KöMaL problem A, 2019

Combinatorics Difficulty 6.0 National Olympiad Find the answer Hungary

A $k\times k$ array contains each of the numbers $1,2,\ldots,m$ exactly once, with the remaining entries all zero. Suppose that all the row sums and column sums are equal. What is the smallest possible value of $m$ if $k=3^n$ ( $n\in\mathbb{N^+}$ )?

We don't reproduce this publisher's solutions. Their own solution is here — work the problem first.

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: KöMaL, licensed Rights held by KöMaL and the MATFUND Foundation. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.