Hints only please!
I am trying to figure this out somehow. A row can have as many girls, and a column can have as many boys.
Proof by contradiction seems like a good technique, but I am not sure how much it would help here.
2026-04-12 02:00:40.1775959240
Proving minimum number of chairs is $567$
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Let $r$ be the number of rows and $c$ be the number of columns. There are $14r$ boys and $10c$ girls along with $3$ empty seats. There are $rc$ seats in all. Thus, $14r+10c+3=rc$.
Subtracting $14r+3$ from both sides shows us that $10c=(c-14)r-3$ or $10c \equiv -3 \pmod r$.
Subtracting $10c+3$ from both sides shows us that $14r=(r-10)c-3$ or $14r \equiv -3 \pmod c$.
Not sure if this will lead you to the solution, but hopefully, these equations help!