Let the chord x + y = n cuts the circle x² + y² = 16 at A and B.

Length of perpendicular from O on AB,

OM = $$\left| {{{0 + 0 - n} \over {\sqrt {{1^2} + {1^2}} }}} \right| = {n \over {\sqrt 2 }}$$

Radius of the circle = 4

From the picture you can see,

$${n \over {\sqrt 2 }}$$ < 4

$$ \Rightarrow $$ n < 5.65

$$ \therefore $$ Possible value of n = 1, 2, 3, 4, 5

Length of chord AB = $$2\sqrt {{{\left( 4 \right)}^2} - {{\left( {{n \over {\sqrt 2 }}} \right)}^2}} $$

= $$2\sqrt {16 - {{{n^2}} \over 2}} $$

= $$\sqrt {64 - 2{n^2}} $$ = $${l}$$

For n = 1, $${l^2}$$ = 62

For n = 2, $${l^2}$$ = 56

For n = 3, $${l^2}$$ = 46

For n = 4, $${l^2}$$ = 32

For n = 5, $${l^2}$$ = 14

$$ \therefore $$ Sum of square of length of chords = 62 + 56 + 46 + 32 + 14 = 210