MathDB

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

Problem Statement