XXVI OBM

Let's count in two ways the number of ordered pairs $(d,l)$, where $d$ is a digit and $l$ is a row or column containing $d$. For simplicity, let a line be a row or a column. Since there are $10$ occurrences of $d$, they are present in at least $7$ lines (the intersections of the rows and columns must cover all ten numbers). So the number of pairs are at least $7\cdot 10=70$. Since there are $10+10=20$ lines, one line must contain at least $\lfloor \frac{70-1}{20}\rfloor +1=4$ different numbers. The

following example shows that the answer is indeed $4$: