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Omni-MATH / Sir Alex plays the following game on a row of 9 cells. Initially, all cells are…

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problem

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: [list=1] [*] Choose any number of the form 2j2^j, where jj is a non-negative integer, and put it into an empty cell. [*] Choose two (not necessarily adjacent) cells with the same number in them; denote that number by 2j2^j. Replace the number in one of the cells with 2j+12^{j+1} and erase the number in the other cell. [/list] At the end of the game, one cell contains 2n2^n, where nn 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 nn. [i]
Plain-text mathematical notation (without MathML)
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:
[list=1]
[*] Choose any number of the form 2^(j), where j is a non-negative integer, and put it into an empty cell.
[*] 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.
[/list]
At the end of the game, one cell contains 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.

[i]
Original LaTeX notation
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:
[list=1]
[*] Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell.
[*] 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.
[/list]
At the end of the game, one cell contains $2^n$, 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 $n$.

[i]

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Source and history

task_id
53f5325d-8c1f-5114-9fa5-25495191a01f
task_key
test--53f5325d-8c1f-5114-9fa5-25495191a01f
task_revision_id
1
config / split
/ test

Official source

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