How to cut a square on $5$ squares?

Question

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...?

Answer 1

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:

Answer 2

If you had glue, it would be easy: