Here is a good shape problem I heard about recently:

There are 3 buns with sprinkles on the top that 4 people want to share. The buns have a radius of 3 inches, 4 inches and 5 inches, and although the people know where the center of each bun is, they don't know anything else about the buns, and all they have is a knife to divide the buns.

What is the fewest number of pieces required to let each person have the same area of bun? (Note that each cut must be from top to bottom; horizontal cuts would result in uneven sprinkle distribution. The cuts don't need to be straight.)

The total area is:

 9 π
16 π
25 π +
50 π inches²

So, for 4 people, each needs ¼ the total or 12.5π inches².

I assume that these buns are cylindrical (or simply circles since we are allowed only vertical cuts).  One can simply divide the big bun into two pieces (leaving two halves of 12.5π units) and give them to two of the people.

Now, we can place the smallest on top of the middle bun, and slice AROUND the smallest bun... leaving two "smallest" buns and a ring of 4 inches outer radius, and 3 inches inner radius.  This ring has 16π - 9π = 7π area.  Cut the ring in half and give half, along with one of the "smallest" buns...  to either of the other two people.

This is at total of 6 pieces.

Posted by ThoughtProvoker on 2004-05-07 16:04:47