An r
× n
Latin Rectangle [LR] based on 1
, …, n
is a 2-dimensional array of r
rows and n
columns, where r
< n
, such that each entry is one of the integers 1
, …, n
and each of these integers occurs at most once in each row and at most once in each column.
Is it true that every r
× n
LR can be extended to an n
× n
LR (a Latin square [LS])? Why or why not?
Okay so my way of checking this is to first see what conditions need to be met that would make a LR unable to conform to a LS.
I will be making an assumption that the only case we need to worry about is the case where r = n-1
because that is the most confined example. And to convert every other example where r < n-1
to a LS is just a series of steps more trivial than the final step of the r=n-1
case.