Inspired by Binary

If we want to minimize the sum, then we have to minimize each of the terms. That means all 10 of the terms are $1\overbrace{000\dots 000}^{d-1~\text{zeroes}}$ . Then, when we all of those terms up: $1\overbrace{000\dots 000}^{d-1~\text{zeroes}}+1\overbrace{000\dots 000}^{d-1~\text{zeroes}}+\dots=10\cdot 1\overbrace{000\dots 000}^{d-1~\text{zeroes}}=1\overbrace{000\dots 000}^{d~\text{zeroes}}$

Therefore, the minimum for the sum will have $d$ zeroes, plus the 1 at the beginning, giving us ${d+1}$ digits. $~\beta_{\lceil \mid \rceil}$

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Note that this works in any base! This is mostly due to the fact that $10_n=n_{10}$ .