Competition problem — Combinatorics

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Combinatorics Difficulty 5.0 AIME Find the answer Netherlands

The number $1$ is written on the blackboard. A turn consists of wiping out the number on the board and replacing it by the double of the number, or by the number one smaller. For example, we can replace $1$ by $2$ (the double) or $0$ (one smaller), and if $5$ is on the board, we can replace it by $10$ or $4$ .
What is the minimum number of turns needed in order to write the number $2021$ on the board?
A) $14$ B) $15$ C) $16$ D) $17$ E) $18$

This was a multiple-choice question, but the options didn't survive into the source we have. The answer given is B, and the solution below works it through.

Solution

B) $15$

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