Putnam 2014 B1

A base $10$ over-expansion of a positive integer $N$ is an expression of the form $$N = d_k 10^k + d_{k-1} 10^{k-1} + \cdots + d_0 10^0$$ with $d_k \neq 0$ and $d_i \in \{0,1,2,\dots,10\}$ for all $i$. For instance, the integer $N = 10$ has two base 10 over-expansions: $10 = 10 \cdot 10^0$ and the usual base 10 expansion $10 = 1 \cdot 10^1 + 0 \cdot 10^0$. Which positive integers have a unique base 10 over-expansion?