Latin squares have been studied by mathematicians for many years; presented here is a variant based on modular arithmetic.

As would be expected in the modular environment, all numbers in this report are integers. Both the remainders and quotients from modular division will be used, with the modulus serving as divisor. Moduli are positive and all other numbers are nonnegative. With that, the following customary practice is observed here:

Given a dividend and modulus, chosen as a remainder will always be the smallest nonnegative value that will satisfy the following equation: dividend − remainder = modulus × quotient As a consequence, the quotient will be unique and nonnegative.

Two notations are helpful:

⇒ LS(s) stands for a ordinary Latin square:

• It has with s rows and s columns.
• It contains s different values.
• Within each row, and within each column, each value must occur once.

⇒ MLS(s, v) stands for a modular Latin square:

• It has s rows and s columns.
• It contains v different values.
• v must be a divisor of s.
• Within each row, and within each column, each value must occur s ÷ v times.

When s = v, an MLS(s, v) becomes an LS(s). Examples of both are labeled below.

Although the matter is not developed here, the extension to three dimensions and more is straightforward.

Square 1a is an ordinary Latin square of size six. Because modular arithmetic will be performed, the numbers chosen for the contents are the smallest nonnegative integers. The rows and columns are labeled with letters for reference.

1a	U	V	W	X	Y	Z
A	5	3	1	0	4	2
B	0	1	4	5	2	3
C	3	5	2	1	0	4
D	4	2	5	3	1	0
E	1	4	0	2	3	5
F	2	0	3	4	5	1
LS(6)

The next squares are various remainders and quotients.

1b U V W X Y Z A 2 0 1 0 1 2 B 0 1 1 2 2 0 C 0 2 2 1 0 1 D 1 2 2 0 1 0 E 1 1 0 2 0 2 F 2 0 0 1 2 1 MLS(6, 3): remainder modulo 3 of square 1a	1c U V W X Y Z A 1 1 0 0 1 0 B 0 0 1 1 0 1 C 1 1 0 0 0 1 D 1 0 1 1 0 0 E 0 1 0 0 1 1 F 0 0 1 1 1 0 MLS(6, 2): quotient modulo 3 of square 1a
1d U V W X Y Z A 1 1 1 0 0 0 B 0 1 0 1 0 1 C 1 1 0 1 0 0 D 0 0 1 1 1 0 E 1 0 0 0 1 1 F 0 0 1 0 1 1 MLS(6, 2): remainder modulo 2 of square 1a	1e U V W X Y Z A 2 1 0 0 2 1 B 0 0 2 2 1 1 C 1 2 1 0 0 2 D 2 1 2 1 0 0 E 0 2 0 1 1 2 F 1 0 1 2 2 0 MLS(6, 3): quotient modulo 2 of square 1a

Square 1f is a NON-example of a modular Latin square. Even though all rows and columns contain the same complement of values, each row and column considered individually is unbalanced, containing 2 zeroes and 2 ones, but only 1 two and 1 three. This fails because four is not a factor of six.

1f	U	V	W	X	Y	Z
A	1	3	1	0	0	2
B	0	1	0	1	2	3
C	3	1	2	1	0	0
D	0	2	1	3	1	0
E	1	0	0	2	3	1
F	2	0	3	0	1	1
NOT MLS: remainder modulo 4 of square 1a

The table below shows how many permutations of symbols can occur within a row or column of some MLSs:

MLS type	permutations	example
LS(1, 1)	1! = 1	0
LS(2, 2)	2! = 2	1 0
LS(3, 3)	3! = 6	2 0 1
LS(4, 4)	4! = 24	0 3 2 1
LS(5, 5)	5! = 120	4 2 0 1 3
MLS(4, 2)	4! ÷ (2!)2 = 6	0 1 0 1 1 0
MLS(6, 2)	6! ÷ (3!)2 = 20	1 0 1 1 0 0
MLS(8, 2)	8! ÷ (4!)2 = 70	0 1 1 0 0 1 0 1
MLS(6, 3)	6! ÷ (2!)3 = 90	2 0 1 1 2 0
MLS(9, 3)	9! ÷ (3!)3 = 1680	1 0 2 2 2 0 0 1 1
MSL(s, v)	s! ÷ ((s ÷ v)!)v

For many applications, an equivalence relation is defined for Latin squares:

• If two rows of a Latin square are exchanged, the result is deemed equivalent to the original.
• If two columns of a Latin square are exchanged, the result is deemed equivalent to the original.

By contrast, if a row is exchanged with a column, the result will probably not be a Latin square.

A standard practice is to exchange rows multiple times until the numbers in the first column are increasing; and then similarly with the columns. When this is done to square 1a, the result is 2a. In this case, the rows were sorted before the colums were; had the columns been sorted first, a different, but equally valid, representation would have been produced.

2a	U	V	Y	Z	W	X
B	0	1	2	3	4	5
E	1	4	3	5	0	2
F	2	0	5	1	3	4
C	3	5	0	4	2	1
D	4	2	1	0	5	3
A	5	3	4	2	1	0
LS(6): sorted version of square 1a

In an ordinary Latin square, where no number is repeated within a row or column, a procedure to sort the rows need consider only the numbers in the first column; to sort columns, only the first row. On the other hand, in a modular Latin square where numbers are repeated, more is required: a lexicographical approach is suggested.

If two rows differ in the first column, the less-than relationship between them is established. If the rows are equal in the first column but are unequal in the second column, the second column prevails. This continues to further columns if necessary. Of course, columns are handled analogously.

As an explicit example, what follows is a lexicographical listing, from least to most, of all the possible rows in a size-six Latin square formed with modulo-two remainders or modulo-three quotients — an MLS(6, 2). Those rows and columns that happen to appear in square 2d, which appears below, are marked.

0 0 0 1 1 1
0 0 1 0 1 1 – D, W
0 0 1 1 0 1
0 0 1 1 1 0 – F
0 1 0 0 1 1
0 1 0 1 0 1 – B, U
0 1 0 1 1 0
0 1 1 0 0 1
0 1 1 0 1 0 – Y
0 1 1 1 0 0
1 0 0 0 1 1
1 0 0 1 0 1 – V
1 0 0 1 1 0 – X
1 0 1 0 0 1
1 0 1 0 1 0
1 0 1 1 0 0 – E
1 1 0 0 0 1 – C
1 1 0 0 1 0 – A
1 1 0 1 0 0
1 1 1 0 0 0 – Z

When a sorted square is modularly reduced, whether by remainder or quotient, it might become unsorted. Below are reductions of square 2a, with listings of inversions of consecutive rows or columns. In 2e, the rows and columns happen to end up sorted for no particular reason.

2b U V Y Z W X row < col < B 0 1 2 0 1 2 C < F Z < Y E 1 1 0 2 0 2 F 2 0 2 1 0 1 C 0 2 0 1 2 1 D 1 2 1 0 2 0 A 2 0 1 2 1 0 MLS(6,3): remainder modulo 3 of square 2a	2c U V Y Z W X row < col < B 0 0 0 1 1 1 F < E D < C W < Z E 0 1 1 1 0 0 F 0 0 1 0 1 1 C 1 1 0 1 0 0 D 1 0 0 0 1 1 A 1 1 1 0 0 0 MLS(6,2): quotient modulo 3 of square 2a
2d U V Y Z W X row < col < B 0 1 0 1 0 1 F < E D < C Y < V W < Z E 1 0 1 1 0 0 F 0 0 1 1 1 0 C 1 1 0 0 0 1 D 0 0 1 0 1 1 A 1 1 0 0 1 0 MLS(6,2): remainder modulo 2 of square 2a	2e U V Y Z W X row < col < B 0 0 1 1 2 2 none none E 0 2 1 2 0 1 F 1 0 2 0 1 2 C 1 2 0 2 1 0 D 2 1 0 0 2 1 A 2 1 2 1 0 0 MLS(6,3): quotient modulo 2 of square 2a

With an ordinary Latin square, it is always possible to sort both the rows and columns into ascending order. On the other hand, this is not generally true with modular Latin squares. For instance, square 3a is in order by rows but not columns; but when 3a is sorted by columns, producing 3b, it is no longer in order by rows.

3a U V W X Y Z row < col < A 0 0 0 1 1 1 none W < V Z < Y B 0 1 0 1 1 0 C 0 1 1 0 0 1 D 1 0 1 0 1 0 E 1 0 1 1 0 0 F 1 1 0 0 0 1 LS(6,2): sorted by rows

3b U W V Z X Y row < col < A 0 0 0 1 1 1 F < E none B 0 0 1 0 1 1 C 0 1 1 1 0 0 D 1 1 0 0 0 1 E 1 1 0 0 1 0 F 1 0 1 1 0 0 LS(6,2): sorted by columns

Square 3c shows that a modular square sorted by both rows and columns might not be symmetrical around the main diagonal; highlighted are the exceptions.

3c	U	V	W	X	Y	Z	row <	col <
A	0	0	0	1	1	1	none	none
B	0	1	1	0	0	1
C	0	1	1	0	1	0
D	1	0	0	1	1	0
E	1	0	1	0	0	1
F	1	1	0	1	0	0
LS(6,2): sorted by rows and columns

More will be said about symmetry later.

Here are some enumerations:

quantities of Latin squares
	total	sorted by row or column	sorted by row and column
LS(1,1)	1	1	1
LS(2,2)	2	1	1
LS(3,3)	12	2	1
LS(4,4)	576	24	4
LS(5,5)	161,280	1,344	56
LS(6,6)	812,851,200	1,128,960	9,408
LS(7,7)	61,479,419,904,000	12,198,297,600	16,942,080
see:	OEIS A002860		OEIS A000315
quantities of modular Latin squares
	total	sorted by row or column	sorted by row and column
MLS(2,2)	2	1	1
MLS(4,2)	90	6	2
MLS(6,2)	297,200	550	25
MLS(8,2)	116,963,796,250	3,330,915	7,776
see:	OEIS A058527
quantities of modular Latin squares
	total	sorted by row or column	sorted by row and column
MLS(3,3)	12	2	1
MLS(6,3)	35,599,500	50,115	873

There is an obvious definition of diagonal symmetry for modular Latin squares; it is the same as used in matrix arithmetic, and is mentioned in connection with square 3c.

Let A be an MLS, and let A(r, c) be the value in the rth row and cth column. If A(r, c) equals A(c, r) for all r and c, then A is diagonally symmetric.

Because modular reduction is a function in the strict sense, a diagonally symmetric MLS remains that way when reduced.

Quite a different kind of symmetry has been investigated by E. I. Vatutin and others, with examples here.

Start with an even number s and an MLS(s, v) named A. Within A, the first row and column are numbered zero, and the last s − 1.

• Suppose that there exist particular values of r, c, x, and y such that A(r, c) = x and A(r, s − c − 1) = y.
• And suppose that for all values of r and c, it is true that A(r, c) = x implies A(r, s − c − 1) = y.
• Then A is said to be horizontally symmetric.

Horizontal symmetry can be defined for not just Latin squares, but adapted to any rectangular arrays of numbers.

An example will be helpful. Square 4a is horizontally symmetric, and 4b is an extract that may make the arrangement clearer. Within any row, if the value 1 appears n cells to the left of the dividing bar, the value 3 will appear n cells to the right of the bar.

4a U V W X Y Z A 0 1 5 2 3 4 B 4 5 3 1 2 0 C 3 4 2 5 0 1 D 5 3 4 0 1 2 E 2 0 1 3 4 5 F 1 2 0 4 5 3 LS(6): horizontally symmetric

4b U V W   X Y Z A   1     3   B     3 1     C 3         1 D   3     1   E     1 3     F 1         3 extract of square 4a

Naturally, the analogous vertical symmetry is achieved by exchanging rows and columns. Horizontal and vertical symmetries are preserved in modular reduction.