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Another user just inquired about possible solutions to the famous $70$x$70$ square puzzle. When I encountered that many years ago and the first idea that came to my mind as to why I wouldn't think it was possible to solve had to do with the $1$x$1$ square. Once you place this square, it appears that it creates a problem and you seem to end up building around that piece endlessly (results don't really change even if you hold off on placing the $1$x$1$).

This got me thinking so I started drawing a few pictures. I couldn't come up with a way to use smaller distinct squares (can't use the same square twice) to create a bigger one. I tried working with a few Pythagorean Triples as they share a similar idea of taking smaller 'squares' and putting them together to make bigger ones, but that didn't offer me anything.

Does anyone know of a example? Or, if it is impossible, a proof to support why not?

I apologize if this is obvious/trivial. Also, I didn't have a good idea how to tag this if someone could be so kind as to correct any misgivings.

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I think this is known as Mrs Perkins' quilt puzzle. (Sorry, I missed the part about needing distinct sizes. Anyway, a lot of related material there- I just remembered that buzzword for search) – Jyrki Lahtonen Jul 23 '14 at 7:07
Is it compulsory to use a $1\times1$ square? If not, see ant11's answer. If it is compulsory then the problem may be harder. – David Jul 23 '14 at 7:40
up vote 7 down vote accepted

I commented on the previous post as well!

There is a good reason you couldn't find an example by hand: the smallest example of what's called a perfect squared square is a $112\times 112$ (link).

There is much more research here.

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