Transcript Latin Squares: For Dummies

Latin Squares:
For Dummies
Lauren Marsh
Chaquana Dunton
7.22.05
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The Beginnings
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A Latin square is an nxn array,
when each member of the
array is one of the numbers
from 1 to n. Each number
being between 1 to n is
written once in each row and
column.
An Orthogonal pair are two
Latin squares combined to
create one array on the
other, each of those pairs (n2)
are written once in each row
and column.
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Latin Squares are more than
200 years old
It is possible to find a pair of
nxn orthogonal Latin squares
for every n except 2 and 6.
Leonhard Euler introduced
Latin squares in 1783 as, a
new kind of magic squares.
Leonhard Euler believed that
there are no orthogonal latin
squares for orders 2, 6, 10,
14, and indeed for all orders
of the form 4n+2. This
Conjecture was later proven
false in 1960 by Bose,
Shrikhande.
Latin Square Usage
 Latin Squares are important in all
types of mathematic forms:
• Experimental Designs
• Geometry
• Graph Theory
• Grouped Multiplication Tables
Prime (easy) – Magic Squares
2,3,5,7
In Between – Magic Squares
4=22,8=23,9=32
Impossible or hard - Magic Squares
6,10,12,14
The Impossible 4x4
*This LaTin square doesn’T have an orThogonaL pair
Because The inverse Would Create All Rows Of
2,3,4,and 1’s no maTTer whaT order They’re in.*
1
2
3
4
2
3
4
1
3
4
4
1
1
2
2
3
Latin Squares
Box..1..Bottoms
4x4 Latin Squares
..Legend..
•(4)Navy Blue - Capri’s
•(4)Hot pink - Black Tee
•(3)Gray - Skirts
•(3)Green - No Sleeves
•(2)Black - Shorts
•(2)Pink - Short Sleeves
•(1)Red – Pants
•(1)Baby Blue - Long Sleeves
Box..2..Tops
Graeco Squares
Box..1..Bottoms
..Legend..
•Navy Blue - Capri’s
•Hot pink - Black Tee
•Gray - Skirts
•Green - No Sleeves
•Black - Shorts
•Pink - Short Sleeves
•Red – Pants
•Baby Blue - Long Sleeves
Box..2..Tops
!!Fun Websites!!
• http://www.grogono.com/magic/makeyourown
4.php
• www.Sudoku.com