Solution To: You Can Have Your Cake And Eat It Too...Once You Figure Out How To Cut It

Here was the original problem:

 Here is the solution:

Let's start with where is says "Cut the cake so that the number of pieces without frosting is equal to eight times the number of pieces that have frosting on three sides".  First we need to figure out how many pieces with have frosting on three sides.  If you think about a cube, the only pieces that would have three outer sides would be the corner pieces.  There are 8 corner pieces in total (four on top and four on bottom).  So, if the number of pieces without frosting equals 8 times the number of pieces with frosting on three sides (which is 8), then the number of pieces without frosting is equal to 8 times 8 which is 64.  Right now we have two of the five questions answered.

This is where things get a wee bit complicated.  Perhaps there is an easier way to solve it that I am just not seeing right now...but this is how my brain processed it.  If you are not a very visual person, you may have difficulty understanding how I solved it.

There are 64 pieces without frosting.  These 64 pieces form a cube on the inside of the cake.  We need the dimensions on this cube.  This cube, we could say, has a volume of 64.  The volume of a cube would be equal to the length of one side cubed.  Since we know the volume is 64, and we want to know the length of the sides, we (working backwards) take the cube root of 64 which is 4.  So each side of the inner cube consists of 4 pieces of unfrosted cake.

Now, this inner cube has a layer of 1 piece of cake surrounding it on all sides.  This is the outer layer whose pieces contain frosting.  One thing we need to know is, how many pieces of cake contain just one side with frosting.  The pieces that contain just one side of frosting are those in the center of each face of the cube (the ones not along the edge).  In other words, it would be the cake that sits directly on top of the face of the smaller, unfrosted inner cube.  One face of the inner cube has an area of 4 times 4 which is 16.  This means that there are 16 pieces of cake on each face with just one side of frosting.  A cube has 6 faces, so in total, there are 96 pieces with just one side of frosting.

Next, we need to know how many pieces have exactly two sides with frosting.  These are going to be the edge pieces of the cake (except for the corners).  If you can visualize the cake thus far, you would see that there are 4 edge pieces between each corner piece.  Now each face has four edges.  So four edge pieces time four edges would equal 16 edge pieces on each face.  There are 6 faces, so 16 times 6 is equal to 96.  BUT, if you think about the cake, each edge piece is shared by 2 faces, so we counted each edge piece twice.  Dividing 96 in half gives us 48, which is the number of pieces with 2 frosted sides.

Here is what we have so far:

Mathematicians served cake without frosting:  64
Mathematicians served cake with only 1 side of frosting:  96
Mathematicians served cake with exactly 2 sides of frosting:  48
Mathematicians served cake with 3 sides of frosting:  8

To figure out how many mathematicians were there in total, just add up the number of pieces of cake served.  This would give you 216.  So there were 216 mathematicians at the banquet.

Phew!  Are you tired?  Do you have a headache?  Take a couple of days to recover and come back next week for some more punishment!  :)

Sincerely,
The Math Freak

Oh, and congrats to anyone who solved it.  You are well on your way to becoming a math freak!