Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Baked brownies

  1. Jul 21, 2004 #1
    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. :smile: :smile:
     
  2. jcsd
  3. Jul 21, 2004 #2

    marcus

    User Avatar
    Science Advisor
    Gold Member
    2015 Award
    Dearly Missed

    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
     
  4. Jul 21, 2004 #3

    Gokul43201

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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" ) ?
     
    Last edited: Jul 21, 2004
  5. Jul 21, 2004 #4

    marcus

    User Avatar
    Science Advisor
    Gold Member
    2015 Award
    Dearly Missed

    Hello Gokul!
    this post was not a reply to you.:smile: 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 :smile:
     
    Last edited: Jul 21, 2004
  6. Jul 21, 2004 #5

    marcus

    User Avatar
    Science Advisor
    Gold Member
    2015 Award
    Dearly Missed

    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.
     
  7. Jul 21, 2004 #6
    That, after solving this one, there is another one that is similar to this one that you could figure it out.

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

    Is Gokul from India? If so, NAMASTAY, Gokul.
     
  8. Jul 21, 2004 #7

    marcus

    User Avatar
    Science Advisor
    Gold Member
    2015 Award
    Dearly Missed

    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
     
  9. Jul 21, 2004 #8

    Gokul43201

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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.

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

    High school, perhaps...not middle school. :wink:

    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 ! :cry:

    Namastay, Imp ! :smile:
     
    Last edited: Jul 21, 2004
  10. Jul 21, 2004 #9

    marcus

    User Avatar
    Science Advisor
    Gold Member
    2015 Award
    Dearly Missed

    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 dont 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 :smile:

    (I guess it does not have to be a finite sheet of paper does it? Could just be the plane itself.)
     
  11. Jul 21, 2004 #10

    marcus

    User Avatar
    Science Advisor
    Gold Member
    2015 Award
    Dearly Missed

    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
     
  12. Jul 21, 2004 #11

    Gokul43201

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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 :

    [tex] \frac {n^2 + n + 2} {2} = \frac {n(n+1)} {2} + 1 [/tex]
     
    Last edited: Jul 21, 2004
  13. Jul 21, 2004 #12

    marcus

    User Avatar
    Science Advisor
    Gold Member
    2015 Award
    Dearly Missed

    Gokul I think it must be your turn to provide a problem, please make it easy
     
  14. Jul 22, 2004 #13

    NateTG

    User Avatar
    Science Advisor
    Homework Helper

    If you allow for rearangements between cuts, then the maxium number of pieces of a convex solid after [tex]n[/tex] cuts is going to be [tex]2^n[/tex]. This is an easy consequence of the fact that cutting any convex solid with a planar cut will always generate two convex solids.
     
  15. Jul 22, 2004 #14

    Njorl

    User Avatar
    Science Advisor

    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
     
  16. Jul 22, 2004 #15

    BobG

    User Avatar
    Science Advisor
    Homework Helper

    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).

    :wink: :eek: :rofl: :yuck: :frown: :surprise: :uhh: :mad: :cry: :redface: :devil: :grumpy: :rolleyes: :biggrin: :tongue2: :smile: :approve:

    Uh, sorry, too much sugar from all those brownies.
     
  17. Jul 22, 2004 #16

    BobG

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
  18. Jul 22, 2004 #17

    marcus

    User Avatar
    Science Advisor
    Gold Member
    2015 Award
    Dearly Missed

    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
     
  19. Jul 23, 2004 #18
    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.
     
  20. Jul 24, 2004 #19
    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?
     
  21. Jul 26, 2004 #20

    NateTG

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Baked brownies
  1. Baking (Replies: 44)

  2. Baked Doritos (Replies: 33)

  3. Baked potatoes (Replies: 78)

  4. Baked beans (Replies: 14)

  5. Baking Pan Patterns (Replies: 6)

Loading...