I've been thinking about the following problem:
We have a $1\times 5$ rectangle: how to cut it and reassemble it such that it forms a square?
Thanks a lot!
I've been thinking about the following problem:
We have a $1\times 5$ rectangle: how to cut it and reassemble it such that it forms a square?
Thanks a lot!
Cut a small square out of it and throw the rest away. If you are not allowed to throw any part away, then crumple the rest into a ball and stack it on the square. If you are not allowed to throw any part away and you are not allowed to overlap pieces, then I don't know the answer.
Edit:
Now there's a standard way to cut two squares into a totasl of four peices that rearrange to form a single square. I wish I knew how to draw pictures.
(source: cut-the-knot.org)
You can do it with four pieces, and translations only (no rotations).
A standard approach to finding solutions to problems like this is to overlay two tilings. In the following image, the bright yellow rectangles give one 5-piece and two 4-piece solutions requiring only translations (one of which is the same as Robert Israel’s above):
$\hspace{1.15in}$
Here's a method that is surely far from optimal.
First, cut it into 5 $1\times1$ squares, and arrange them into a $2\times2$ square sitting next to a $1\times1$. Now there's a standard way to cut two squares into a total of four pieces that rearrange to form a single square. I wish I knew how to draw pictures. Anyway, let the small square be $ABCD$ With $C$ a vertex of the big square and $CD$ along the side $CEFG$ of the big square. Find $H$ on $CG$ such that $GH=AB$. Cut along $FH$ and along $AH$. Then the bits $FHG$, $ABH$, and $ADEFHA$ can be moved to form a square.