# How to cut a square on $5$ squares?

We can cut any square on $$n$$ squares if $$n>5$$ and $$n=4$$.

The proof is easy by induction. Base cases $$n=6,7,8$$ are easy to find and then since we can cut a square on $$4$$ squares we get $$3$$ new squares, so we go from $$n\to n+3$$ and we are done.

But I can not find a proof that we can't cut it on $$5$$ squares. I suppose we should search for some contradiction, but...?

• Here is the link for $n>5$. For $n=5$ see here. Sep 22 '18 at 9:01
• @DietrichBurde: The given link for $n=5$ is about decomposing and reassembling. I think that the latter is forbidden here. Sep 22 '18 at 9:31

The book Mathematical Olympiad Challenges by Titu Andreescu and Razvan Gelca (Birkhäuser 1967) contains a proof for the case $$n=5$$. Here is a screenshot from page 128: