 3) of the table above carries class R: B would be 6×2 + 3 + 1 = 16, so R = 37 − 16 = 21, which is the value shown. No other cell was needed.

Three conditions bear on the labelling. The first, on positions, is necessary and sufficient for all thirty-six numbers to appear once. The other two are equalities of class counts per row and per column, and it is the second one that selects, at order 6, exactly the interesting labellings.

The two ansate-cross figures

The off-diagonal cells pair up two by two: horizontally when the two classes are partners under B ↔ J or R ↔ V, vertically otherwise. The traits so drawn form a figure, and at order 6 there are only two — proved, and re-checked by complete enumeration.

The two are exactly complementary: where one carries a horizontal trait, the other carries a vertical one. This is not a mirror symmetry but an antinomy — the only two ways to hold the constraints, and they exclude each other cell by cell. The grey dots mark the twelve diagonal cells, which carry no trait.

Every row carries a single horizontal trait and every column a single vertical one: odd numbers, as the criteria require. That odd parity is precisely what forbids the figure at orders 8, 12, 16 — and what allows it at 6, 10, 14, 18. See the page on even orders in general.

How many there are

The counts are exact and obtained by complete enumeration, in Python with no dependency.

At order 6, by exhaustive enumeration

magic labellings of the protocol	18,432
of which those meeting the per-column count equality	8,192
of which those that tile by parity alternation	8,192
of which those carrying an ansate cross	8,192
distinct ansate-cross figures	2

The three sets of 8,192 coincide, checked element by element and not merely in cardinality: at order 6 the column condition is exactly the tiling criterion, and the ansate cross is its drawn form. At higher orders that equivalence is not established. And 8,192 labellings carry only 2 figures: each figure is realized by 4,096 labellings, an exactly equal split.

The other 10,240 magic labellings are magic without carrying a cross: they violate the per-column count equality, which is therefore not a condition of magicity.

What the classical methods do

Order 6 is the smallest singly even order — even but not divisible by 4 — and it is the case held to be difficult. The known methods are procedures:

LUX (Conway)

Build an odd square of order 3, then replace each cell by a 2 × 2 block of type L, U or X according to its position. It works, but it has to be executed: nothing gives a cell's value from its coordinates.

Strachey

Split the square into four order-3 quadrants, fill them by the odd-order method, then swap certain columns between quadrants. Same remark: a sequence of operations, with special cases on the central rows.

The pattern method

With two classes only it is well known and well formalized, and it does give a genuine pointwise formula — but it is documented only for doubly even orders, 4, 8, 12. The move to four classes is what unlocks order 6.

The practical difference: here, to know cell (4, 2), you read its class and apply the formula. The other thirty-five are not needed.

Code and data

Everything is checkable with nothing installed, in plain Python:

python tools/compte_etiquetages.py --ordre 6   # the 18,432
python tools/enum6.py                          # the 8,192 that tile
python tools/croix_auto.py --ordre 6           # the triple equivalence
python tools/construction.py --toutes          # the 2 figures, constructed

The scripts, the data and the confidence map — every statement classified by its status — are in the code repository. The corpus of 256 order-6 labellings all of this came from is identified on plate 047 of the treatise, doi:10.5281/zenodo.22722485.