The number of pairs \((x, y)\) of integers satisfying the inequality \(|x - 5| + |y - 5| \leq 6\) is:

There are 85 integer pairs. Translating the centre to $(5,5)$, there is one pair at distance zero under the sum of absolute differences and $4d$ pairs at each integer distance $d=1,\ldots,6$. The total is $1+4(1+2+3+4+5+6)=85$, and omitting the centre gives 84.