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.)

This set of 'buns' is quite interesting.

The radii are 3, 4 and 5 and their areas are the squares 9, 16 and 25 (times pi) units. 

I drew the three circles in ACAD so that the circumference of each pair of circles touched.  A triangle formed using their respective centres had sides of 7, 8 and 9 units; dismissing the 8, the 7 and 9 are the differences between the areas of the buns.

But that's a little beside the point.  I drew tangents to centres, created common tangents and measured angles but nowhere could I find an angle of 78.75 Deg (7/32 of 360 Deg).  I was looking for some construction that might have yielded something like a line that was both a tangent and a radius to two circles; fruitless.

The closest I could achieve was the included angle of the 7 and 8 unit sides, being 73.4 Deg.

 

Posted by brianjn on 2004-05-17 01:52:27