(a) Assume that $ n=d^{2}_{1} +d^{2}_{2} +d^{2}_{3}$ and $ n$ is not divisible by $ 3$. Then $ d_{i}$ is not divisible by $ 3$. So $ d_{i} \equiv \pm 1\pmod 3$ for $ i=1,2,3$. Note that $ n-3=( d_{1} +1)( d_{1} -1) +( d_{2} +1)( d_{2} -1) +( d_{3} +1)( d_{3} -1)$. Since $ \text{RHS}$ is divisible by $ 3$, we have $ \text{LHS}$ is also divisible by $ 3$. It follows that $n$ is divisibly by $3$ , which is a contradiction. (b) $ n=30k^{2} =k^{2} +( 2k)^{2} +( 5k)^{2}$