Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / R / R2 / iv

The missing sets in the five columns are $\{4,5\}$, $\{1,3\}$, $\{4,5\}$, $\{2,3\}$ and $\{1,2\}$. One choice of distinct representatives is $(4,1,5,3,2)$, giving the fourth row. Each column then has exactly one missing number, producing the fifth row $(5,3,4,2,1)$. A completed square isEvery row and column contains each number from one to five once. The construction explicitly chooses from the missing sets, so it preserves all entries of the given rectangle.