Should it be "circle having diameter greater than $1$" or "square of side $2$"

Sorry. Fixed. Edit: @below, I am not sure, I'll have to check .

Can the two pieces intersect?

just to make sure, is it necessary for the cut to be a straight line?

I think the cut doesn't need to be a straight line. If it does, then certainly the answer will be no.

It should be read as: Is it possible to partition a square of side $1$ into two parts and cover with them a circle having diameter greater than $1$? That's: yes, they can overlap and we don't impose the square to be cut by a straight line, just to be presented as a union of two non-intersecting sets.

msinghal and I were working on the problem with the added restriction that the pieces cannot overlap, but all promising constructions have yielded a maximum circle size of diameter 1... any ideas here?

For the problem of when they can overlap, it's not hard, just Click to reveal hidden text

bobthesmartypants wrote: ...aim to use some extra pieces from the corners to cover an extra epsilon near the middle of two adjacent sides of the square So, as I see the parts in that construction don't overlap. In fact the given by the author solution is actually the same.

dgrozev wrote: bobthesmartypants wrote: ...aim to use some extra pieces from the corners to cover an extra epsilon near the middle of two adjacent sides of the square So, as I see the parts in that construction don't overlap. In fact the given by the author solution is actually the same. Can you please give a drawing?

Ok, open this pdf document: http://sasja.shap.homedns.org/Turniry/TG/38os.pdf go at the 6-th page- at the bottom, look at the right drawing. The two red triangles are cut off and go to the place where the green ones are. Or simpler, one can cut off the four corners and glue them at the middle of the four sides of the square.

dgrozev wrote: Ok, open this pdf document: http://sasja.shap.homedns.org/Turniry/TG/38os.pdf go at the 6-th page- at the bottom, look at the right drawing. The two red triangles are cut off and go to the place where the green ones are. Or simpler, one can cut off the four corners and glue them at the middle of the four sides of the square. Oh, but they do overlap: the skinny green strip of one piece overlaps with the other piece. It's a much harder problem to not have any overlaps at all.

I thought the english word "overlap" meant they had a common interior point. In that case they touches each other, that's they have only common points on their contours. If you want the two parts to be isolated from each other, it's impossible.

But there's a common interior point in the red circled region:

Come on, don't be so punctual. As I said, just cut off a small piece of every corner and glue them at the middle of the four sides of the square(at their outer sides). The problem is not so smart as you think.

But then you don't have two connected pieces, you have 5...

Can someone share some idea how to solve the problem or tell where I can (if possible) see the official solution?