Slicing a Square/Rectangle into 8 Pieces with 3 Cuts

[Imparcticle]

Imagine you've just baked brownies. The container in which you have baked brownies is either a square or rectangle AND so is the contents of the container. Is it possible to devise a way to cut only three times and make 8 pieces?
In other words, can you divide a square/rectangle into eighths with only 3 lines?

IF SO, how?
There is a corresponding problem for this one, BTW.

 

[marcus]

Imagine you've just baked brownies. The container in which you have baked brownies is either a square or rectangle AND so is the contents of the container. Is it possible to devise a way to cut only three times and make 8 pieces?
...

the contents is not a 2D rectangle, but rather a 3D rectangular solid analogous to a brick, or a two-story rectangular office building


1. first cut: slice it horizontally into two layers, like the first floor and the second floor of a building

2. second and 3rd cut, into quarters from the roof down

that divides it exactly into eighths

 

[Gokul43201]

Why does the container have to be a square or rectangle ? Marcus' solution would work for any prismatic shape (as long as you know the shape) as well as for spheres and ellipsoids.

This problem is just the same as asking for 3 planes that divide a sphere into 8 equal parts...wot ?

PS : What do you mean by that last line ("There is a corresponding problem for this one, BTW" ) ?

 

[marcus]

Hello Gokul!
this post was not a reply to you. I hadnt noticed you had answered.
I was just expanding on my answer to clarify.
---
If one just has a flat 2D rectangle or square paper and is dividing it by drawing straight pencil lines
then one can make at most 7 pieces

if one divides it by lines which are allowed to curve then one can
make as many pieces as one wants with just two lines

with the brownies, assuming one is slicing with a straight knife and dividing it by planes
then the only way one can cut it into eighths is to notice that it
is actually 3D
slice horizontally to make two layers
and then quarter it

this is an opinion

 

[marcus]

Gokul, as OSU graduate student you probably know some topology puzzles

how about a real easy one for us?

(I hate hard problems)

Like...with 2 cuts one makes at most 4 pieces (of a convex piece of paper)

with 3 cuts one makes at most 7 pieces

what about with N cuts
how many pieces can one make?

In India you probably do problems like that in Middle School, so come on.

 

[Imparcticle]

("There is a corresponding problem for this one, BTW" ) ?

That, after solving this one, there is another one that is similar to this one that you could figure it out.

the contents is not a 2D rectangle, but rather a 3D rectangular solid analogous to a brick, or a two-story rectangular office building


1. first cut: slice it horizontally into two layers, like the first floor and the second floor of a building

2. second and 3rd cut, into quarters from the roof down

that divides it exactly into eighths

Sorry about that. Keep it confined to a 2D parallelogram.

In India you probably do problems like that in Middle School, so come on.

Is Gokul from India? If so, NAMASTAY, Gokul.

 

[marcus]

you get to MOVE the pieces before the third cut

you cut into equal quarters
and then line the quarters into a long row
and cut down the middle
so you have eight brownies now

 

[Gokul43201]

Gokul, as OSU graduate student you probably know some topology puzzles

I didn't know there's a connection betewen Ohio State and Topology. College Football , yes ! Topology...I really wouldn't know...I'm just a poor grad student in the Physics Dept.

how about a real easy one for us?

(I hate hard problems)

Like...with 2 cuts one makes at most 4 pieces (of a convex piece of paper)

with 3 cuts one makes at most 7 pieces

what about with N cuts
how many pieces can one make?

You wanted a good problem...isn't that one such. Or are you asking me to solve it ?

In India you probably do problems like that in Middle School, so come on.

High school, perhaps...not middle school.

Anyways...here's my best shot at N straight line cuts on a convex piece of paper :

I used the Euler relation for a closed sheet, F+V-E = 1 or F = 1+E-V (Faces, Vertices, Edges)

It's not hard to see that V(n) = 2n + n(n-1)/2 = n(n+3)/2

Also, E(n) = 2n + n^2 = n(n+2)

So, F(n) = 1 + n(n+2) - n(n+3)/2 = (2 + n + n^2)/2

Thus, F(1) = 2, F(2) = 4, F(3) = 7, F(4) = 11, etc...

Of course, this is the maximum number of pieces that can be cut. The minimum is just n+1.

EDIT : This approach doesn't allow for nifty rearrangements between cuts !

Namastay, Imp !

 

[marcus]

Gokul, I was not thinking of rearranging the pieces
Only a piece of paper and you get to draw N straight pencil lines on
it.

And you know what? I was in such a hurry that I forgot to do it myself. I just wanted to think of a problem to give you an example of what sort of problem I wanted you to give us!

I had to go out in a hurry and just got back.

I actually don't know the answer to my own problem! (the N pencil lines one)

so I will give that one to you

you decide if you get the prize or not, for solving it

(I guess it does not have to be a finite sheet of paper does it? Could just be the plane itself.)

 

[marcus]

Gokul, this is an example of "elegant"
to use the Euler relation as you did
(mathematicians word for cool, they were discussing it, but not
too helpfully, in philosophy forum a few days ago)


Also another thing, obviously you have found that the number of
regions is always one more than the
bowling pin numbers

1,3,6,10,15,...

the bowling pin numbers occur when you set up bigger and bigger wedges of bowling pins
always adding a back row that is one more than the row before
you knew this but didnt mention it, having already said Euler and
not wanting to make the soup too rich

 

[Gokul43201]

The "bowling pin numbers" are also called triangular numbers, for obvious reasons. The general form of a triangular number is n(n+1)/2. This is true because, (i) it is the sum of natural numbers up to n, OR (ii) it is half the area of a rectangle of sides n, n+1. There are possibly several other ways of looking at it.

Yes, the number of pieces cut from a sheet of paper is always 1 more than a triangular number, since :

$$\frac {n^2 + n + 2} {2} = \frac {n(n+1)} {2} + 1$$

 

[marcus]

Gokul I think it must be your turn to provide a problem, please make it easy

 

[NateTG]

EDIT : This approach doesn't allow for nifty rearrangements between cuts !

If you allow for rearangements between cuts, then the maxium number of pieces of a convex solid after $$n$$ cuts is going to be $$2^n$$. This is an easy consequence of the fact that cutting any convex solid with a planar cut will always generate two convex solids.

 

[Njorl]

How about if I tape the paper up so that it makes a cylinder , then make one helical cut and one longitudinal cut. Then I can have as many pieces as I want!

Njorl

 

[BobG]

Imagine you've just baked brownies. The container in which you have baked brownies is either a square or rectangle AND so is the contents of the container. Is it possible to devise a way to cut only three times and make 8 pieces?
In other words, can you divide a square/rectangle into eighths with only 3 lines?

IF SO, how?
There is a corresponding problem for this one, BTW.

If top is side t
Bottom is side b
north side is n
east side is e
west side is w
south side is s

Cut from top, east to bottom,west - plane defined by ten,tes,bwn,bws.
Cut from top, west to bottom,east - plane defined by twn,tws,bwn,bws.
Cut from top north to bottom south - plane defined by twn,ten, bwn,ben.

Wind up with a top piece, bottom piece, and six other pieces.

I think.

It's easier with an apple - then you cut 60 degrees (inclination) rotating the 60 degree cut 120 degrees each time (right ascension).

:surprise:

Uh, sorry, too much sugar from all those brownies.

 

[BobG]

the contents is not a 2D rectangle, but rather a 3D rectangular solid analogous to a brick, or a two-story rectangular office building


1. first cut: slice it horizontally into two layers, like the first floor and the second floor of a building

2. second and 3rd cut, into quarters from the roof down

that divides it exactly into eighths

Geez, how boring is that!? No one will ever invite you to cut the cake at parties.

(I can't believe I couldn't see the easy solution)

In 2D, you can't get more than 7.

 

[marcus]

Regret to inform sir that it is
unfortunately not true that "in 2D you can't get more than 7"
See this post from yesterday evening

you get to MOVE the pieces before the third cut

you cut into equal quarters
and then line the quarters into a long row
and cut down the middle
so you have eight brownies now

 

[vikasj007]

hey its nice to see that there are a lot of people on this forum who are from india.

and gokul can you tell me, from where are you doing your PhD.

 

[Imparcticle]

Okay, Marcus. I'll give you the second problem:

Imagine you have a rectangle. In the rectangle, imagine that there is a significantly smaller parallelogram (like another rectangle). Let this be anywhere in the rectangle. Is it possible to draw a line across the large rectangle such that there is always always a line of symmetry in the rectangle, regardless of the position of the smaller rectangle?

 

[NateTG]

Okay, Marcus. I'll give you the second problem:

Imagine you have a rectangle. In the rectangle, imagine that there is a significantly smaller parallelogram (like another rectangle). Let this be anywhere in the rectangle. Is it possible to draw a line across the large rectangle such that there is always always a line of symmetry in the rectangle, regardless of the position of the smaller rectangle?

What do you mean by line of symetry?

For a general rectangle, there are only two lines of symetry, and an additional two for a square. As long as the smaller rectangle is placed so that it intersects none of those lines, there is no line of symetry for both.