2014 Putnam 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?

正整数 $N$ 的 *base $10$ 过度扩展* 是以下形式的表达式

$$

N = d_k 10^k + d_{k-1} 10^{k-1} + \cdots + d_0 10^0

$$

对于所有 $i$，$d_k \neq 0$ 和 $d_i \in \{0,1,2,\dots,10\}$。

例如，整数 $N = 10$ 有两个以 10 为底的过度扩展： $10 = 10 \cdot 10^0$

以及通常的以 10 为底的扩展 $10 = 1 \cdot 10^1 + 0 \cdot 10^0$。

哪些正整数具有唯一的以 10 为底的过度扩展？