Problem

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Tags: combinatorics



Alberto chooses $2022$ integers $a_1,\,a_2,\dots,\,a_{2022}$ (not necessarily positive and not necessarily distinct) and places them on a $2022\times 2022$ table such that in the $(i,j)$ cell is the number $a_k$, with $k=\max\{i,j\}$, as shown in figure (in which, for a better readability, we have denoted $a_{2022}$ with $a_n$). Barbara does not know the numbers Alberto has chosen, but knows how they are displaced in the table. Given a positive integer $k$, with $1\leq k\leq 2022$, Barbara wants to determine the value of $a_k$ (and she is not interested in determining the values of the other $a_i$'s with $i\neq k$). To do so, Barbara is allowed to ask Alberto one or more questions, in each of which she demands the value of the sum of the numbers contained in the cells of a "path", where with the term "path" we indicate a sorted list of cells with the following characteristics: • the path starts from the top left cell and finishes with the bottom right cell, • the cells of the path are all distinct, • two consecutive cells of the path share a common side. Determine, as $k$ varies, the minimum number of questions Barbara needs to find $a_k$.


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