Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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!

share|improve this question
1  
Are you allowed to overlap the pieces? Do you require that the resulting square have the same area as the rectangle? –  user2468 Jan 5 '12 at 22:51
    
yes... total area of the square should be 5 –  Amihai Zivan Jan 5 '12 at 23:04
1  
The inverse of this question can be found at mathoverflow.net/questions/15181/…. –  TonyK Jan 10 '12 at 22:44
add comment

5 Answers 5

up vote 0 down vote accepted

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.

squares

share|improve this answer
5  
When you copy something from another post, common practice is to acknowledge the source. But thanks for supplying the picture. –  Gerry Myerson Jan 5 '12 at 23:17
add comment

enter image description here

share|improve this answer
    
While this is pretty impressive, how would you figure this out if you didn't already know the answer? Is there a general technique being employed here, or is this problem-specific? –  templatetypedef Jan 6 '12 at 2:08
6  
The area of your original rectangle is 5. A square with the same area must have sides of length $\sqrt{5}$. So thinking of the pythagorean theorem, we need $a^2+b^2=5$. The only positive integer solution is $a,b=1,2$. This led me to chop off a rectangle of size $1\times 2$ and slice it diagonally. It then made sense to do it again and then sliding things around we get the above answer. That's how I thought about it anyway. The hard part was drawing it in Microsoft Paint :) –  Bill Cook Jan 6 '12 at 2:50
add comment

You can do it with four pieces, and translations only (no rotations).

enter image description here

enter image description here

share|improve this answer
    
Very optimal. +1! –  Patrick Da Silva Jan 6 '12 at 16:24
add comment

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):

$\hspace{1.15in}$ tilings

share|improve this answer
add comment

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.

share|improve this answer
    
Googling for Pythagoras' theorem gives a couple of usefull pictures, e.g this one: goo.gl/PuAz7 –  Myself Jan 5 '12 at 22:59
    
... or this one: goo.gl/DcLWS –  Myself Jan 5 '12 at 23:04
    
If you want to draw pictures on MSE, draw them on anything you want as long as you can get them in a file format that allows you to upload them here. There's a button to upload images directly in your answers when you're answering... find it =) personally I am a fan of picturing my blackboard with my Android phone! –  Patrick Da Silva Jan 5 '12 at 23:26
    
@Patrick, thanks, but I'm on the road, using whatever computers I can find, and the only thing I can draw on is a napkin. But I'll keep your advice in mind. –  Gerry Myerson Jan 6 '12 at 12:50
    
@Gerry Myerson : You have no idea what can come out of a napkin... =P –  Patrick Da Silva Jan 6 '12 at 16:24
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.