If you've ever asked a young child to cut a pie for several people, the child will likely cut a piece for himself or herself and then pass you the knife. Perpendicular cuts across the length of the pie come naturally to older children for the sake of overall efficiency. Young children may find this to be too altruistic. This puzzle is an attempt to help you recapture that inner child.
You will design a series of cuts to partition a square pie into equal pieces to share among several people. The piece finally offered to each person is called a "final piece." Here are the rules.
1. All cuts must be straight and vertical.
2. All final pieces must contain the same amount of pie volume (and, because of the first rule) the same amount of the top of the pie.
3. Child-cut rule: Every cut should yield one final piece except the last one, which yields two final pieces.
The goal, which admittedly might not occur to a child, is to minimize the sum of the perimeters of all the final pieces.
We start with the square pie in figure 1. We could cut that square pie into three pieces using two parallel cuts (A) or using two perpendicular cuts (B). Both satisfy the three rules: equal size final pieces, vertical cuts and child cut rule. Which yields a smaller total perimeter?
Solution to Warm-Up:
In the parallel cut case (A), each final piece has a perimeter of length 1 + 1/3 + 1 + 1/3 = 8/3. So the three final pieces altogether have a total perimeter of 8. Two perpendicular cuts (B) yield a first final piece having perimeter 8/3 and the remaining two final pieces each having perimeter: 2/3 + 1/2 + 2/3 + 1/2 = 7/3. So the total perimeter is 8/3 + (2)(7/3) = 7 1/3.
Now it's your turn.
1. Find a cut design obeying the above three rules that minimizes the total length of the perimeters for five pieces.
For the rest of the problems, suppose we drop the child cut rule, but keep the first two (equal volume and vertical cut).
2. How much better can you do for five pieces in this case, assuming your cuts must be parallel to the original edges of the square?
3. What about for nine pieces under the same assumption? (Hint: there is a Russian doll feeling about this one.)
4. Try five pieces again if the cuts need not be parallel to the edges of the square.
The generalization to more pieces involves some elegant recurrences. If you are looking for a challenge, give it a try.