Fraction of Fraction

Make each variables have closest value to the result.

$a = \frac{198}{99} = 2$

$\frac{1}{b + \frac{1}{c + \frac{1}{d}}} = \frac{31}{99}$

$b + \frac{1}{c + \frac{1}{d}} = \frac{99}{31}$

$b = \frac{93}{31} = 3$

$\frac{1}{c + \frac{1}{d}} = \frac{6}{31}$

$c + \frac{1}{d} = \frac{31}{6}$

$c = \frac{30}{6} = 5$

$\frac{1}{d} = \frac{1}{6}$

$d = 6$

So,

$a^{2} + b^{2} + c^{2} + d^{2} = 2^{2} + 3^{2} + 5^{2} + 6^{2} = 4 + 9 + 25 + 36 = \boxed{74}$