two shapes in a $2n\times 2n$ grid sheet, can we pick third one? Can anyone help me with this problem? It just popped to my mind!!!
we have a $2n\times 2n$ grid sheet and a connected shape $L$ consisting of $2n-1$ grid squares. we've cut two copies of $L$ out of the sheet. Is it always possible to cut a third copy of $L$?
I think the answer is yes, but I couldn't solve it. any Ideas?
 A: Asked and answered at https://math.stackexchange.com/questions/111011/two-shapes-in-a-2n-times-2n-grid-sheet-can-we-pick-third-one/111196#111196
EDIT: I was hoping that my answer for the size 8 grid would generalize readily to larger squares, but so far the only ones I can handle are size $6k+2$, $k=1,2,\dots$. The piece is a cross with mutually bisecting lines of $4k+1$ and $2k+1$ squares. The two crosses are placed in the same orientation and in such a way that the short lines of squares meet at a single point, that point being the center of the big square. An argument similar to the one given at m.se shows that there are very few places to put a third line of $4k+1$ squares: 


*

*You can put it too close to the edge of the big square to allow for the line of $2k+1$, or 

*You can put it in either of two places where it runs from an edge of the big square just past a long arm of one piece and just short of a long arm of the other piece. But this third line can't be part of a third piece because its short arm would overlap a long arm of one of the first two pieces. 
[Added by J.O'Rourke:]
          

