The sum of the squares of the lengths of the chords intercepted on the circle $x^2+y^2=16$ by the lines $x+y=n,n\in N$ is

Let $d$ be the perpendicular distance from $C(0,0)$ to line $x+y=n$

$d=\frac{n}{\sqrt{2}}$

here $d<r\Rightarrow \frac{n}{\sqrt{2}}<4\Rightarrow n<4\sqrt{2}$, hence the $n$values are $1,2,3,4,5.$

length of chord is $=2\sqrt{16-\frac{n^2}{2}}$

square of intercept $=4\times (16-\frac{n^2}{2}),n\in N=2\left(32-n^2\right)$

sum of squares of intercepts

 $$\begin{gathered} =2\times \left(32-1+32-4+32-9+32-16+32-25\right) \\ =2\times \left(31+28+23+16+7\right) \\ =2\times \left(105\right) \\ =210 \end{gathered}$$