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.
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.