Magic Squares of Odd Order

Magic Squares of odd order

Magic Squares of odd order by Marios Mamzeris

This page presents the Mamzeris Method: a systematic approach to constructing associative magic squares of any odd order using simple two-pass transformations or direct mathematical formulas.

The Origin Story

I developed this method in 1988 while studying computer science. When my university's mathematics faculty presented what they called an 'unsolved problem' in magic square construction, they challenged me to apply my programming background to find a systematic solution. What emerged was a universal algorithmic method that works for any odd-order magic square. I used this method personally for over three decades before publishing it in 2020, along with a new universal closed-form formula that enables direct calculation of any cell position.

A Magic Square is an n × n square grid (where n is the number of cells on each side) filled with distinct positive integers in the range 1,2,...,n² such that each cell contains a different integer and the sum of integers in each row, column, and diagonal is equal.

An associative magic square has an elegant property: pairs of numbers positioned symmetrically opposite the centre always sum to n² + 1. For example, in a 7×7 square (n²=49), opposite pairs sum to 50. These are also called symmetric magic squares, and all associative magic squares are self-complementary. (Wikipedia)

The Mamzeris Method offers two complementary approaches:

1. Algorithmic Construction (detailed below): A two-pass table transformation using row and column shifts. Best for learning, teaching, or manual construction.

2. Universal Closed-Form Formula: A direct mathematical formula that computes the value of any cell position (x, y) for any odd order N. This formula was first published in 2025 and enables instant calculation without iterative steps. Best for programming, large squares, or individual cell calculations. View the complete mathematical formula

Algorithmic Construction Method:

To create any odd-order associative magic square quickly and efficiently using the algorithmic approach, we need to perform only two simple passes, which I call table transformations (shifts). The three tables below demonstrate the construction of a 31×31 magic square.

Pass 1: Begin with a table filled sequentially with all numbers from 1 to n² (1^st table below, with cells shifting shown in green). Perform a left shift of all rows, skipping the middle one: shift the first row by one cell, the second row by two cells, and so on, until the last (n^th) row is shifted by n−1 cells.

Pass 2: Apply a similar transformation to columns instead of rows. Shift cells upward, starting with one cell in the leftmost column, two cells in the next column, and so on, until the last (n^th) column is shifted by n−1 cells, skipping the middle column. This is shown in the 2^nd table below with cells shifting in blue.

That's it. A magic square is ready. The completed square is shown in the 3^rd table below, with alternating grey cells and highlighted yellow diagonals.

1. Initial Table: Begin with numbers from 1 to n² arranged sequentially
The green arrow indicates the direction of the upcoming cell shift. Green-highlighted cells show which cells will move.

 

2. After First Pass: Rows shifted, columns ready for transformation
The blue arrow indicates the direction of the second shift. Blue-highlighted cells show which cells will move.

 

3. Completed Magic Square: Final result after both transformations
All rows, columns, and diagonals now sum to the magic constant. Grey cells alternate for visibility, and yellow highlights show the main diagonals.

 

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Alternative Approach: Bidirectional Shifting

Instead of shifting all rows in one direction and all columns in another, you can shift each half of the table in opposite directions (always skipping the middle row or column). This produces the same result using a mirror-symmetry pattern.

Initial square with numbers 1 - n² in sequence

After the first pass: Notice how each column is equal to Σ

 

Second pass completed: Now each row is also equal to Σ. The Magic Square is ready!

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Universal Closed-Form Formula

In addition to the algorithmic construction method described above, the Mamzeris Method includes a universal closed-form formula that directly computes the value at any cell position without performing iterative shifts. This formula is set out in my paper, Universal Closed-Form Construction for Odd-Order Magic Squares (The Mamzeris Method).

Key advantages of the closed-form formula:

• Direct computation: Calculate any individual cell value without constructing the entire square
• Computational efficiency: Ideal for programming, large squares, or sparse calculations
• Mathematical completeness: Provides explicit mathematical expression for the entire construction
• Validation tool: Verify algorithmic constructions or explore specific patterns

The formula

Given an odd order N ≥ 3 and a position (x, y) with 1 ≤ x, y ≤ N, where x is the column and y is the row, both counted from one:

Note the order. The column shift is undone first, which produces Y, and the row shift is then undone using s(Y), not s(y). The construction shifts the rows first and the columns second, so to find what ends up at a given position the two operations must be undone in the reverse order. This is the single point at which readers most often go wrong.

A full derivation, eight worked examples covering every branch of the shift function, the completed 5 × 5, 7 × 7 and 11 × 11 squares, and the zero-based form of the same result are given in the published paper.

Worked example. The value at column 3, row 4 of a 5 × 5 square:

Reading row 4, column 3 of the completed 5 × 5 square shown further down this page confirms the value 16. Remember that x is the column and y is the row; reading the pair the other way round will give the wrong cell.

The formula has been checked cell by cell against the algorithmic construction for every odd order from 3 to 121 inclusive. Every entry agrees, and every resulting square is a genuine associative magic square. The two approaches are therefore complementary rather than alternative: the algorithm is the better way to understand the construction, and the formula is the better way to compute with it.

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Understanding How the Method Works

This section explains the mathematical insight behind why the two-pass transformation creates a magic square of odd order. Understanding the balance of row and column sums reveals the elegance of the method.

Key Variables:

n = the square's dimension (for a 7×7 square, n = 7)
M = (n² + 1)/2 = the middle number (located at the center)
Σ = M×n = the magic constant (target sum for each row, column, and diagonal)
r = distance from the middle row or column

The Balancing Principle:

When we fill the initial square (1) with numbers 1 to n² in sequence, each row and column has a different sum based on its distance from the center. This creates predictable "weight" imbalances:

For columns:
• Moving left from center: each column's sum = Σ - r×n
• Moving right from center: each column's sum = Σ + r×n

For rows:
• Moving up from center: each row's sum = Σ - (r×n²)
• Moving down from center: each row's sum = Σ + (r×n²)

Pass 1 - Balancing the Rows:

The first pass shifts cells left by increasing amounts (row 1 shifts by 1, row 2 by 2, etc.), skipping the middle row. This creates square (2) where all row sums equal Σ.

The key insight: cells shifted off the left edge wrap to the right, adding exactly the right amount to compensate for each row's initial imbalance. The top row gains +1, the next gains +2, and so on, perfectly counterbalancing the initial deficits.

Pass 2 - Balancing the Columns:

The second pass applies the same logic to columns, shifting cells upward by increasing amounts. This transforms square (2) into square (3), where all column sums also equal Σ, creating a perfect magic square.

Visual Guide to the Tables Below:

Left tables: Show each cell's value relative to n (the algebraic view)
Right tables: Show actual numeric values as the square transforms
Green cells: Display row and column sums (diagonals always equal Σ)
Blue cells: Track the middle row and column positions through both passes
Square (3): The completed magic square with all properties satisfied

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Variations

The Mamzeris Method is remarkably flexible. From the same starting table it produces four different magic squares of the same order. Two things may be varied, and each has two settings:

• The shift directions: rows left with columns up, or rows right with columns down
• The order of the two passes: rows shifted first, or columns shifted first

Two direction settings multiplied by two pass orders give four distinct squares. All four are shown below, first for order 5 and then for order 19.

A warning about the shift directions. Only two of the four possible direction pairs give a magic square: rows left with columns up, and rows right with columns down. The mixed pairs, rows left with columns down and rows right with columns up, do not. They give correct column sums and correct diagonal sums, and the result is still associative, so a partial check appears to pass. The row sums are wrong. Anyone verifying the method must check the rows as well as the columns.

The order of the two passes. Shifting the rows first and then the columns gives one square. Shifting the columns first and then the rows gives a genuinely different square. This is the second source of variation, and combined with the two working direction pairs it accounts for all four squares.

The two pass orders are not related to one another by any rotation or reflection. Verified by enumeration at orders 5, 7 and 9, the square obtained by shifting the columns first never appears among the eight rotations and reflections of the square obtained by shifting the rows first.

The same four variations at order 19, with every number shown. Each cell is shaded on a single scale running from red at the lowest values through white at the middle to blue at the highest, so the pattern each variation creates can be seen at a glance. Every row, column and main diagonal in all four squares totals 3,439, and every pair of cells placed symmetrically about the centre totals 362.

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Permutations

Beyond the four variations, the method produces many more distinct squares, because the initial table itself may be rearranged before the two shifts are applied. Its rows may be permuted, and so may its columns.

Not every rearrangement is allowed. The finished square must be associative, which means that any two cells placed symmetrically about the centre must add up to n² + 1. That holds only if the permutation preserves the central pairing, so that row i and row n + 1 − i stay together as a pair and the middle row stays in the middle. The same condition applies to the columns. A permutation that meets it is called admissible. Every admissible pair of permutations, one for the rows and one for the columns, yields a valid associative magic square, and no two pairs yield the same square.

How many admissible permutations are there

Write m = (n − 1) / 2 for the number of pairs on either side of the centre. Each pair may be placed in any of m positions and either way round, which gives 2^m × m! admissible permutations along a single axis. Applying this independently to the rows and to the columns squares that figure:

$${({\mathbf{2}}^{\mathit{m}}\times \mathit{m}!)}^{\mathbf{2}}\text{ where }\mathit{m}=\frac{\mathit{n}-\mathbf{1}}{\mathbf{2}}$$

For a 7×7 square that is 48 admissible permutations of the rows and 48 of the columns, so 2,304 different initial tables, each of which produces a different magic square.

The unit P

The published papers carry a single symbol, P, so that the various totals can be written compactly. It is defined as:

$$\mathit{P}={\mathbf{2}}^{\mathit{n}-\mathbf{3}}\times {(\frac{\mathit{n}-\mathbf{1}}{\mathbf{2}}!)}^{\mathbf{2}}$$

P is a unit of measurement rather than a count of anything on its own. Its only purpose is to keep the totals short. The quantity that genuinely counts something is the number of admissible initial tables, and that is exactly four times P, since (2^m × m!)² = 4P for every odd n. At order 7, P = 576 and the 2,304 initial tables counted above are 4P.

The rising ladder

Every square in the three steps below is a different square. Nothing is merged, and nothing is treated as equivalent to anything else. If the squares were printed on paper and laid side by side, no two would look the same.

4P  —  one setting. Fix the pass order, say the rows are shifted first, and fix the direction, say rows to the left and columns upwards. Running every admissible initial table through that one setting gives 4P squares. At order 7 that is 2,304.

8P  —  both working directions. Keep the same pass order but allow both of the direction pairs that work, rows left with columns up, and rows right with columns down. The two sets have no square in common, so the count doubles. At order 7 that is 4,608.

16P  —  both pass orders as well. Now allow the columns to be shifted before the rows as well as after. Again the new set shares no square with the old one, so the count doubles a second time. At order 7 that is 9,216.

In one line, the factor of sixteen is:

4P admissible initial tables  ×  2 working shift directions  ×  2 pass orders  =  16P

So 16P is the complete number of distinct magic squares the Mamzeris Method can construct at any odd order from 5 upwards. The four families are entirely separate: none of the six possible overlaps between them contains a single square. Every figure in the table below has been confirmed by exhaustive enumeration at orders 5, 7 and 9.

A note on reading the paper alongside this page. Earlier releases of the closed-form paper give these counts for the rows-first pass order only, and they treat a square and its mirror image as one square. Under that convention the same results read 4P and 8P. This page counts every square that looks different on paper, and it admits both pass orders, which is why the total here is 16P. The two are not in conflict. They answer different questions, and the ladder above is the complete one.

Order 3 is the exception. There the four families fall on top of one another instead of staying apart, and the true total is 8 rather than 16P. That figure is not a shortcoming. It is every 3×3 magic square that exists, since the 3×3 has exactly one square in eight orientations and nothing else. The method therefore produces all of them.

Two things that are deliberately not counted

Rearranging a finished square. Permuting the rows and columns of a square that has already been built is a different operation from permuting the initial table, and it is not part of the method. Almost none of the squares it produces can be built by the method at all, so they do not belong in the total. The two families are printed side by side in the published paper so the difference can be seen directly.

Filling the initial table down the columns. Writing 1 to n² down the columns instead of across the rows does produce further magic squares, but they turn out to be nothing more than the 16P squares reflected along the main diagonal. They are the same squares viewed sideways, so counting them would be counting twice.

Rearrangements that break the central pairing occasionally still give a magic square, but never an associative one, and the number of them follows no regular pattern from one order to the next. Associativity is a defining property of the method, so these are excluded.

Twelve examples at order 7 follow, each produced by permuting the rows and the columns of the initial table and then applying the two shifts. These are members of the 4P family counted above. The first is the plain case, with the initial table left in reading order. Cells are shaded by the row of the initial table each number came from, so the effect of the permutation can be followed by eye. Every square shown is a genuine associative magic square with all lines totalling 175.

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Rotations and Reflections

This section answers the question every reader asks next. If the method builds 16P squares, surely turning them round and mirroring them produces still more. The answer is no, and the reason is worth seeing.

Any magic square can be turned through a quarter, a half or three quarters of a turn, and each of those can be mirrored, giving eight orientations in all. Every one of them is still a magic square, because rotating or mirroring a grid moves whole rows, columns and diagonals about without changing what is in them. The four rotations of the 3×3 square are shown below.

 

 

What the orientations do to a Mamzeris square

Apply all eight orientations to every one of the 16P squares and a sharp division appears. Four of the eight always land back on a square the method already builds. The other four always land on a square the method cannot build at all. There are no exceptions and no borderline cases. This has been checked on every square at orders 5, 7 and 9.

Orientation	Result	Already in the 16P total?
Leave as it is	a Mamzeris square	Yes
Turn a half turn (180°)	a Mamzeris square	Yes
Mirror left to right	a Mamzeris square	Yes
Mirror top to bottom	a Mamzeris square	Yes
Turn a quarter turn (90°)	a magic square, but not one of ours	No
Turn three quarters (270°)	a magic square, but not one of ours	No
Reflect along the main diagonal	a magic square, but not one of ours	No
Reflect along the other diagonal	a magic square, but not one of ours	No

Why the first four cannot escape. Turning an associative square through a half turn produces exactly its complement, the square in which every value v has been replaced by n² + 1 − v. That follows straight from associativity, because the value opposite any cell is already its complement, and the half turn is what brings it into place. The method always builds a square together with its complement, so the half turn has nowhere new to go. Mirroring behaves in a similar way: it carries the left-and-up family onto the right-and-down family, and both are inside the total already. The two mirrors and the half turn between them account for the whole of that group.

Why the other four fall outside. A quarter turn, or a reflection along a diagonal, interchanges the roles of the rows and the columns. The method treats the two axes differently, one shifted before the other, so the result is a magic square built by some other means, not by this construction. Not one square out of the 9,216 at order 7 survives a quarter turn and stays in the set.

The consequence for the total. Rotations and reflections add nothing. Four of them are already counted and the other four produce squares outside the method, which the method does not claim. The total stands at 16P.

Order 3 once more. At order 3 all eight orientations stay inside, because the total there is 8 and that is every 3×3 magic square in existence. The 3×3 is the only order at which the eight orientations and the method's own total are the same thing, which is why it makes a clear picture but a misleading rule.

The four variations are not orientations. This is worth stating plainly, because the two ideas are easy to confuse. The orientations above are applied to a square that is already finished, and they work on any magic square ever written down. The four variations of the Mamzeris Method come from the construction itself, from the shift directions and the order of the two passes. The four are genuinely different squares: none of them is a rotation, a reflection or a diagonal reflection of any other. That has been checked at orders 5, 7, 11, 19 and 31.

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For any comments or questions, please email me at contact@oddmagicsquares.com

Smaragdos (Marios) Mamzeris - Magic Squares