# Filling a sudoku/latin square like array

Let us build a square array in the following manner, which I would like to call a modified sudoku:

1) Every row and column contains only one copy of a positive entry and there are exactly $$t$$ such entries.

2) The zeros must repeat a fixed number of times, equal to $$n-\text{number of distinct positive entries}=n-t$$(the entries left unfilled by positive entries is filled by zeros).

3) Every row should have a distinct vector from each of its corresponding column vectors to positive entries.

4) The positive entries are from a set of given positive integers which is atmost the order of the array, i.e, the vectors(t-tuples of positive entries) are chosen from a set of an $$t+1$$ positive entries where $$t+1$$ is atmost the order of array$$=n$$.

What could be a probable algorithm to build such an array? One probable combination satisfying the above rules for order of array$$=4$$, and cardinality of the set of positive integers is $$4$$: $$\begin{pmatrix}1&2&3&0\\4&3&0&1\\2&0&4&3\\0&4&1&2\end{pmatrix}$$

Another example involving the order of array $$=6$$ and number of positive integer options $$=4$$, i.e length of tuple of positive entries $$=3$$ is: $$\begin{pmatrix}1&2&3&0&0&0\\4&3&0&0&0&1\\2&0&0&0&4&3\\0&0&0&4&3&2\\0&0&4&2&1&0\\0&4&1&3&0&0\end{pmatrix}$$

Also, a very important question is, whether such a n array can be built for every $$n$$ and $$t$$? Or, are there any constraints on $$n,t$$? I find that barring the case of $$t=2$$, other cases are possible, by combinatorial arguments, since we need to choose at most $$t=1$$ distinct vectors, which is possible as we are choosing the $$t$$ positive entries from $$t+1$$ entries. Any hints? Thanks beforehand.

• I don't understand what point (3) means. Could you explain it in a different way, please? – hexomino Feb 7 '19 at 11:50
• @hexomino What point (3) means is that suppose you have a vector say, of $6$ entries with $3$ positive entries, like $\begin{pmatrix}1&2&3&0&0&0\end{pmatrix}$, then each corresponding column vector to entries $1$, $2$ and $3$ should have different positive entries than $1,2,3$. Like, the corresponding column vectors can be $\begin{pmatrix}1\\2\\4\\0\\0\\0\end{pmatrix}$ corresponding to $1$, contd... – vidyarthi Feb 7 '19 at 12:05
• $\begin{pmatrix}2\\4\\1\\0\\0\\0\end{pmatrix}$ corresponding to $2$ and $\begin{pmatrix}3\\0\\1\\4\\0\\0\end{pmatrix}$ corresponding to $3$, but any of the corresponding column vectors should contain their positive entries as $1,2,3$ or their permutations, like $\begin{pmatrix}2\\0\\0\\0\\1\\3\end{pmatrix}$ corresponding to $2$ – vidyarthi Feb 7 '19 at 12:09