Problem

Source: 2023 Indonesian TST Stage 1: Test 3 - Combinatorics

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Let $n$ be a positive integer. Each cell on an $n \times n$ board will be filled with a positive integer less than or equal to $2n-1$ such that for each index $i$ with $1 \leq i \leq n$, the $2n-1$ cells in the $i^{\text{th}}$ row or $i^{\text{th}}$ collumn contain distinct integers. (a) Is this filling possible for $n=4$? (b) Is this filling possible for $n=5$?