Eighteenth STARS OF MATHEMATICS Competition

Assume the square in the upper-left corner is white. By the condition in statement, a set $\{a,b\}$ in $P$ is a horizontal domino if $|a-b|=1$ and a vertical domino if $|a-b|=506$, with the possible exceptions $\{506,507\}$, $\{1012,1013\}$ and $\{1518,1519\}$. As this latter is an all-black member of $P$ and dominoes are bicolour, $P$ must have some all-white member. The only such is $\{1012,1013\}$. Hence, to form a set in $P$, the number $506$ pairs off with exactly one of $505$ and $507$. As $\{506,507\}$ is all-black and there are no more all-white sets left, $506$ must pair off with $505$, as stated.