South African Mathematics Olympiad

If we call the three numbers $n-1$, $n$, $n+1$, then $(n-1{)}^2+n^2+(n+1{)}^2=770$, which simplifies to $3n^2+2=770$. Therefore $n^2=768/3=256=16^2$, so $n=16$ (since $n>0$), and the largest number is $n+1=17$.

Without doing any algebra, it is clear that $n^2\approx 770/3=256\frac{2}{3}$, and the nearest perfect square is 256.