Latin Squares

Jacky Ligon wrote:

>One further interesting thing is that it is impossible to create a LS
>like the symmetrical one above with the number 14. Until I found out
>that Euler couldn't do it, I thought I was mentally deficient,
>because I'd tried on and off for a long time to crack the code.

I suspect your memory is playing tricks on you. Latin squares of any size are
easy to create. A simple cyclic shift like

1 2 3 4 5 6 7 8 9 A B C D E
2 3 4 5 6 7 8 9 A B C D E 1
3 4 5 6 7 8 9 A B C D E 1 2
4 5 6 7 8 9 A B C D E 1 2 3
5 6 7 8 9 A B C D E 1 2 3 4
6 7 8 9 A B C D E 1 2 3 4 5
7 8 9 A B C D E 1 2 3 4 5 6
8 9 A B C D E 1 2 3 4 5 6 7
9 A B C D E 1 2 3 4 5 6 7 8
A B C D E 1 2 3 4 5 6 7 8 9
B C D E 1 2 3 4 5 6 7 8 9 A
C D E 1 2 3 4 5 6 7 8 9 A B
D E 1 2 3 4 5 6 7 8 9 A B C
E 1 2 3 4 5 6 7 8 9 A B C D

will work for any number and for a composite number like 14 you can also create
product squares by taking an m by m Latin square and replacing the individual
entries with an n by n Latin square to create an mn by mn Latin square. Since
you mentioned Euler, I suspect you were thinking of the Greco-Latin square problem.
In that case, the problem was to arrange the n^2 ordered pairs into an n by n
square such that each pair was used exactly once and each first component was
used exactly once in each row and column and each second component was used exactly
once in each row and column. Euler was able to construct solutions for all cases
except those where n was even but not a multiple of 4. In the late 1950s some
American mathematicians found a 10 by 10 Greco-Latin square and since then I believe
Greco-Latin squares have been found for all sizes but 2 and 6.

David Bowen