Track / Stage 4 / 288 of 340 #548 of 1964

Problem 548

AMC 12 late, AIME early

Combinatorics Difficulty 5.0 Find the answer Dutch Mathematical Olympiad · 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, so there is nothing here to pick from. Work it on paper and mark yourself against the solution below.

Official solution

B) $15$

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