  A Latin square $L(n,k)$ is a square of order $n$ with its entries colored
with $k$ colors so that all the entries in a row or column have different
colors. Let $d(L(n,k))$ be the minimal number of colored entries of an $n
\times n$ square such that there is a unique way of coloring of the yet
uncolored entries in order to obtain a Latin square $L(n, k)$. In this paper we
discuss the properties of $d(L(n,k))$ for $k=2n-1$ and $k=2n-2$. We give an
alternate proof of the identity $d(L(n, 2n-1))=n^2-n$, which holds for even
$n$, and we establish the new result $d(L(n, 2n-2)) \geq
n^2-\lfloor\frac{8n}{5}\rfloor$ and show that this bound is tight for $n$
divisible by 10.
