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.)

 

Ignore Pi, as it is a constant for all buns, they each have 9, 16 and 25 units of area, 50 in total. For equal shares, each of 4 people must have 12 1/2 units. This means that the smallest must remain intact. The largest will yield 2 equal shares of that amount when cut across the diagonal.  Therefore the remaining bun needs to be divided such that part of it, along with the smallest bun, provides an equal share whilst what remains is an equal share. Such division would be in the ratio of 7:25 so that the smaller bun and 7/32 of the middle sized bun form the third equal share whilst the remaining 25/32 becomes the fourth.<o:p></o:p>

(From the centre make a radial cut, determine a place on the circumference for a quarter, and proceed to determine an eighth and then a sixteenth within the sector furtherest from the cut, and similarly determine the thirty-second and CUT).<o:p></o:p>

The question did not mention the number of cuts; this would take three. For those wanting two, a second cut would need to divide the middle sized bun into segments of area of the ratio of 7:25.

 No matter how the cuts are made, the minimum number of pieces to satisfy the question is 5.

Posted by brianjn on 2004-05-08 00:29:48