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...?

@DietrichBurde: The given link for $n=5$ is about decomposing and reassembling. I think that the latter is forbidden here. — Christian Blatter, Sep 22 '18 at 09:31

score 3 · Accepted Answer · answered Sep 22 '18 at 09:43

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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:

answered Sep 22 '18 at 09:43

Christian Blatter

226,825

user376343 · Answer 2 · 2023-03-15T11:46:02.170

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If you had glue, it would be easy:

edited Mar 15 '23 at 11:46

answered Sep 22 '18 at 10:59

user376343

8,311

2

Thanks...........! +1 – nonuser Sep 22 '18 at 11:05

How to cut a square on $5$ squares?

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