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| Title: | Mathematics at DEC |
|
| Moderator: | RUSURE::EDP |
|
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
421.0. "Latin and Orthogonal squares" by MENTOR::KOSTAS () Fri Jan 10 1986 15:48
The following will create several problems in the topic of Latin and
Orthogonal Latin Squares. This note has information relevant to Note # 58,
in this notes file.
Definition 1.0:
---------------
Latin Square: We define a Latin square of order k, (i.e. k by k), if the
assignment of the numbers 1,2,3,4,...,k-1,k in a k by k array appear
exactly once in each row and each column.
(Please note that the multiplication table of finite groups also defines
a Latin square).
Example 1.0 of a Latin square or order k, (k by k)
--------------------------------------------------
L
-------------------------------
1 2 3 . . . k-1 k
2 3 4 . . . k 1
3 4 5 . . . 1 2
. . . . . . . .
. . . . . . . .
. . . . . . . .
k-1 k 1 . . . k-3 k-2
k 1 2 . . . k-2 k-1
Example 1.1 of a Latin square or order 2, (2 by 2)
--------------------------------------------------
M
-----
1 2
2 1
Example 1.2 of two Latin squares or order 3, (3 by 3)
-----------------------------------------------------
N O
----------- -----------
1 2 3 1 3 2
2 3 1 3 2 1
3 1 2 2 1 3
Definition 2.0:
---------------
Orthogonal Latin Squares: We say that two distinct k by k Latin squares
2
A = (a ) and B = (b ) are Orthogonal if the k (reads k squared)
ij ij
pairs (a , b ) where (i,j = 1,2,3,...,k-1,k), are all different.
ij ij
Example 2.0 of two (4 by 4) Latin squares that are Orthogonal
--------------------------------------------------------------
A B
---------- ----------
1 2 3 4 4 1 2 3
2 1 4 3 3 2 1 4
3 4 1 2 1 4 3 2
4 3 2 1 2 3 4 1
Combined Design of A and B
--------------------------
(1,4) (2,1) (3,2) (4,3)
(2,3) (1,2) (4,1) (3,4)
(3,1) (4,4) (1,3) (2,2)
(4,2) (3,3) (2,4) (1,1)
Note that A and B are Orthogonal since all ordered pairs are different.
Example 2.1 of two (4 by 4) Latin squares that are NOT Orthogonal
------------------------------------------------------------------
C D
---------- ----------
2 1 4 3 2 3 1 4
3 2 1 4 1 4 2 3
4 3 2 1 3 2 4 1
1 4 3 2 4 1 3 2
Example 2.2 of two (2 by 2) Latin squares that are NOT Orthogonal
------------------------------------------------------------------
E F
---------- ----------
1 2 2 1
2 1 1 2
Problems:
---------
1. Since there does not exist a pair of order 2 Orthogonal Latin squares
can you find any of orders: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ... , 16.
2. Are there any triple Orthogonal Latin squares or orders: 3, 4, 5, ...
3. Are there any n-tuple Orthogonal Latin squares or orders: 3, 4, 5, ...
for n >3.
Please note that I have a pair of Orthogonal Latin squares of order 10, and
of order 12.
Enjoy,
Kostas G.
<><><><><>
| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 421.1 | | MENTOR::KOSTAS | | Thu Jan 23 1986 10:16 | 65 |
| The following are examples of pairs of Latin squares of orders 3, 4, 5,
and 10.
Example 1.0 of two (3 by 3) Latin squares that are Orthogonal
--------------------------------------------------------------
A B
---------- ----------
1 2 3 1 2 3
2 3 1 3 1 2
3 1 2 2 3 1
Example 1.1 of two (4 by 4) Latin squares that are Orthogonal
--------------------------------------------------------------
C D
---------- ----------
1 2 3 4 4 1 2 3
2 1 4 3 3 2 1 4
3 4 1 2 1 4 3 2
4 3 2 1 2 3 4 1
Combined Design of C and D
--------------------------
(1,4) (2,1) (3,2) (4,3)
(2,3) (1,2) (4,1) (3,4)
(3,1) (4,4) (1,3) (2,2)
(4,2) (3,3) (2,4) (1,1)
Note that C and D are Orthogonal since all ordered pairs are different.
Example 1.2 of two (5 by 5) Latin squares that are Orthogonal
--------------------------------------------------------------
E F
------------- -------------
5 1 2 3 4 4 5 1 2 3
4 5 1 2 3 2 3 4 5 1
3 4 5 1 2 5 1 2 3 4
2 3 4 5 1 3 4 5 1 2
1 2 3 4 5 1 2 3 4 5
Example 1.3 of two (10 by 10) Latin squares that are Orthogonal
----------------------------------------------------------------
G H
---------------------------- ----------------------------
1 2 3 4 5 6 7 8 9 0 1 2 0 3 4 9 5 7 6 8
4 5 1 2 3 8 0 9 7 6 7 8 9 0 6 3 4 2 1 5
5 4 2 3 1 0 8 7 6 9 0 4 8 5 7 6 9 3 2 1
2 3 5 1 8 9 6 4 0 7 4 9 3 6 5 8 0 1 7 2
3 1 4 8 6 7 9 0 5 2 2 5 6 1 8 4 7 0 9 3
6 8 7 0 9 4 5 2 3 1 3 6 7 2 0 5 1 9 8 4
9 0 8 6 7 2 3 1 4 5 5 1 2 4 9 7 3 8 0 6
7 6 0 9 2 5 4 3 1 8 6 7 5 9 1 2 8 4 3 0
0 9 6 7 4 1 2 5 8 3 9 3 1 8 2 0 6 5 4 7
8 7 9 5 0 3 1 6 2 4 8 0 4 7 3 1 2 6 5 9
Enjoy,
Kostas G.
<><><><><>
|
| 421.2 | an algorithm will do . . . | THEBUS::KOSTAS | Wisdom is the child of experience. | Thu Dec 11 1986 10:57 | 8 |
|
well,
since Orthogonal Latin Squares do not interest anyone.
What about any algorithms to determine if a pair of Latin squares
or order n are Orthogonal?
-kgg
|