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Find the greatest $n$

Source: Caucasus MO 2023

July 16, 2023

Problem Statement

Let n≤100n \leq 100 be an integer. Hare puts real numbers in the cells of a 100×100100 \times 100 table. By asking Hare one question, Wolf can find out the sum of all numbers of a square n×nn \times n, or the sum of all numbers of a rectangle 1×(n−1)1 \times (n - 1) (or (n−1)×1(n - 1) \times 1). Find the greatest nn{} such that, after several questions, Wolf can find the numbers in all cells, with guarantee.
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