Question about cutting cake?

Question

1. After a cake (any arbitrary rectangle) has been cut into three equal pieces, as shown below, a hostess
discovers that four people each want an equal share of the cake. How can she make one more straight cut
so that each of the FOUR people get the same amount of cake? Note: The WHOLE cake needs to
be eaten.

This is the full question, as I can only have 160 characters above. I also have another question:

2. Suppose now that n people want to equally share the cake where n is NOT a multiple of 3. Can the
hostess still make 1 straight cut? If not, what is the least amount of cuts that she needs to make?
Hint: There will be two different formulas.

 

Thank you!

 

Answer 1

Kind of hard to describe without the picture.  I'll assume that the 3 cuts are going vertically parallel to each other other (so the width of the cake is cut into thirds, but the height is intact).

If we now want to share the cake among 4 people think about a new cut going horizontally across.  If we do it right in the middle we get 6 equal pieces (not what we want), but let's say we did it slightly off-center (but still nice and horizontal).

If we did it 1/3 of the way up from the bottom the top pieces are 2/3 the size of the original 3 pieces and there are 3 small pieces which are each 1/3 of the size of the original 3 pieces, but together that would be 3/3 = 1 original sized piece which is slightly too much.

Think about what would happen now if we cut 1/4 of the way up from the bottom.  3 big pieces, 3 small pieces, but the small pieces together should be exactly the same size as the big ones are.  Each is 3/4 of the size of the original 3 pieces.

 

For part 2, since we're ignoring multiples of 3, either n is a multiple of 3 plus 1 (like 4, 7, and 10) or it's a multiple of 3 plus 2 (like 5, 8, and 11).  Think about what would need to be done for each of those numbers (start small) to make the pieces even out.  Then try and generalize the process to make the formulas.

Answer 2

If the current cuts are parallel, then 1 perpendicular cut that takes 1/4 of the total cake will divide the cake into fourths -- 1/4 is new (with 3 new pieces), and 3 old slices are reduced to 1/4 of the total (when they each used to be 1/3 of the total).

 

Note that "the same amount of cake" does not mean that the new cut produces one additional piece.

 

To start with three pieces and make fifths, two new cuts are needed.  They cut fifths of the total cake (as though the cake had not yet been cut) and reduce the one-third-cake pieces to one-fifth-cake pieces.  Note that the new cuts may be on the same side or on opposite sides (this now makes the problem more interesting).

 

Why “not a multiple of 3” and “two different formulas” for the least amount of cuts.  Let’s see –

      For a total of n people, the amount of new cuts (c) is:

    c = (n-3)     [to be minimum, must be made on the sides]

  

But if n is a multiple of 3, the minimum number of cuts is less (because they may be made from the middle).  Here are examples:

      People   new cuts

         6            1

         9            2

        12           3

        15           4

        18           5

     this formula is:

      c = (n-3)/3

 

The solution is:

    c = (n-3)    [for n not a multiple of 3]

    c = (n-3)/3    [for n is a multiple of 3]

 

PLZ note that, not seeing a picture, I assumed perpendicular cuts of the rectangular cake.

 

- - - - - EDITS made due to Comments.

THX for clarifying problem.

 

Here are the minimum number of cuts that can turn 3 pieces into n equal parts (regardless of number of pieces):

 

      people(P)   min. additional cuts(C)       num. of actual pieces

         3                   0                                      3

         4                   1                                      6

         5                   2                                      9

 

         6                   1                                      6

         7                   2                                      9

         8                   3                                     12

 

         9                   2                                       9

        10                  3                                     12

        11                  4                                     15

 

        12                  3                                     12

        13                  4                                     15

        14                  5                                     18

 

Now, a formula for this:   C = INT((P-3)/3)+MOD(P,3)

          where INT is integer division (throw away the remainder) and MOD is just the remainder