2017 IMO Shortlist C3

Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations: (1) Choose any number of the form $2^{j}$, where $j$ is a non-negative integer, and put it into an empty cell. (2) Choose two (not necessarily adjacent) cells with the same number in them; denote that number by $2^{j}$. Replace the number in one of the cells with $2^{j+1}$ and erase the number in the other cell. At the end of the game, one cell contains the number $2^{n}$, where $n$ is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of $n$. (Thailand)

亚历克斯爵士在一排 9 个格子上玩以下游戏。最初，所有单元格都是空的。在每次移动中，Alex 爵士都可以执行以下两种操作之一： (1) 选择 $2^{j}$ 形式的任意数字，其中 $j$ 是非负整数，并将其放入空单元格中。 (2) 选择两个(不一定是相邻的)数字相同的单元格；用 $2^{j}$ 表示该数字。将其中一个单元格中的数字替换为 $2^{j+1}$ 并删除另一个单元格中的数字。游戏结束时，一个单元格包含数字 $2^{n}$，其中 $n$ 是给定的正整数，而其他单元格为空。确定 Alex 爵士可以采取的最大步数，以 $n$ 为单位。 (泰国)