Lord of the files

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Source: Mathematical Minds 2024 P3

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209 posts

#1 h Sep 29, 2024, 10:31 AM • 1 Y

Y by oVlad

On the screen of a computer there is an $2^n\times 2^n$ board. On each cell of the main diagonal there is a file. At each step, we may select some files and move them to the left, on their respective rows, by the same distance. What is the minimum number of necessary moves in order to put all files on the first column?

Proposed by Vlad Spătaru

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#2 h Sep 30, 2024, 5:11 PM

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Without loss of generality, assume that the minimum moves in order are moving by distances $a_1 < a_2 < ... < a_m$ of some files. Each number from 1 to $2^n - 1$ (corresponding to the distance of each file to the first column) is represented by the sum of some distinct move distances. From the above $m$ moves, we can form at most $2^m - 1$ distances. Hence $2^m - 1 \geq 2^n - 1$ , or $m \geq n$ . Equality happens for example when the moves are by distance $1, 2, 2^2, ..., 2^{n-1}$ in order, and each file moves $2^i$ steps when the representation in base 2 of its distance to the first column contains $2^i$ .

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#3 h Apr 13, 2025, 7:46 AM • 1 Y

Y by TheMathletic

Let's play football!!.....ofc after solving the problem.... :yinyang:

:yinyang:

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