This demonstration shows how to find all possible reduced latin
squares of size $4$. Because we are only considering reduced
squares, the first row and the first column are $(1,2,3,4)$.
In position $(2,2)$ we cannot have a $2$, because of the $2$s
directly above and directly to the left. However, we can choose
between $1$, $3$ and $4$. In the bottom right of the box we have
written 0*: the
0 indicates that this is step
$0$, and the * indicates that
we have several choices.
We now have only one choice for the entry in position
: the only way we can avoid duplicates
in the corresponding row and column is to place a
there.
There are two possible choices for the entry in position
, namely
and .
We now backtrack to position , which is
the last place where we had any choice.
We now have a reduced latin square of size $4$.
We have now found all four of the possible reduced latin squares
of size $4$.