Monday, July 20, 2009

Rectangle Tangle

A rectangle is cut out from another bigger rectangle. How can you divide the remaining portion into two equal parts ? Of-course, you don't have the measuring tape, where the heck these things are when you need them the most ?

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16 Comments:

Anonymous Anonymous said...

depending on where the rectangle cut is the division can be either easy or a lil hard... first some details need to be set, if the rectangle is cut out with one cut (meaning straight, no turning of the scissors) u'll be left with a rectangle or a square, which is simple, fold it in half and cut it, if it's cut from let's say the edges, but within the center forming a lowercase n, fold it in half vertically when looking at the n, and cut there, if it's cut in the center, forming a picture frame, again fold in half and cut, it's when it's cut from a corner making an L is when it gets tricky... that L i can't figure out, so gl to those that try. btw first :-P

July 21, 2009 12:42 AM  
Anonymous A said...

I dont get the above guys explanation, anon, you r taking only cuts where the left over bit is symmetrical,
For example: the small rectangle can be cut such that is is only 2 cm from the right, but 5 cm from right.(but u dont know the actual measurements)

Also, if it is L shaped, and again assuming that the ratio of lengths and breadths on the rectangles are equal, then it is easy, fold it along the line joining the corner in the mid of the paper (of the small rect), and the same corner for the bigger one. it will be symmetrical in this case also but i dunno wht to do if the cut is NOT symetrical(refer my above example)

July 21, 2009 2:50 AM  
Anonymous matt said...

hang on this is quite easy
if you cut a rectangle out of another one then your left with somrthing like this:
!!!!!!!!!!!!!!!!!!!!
!..................!
!..................!
!..................!
!!!!!!!!!!!!!!!!!!!!

(the dots are where the rectangle has benn cut out and the !s are where the remaining part is)

and you would have to cut a line in the remaining part so you could get to the middle so the remaing part has a line in it already so you just need to cut another line opposite that and its in two equal parts

simple

July 21, 2009 3:21 AM  
Anonymous Anonymous said...

Draw a dot in the middle of the empty space in the left over bit and then draw another dot in the middle of the original rectangle. Then join the two dots and cut on that line.

July 21, 2009 5:19 AM  
Anonymous Pete said...

First, let's clarify what's meant by "two equal parts" ???

if it means identical, as in shape and size then there are many scenarios where that's NOT possible - the rectangle cut-out needs to be positioned specifically to get a symmetrical shape or one that can be divided into identical/symmetrical shapes.

Next question is - can the smaller rectangle be cut out completely on the INSIDE of the larger one so that all 4 sides of it leave parts of the large rectangle ? Or can it only be cut out from the edge so that the scissors don't have to punch a whole in the large rect?

Perhaps the intention was to come up with a generic solution, regardless of the position or rotation of the inner rectangle but as long as we can show how to divide the remaining portion into TWO shapes of IDENTICAL surface area (not necessarily shape) ?

I think that becomes a lot more interesting than a couple of specific scenarios mentioned above.


There are still cases where this will not be possible - for example if the inner rectangle is rotated and sized so that two of its corners are placed on the edges of the larger rectangle - that by itself divides the remaining portion into two, not necessarily equal parts.

I'm not sure yet how to solve it for a generic case ... perhaps I'll address first a generic case without any rotation where it should be possible to always divide the remaining portion into two of equal surface area.

July 21, 2009 5:48 AM  
Anonymous Euclid's Brother said...

I agree with last anonymous. A straight line that croses the center of the rectangle, will divide that rectangle's area exactly in half.

So if you draw a straight line that crosses the center of BOTH rectangles, then you're dividing both exactly in half.

Doesn't matter where the smaller cutout rectangle is within the larger. It doesn't even have to be oriented the same way.. the cutout rectangle can be tilted.

How do you determin the centers without a ruler? folding in half can help. ;)

July 21, 2009 8:22 AM  
Anonymous Anonymous said...

fold it in half then cut on the line.

Jesse

but really tho it is Jesse

July 21, 2009 8:44 PM  
Anonymous Anonymous said...

Cut it whatever rectangle you want
then cut it diagonally.

July 24, 2009 1:38 PM  
Blogger Daniel said...

the center-center lie idea works extremely well. Any line that passes through the center of a rectangle will create two congruent pieces. therefore a line that passes through the center of both rectangles will create two equal pieces of the larger rectangle with equal pieces removed.

NOTE: The end result will not be two congruent shapes but it will be two pieces of equal area.

The simple way to determine the center of the rectangle is two draw two lines corner-to-corner which make an "X" --the X crosses in the center

July 25, 2009 1:18 AM  
Anonymous Anonymous said...

if you cut a rectangle from a bigger rectangle. it sounds to me like you broke the rectangle.
i'm pretty sure if you sent it to jim that he'd do something about it

July 25, 2009 10:40 AM  
Anonymous Anonymous said...

How about this...

__________________
1 1 1
1___________1 1
1 1 1
1___________1____1

July 25, 2009 11:54 PM  
Anonymous Anonymous said...

Cut along the diagonal. Talk about overengineering. Take a step back, guys...

August 3, 2009 6:50 PM  
Anonymous Anonymous said...

fold it in half and cut?

August 5, 2009 3:54 AM  
Anonymous Chris said...

The line through the centre of the removed rectangle and original rectangle is a beautiful solution.

Anonymous before last: what diagonal are you referring to? Before answering, check your result. Take a sheet of paper and remove an arbitrary (both centre postion and orientation) internal rectangle from it. Now find a way to split the leftover piece into two equal areas. Then realise that the second anonymous, Euclid's Brother and Daniel have the best possible answer.

August 10, 2009 3:47 AM  
Anonymous TristanJvR said...

______________________
| 1 . - 2 |
| . " - - , |
|. --- |
|. - |
| - . /|
| 2 - . _ / |
|___________-_-__/__1|

The mistake that you all have been
making is that you assumed that you
have to cut a rectangle out along
the horizontal plain.
If you cut it out at an angle like
indicated above, then you will find
that you will be left with 4 triangles. And you will find that
you will have two sets of identical
triangles. So set 1 + 2 will be the same as the other 1 + 2 set - giving you your two equal portions.
:)

January 22, 2010 6:54 AM  
Anonymous TristanJvR said...

ok the drawing didn't come out too well...
lets try that again...

.....######...2...........
..1.#OOOOOO#####..
...#OOOOOOOOOOO#
..#OOOOOOOOOOO#.
.#####OOOOOOO#.1
..2...###########.......

January 22, 2010 7:01 AM  

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