# The Watermelon Problem

Suppose you have gone bicycle camping and carried a small, but hefty, watermelon to enjoy by a fire1. Unfortunately, the watermelon is too big to eat in one sitting. And, unfortunately, since you don’t own any one-time-use plastic, you didn’t bring anything with which to protect a cut side.  Cleverly, you think “Can I cut2 this watermelon so that I can put the remnant pieces together such that the inner fruit is protected?”. Making two cuts along planes equal distances from and parallel to any plane of reflective symmetry will do the trick3: eat the middle piece and seal the two remaining pieces together.  This has the nice property that you can choose to eat exactly the amount of watermelon you want to eat – assuming the watermelon has a plane of reflective symmetry.

Question One: Are there other ways in which this can be done? One could repeat the above procedure until there are no reflective planes of symmetry left. If the watermelon is an ellipsoid, you can certainly repeat the above for each of the three planes of symmetry, but is there a more complicated (ingenious?) way? Is there a way to create pieces that lock together?  Presumably when taking away that first slice, you could create a zig-zagged cut that would lock into itself.

Question Two: Suppose you have already started cutting and eating the watermelon before you realize that you will run into the problem of protecting the leftovers.  Are there remnants that cannot be rescued, that is, that cannot be cut to fit together nicely in a protective cover of rind (while eating only a tiny fraction more of watermelon)?

By the way, throwing watermelon rind on a fire really kills it.