Similar to last month, this week I just want to present a couple of problems to consider. Once again, neither of these is new, nor particularly difficult. But I think they are interesting, and as before, these problems have a point, being related both to each other as well as to a more detailed discussion next week.
Problem 1. On his death-bed, your grandfather tells you of a time when he and another man were captured by pirates. One night, the two men escaped overboard from the pirates’ ship, taking with them a priceless golden statue that the pirates had stolen. They eventually reached a small deserted island; however, the other man was mortally wounded during the escape attempt, and died shortly after reaching the island. Your grandfather buried the other man under a simple marked gravestone.
He also buried the golden statue– elsewhere on the island– and it has remained there to this day. Your grandfather gives you the following directions to find the statue: in addition to the gravestone, there are two other landmarks on the island, a maple tree and a palm tree. You are to begin at the gravestone, walk directly to the maple tree, turn left 90 degrees, walk that same distance again, and mark the spot with a stake. Then return to the gravestone, walk from there directly to the palm tree, turn right 90 degrees, walk that same distance again, and mark the spot with a stake. The statue is buried at the midpoint between the two stakes.
After your grandfather’s funeral, you travel to the island, and find the maple tree and palm tree… but the gravestone is nowhere to be found. How can you find the buried golden statue?
Problem 2. Suppose that you drill a cylindrical hole six inches long through the center of a sphere. That is, after drilling the hole, the distance between the two circular rims of the hole is six inches. What is the volume that remains? (Sorry, I couldn’t figure out how to work pirates into this one.)
Problem 3. What do the two problems above have in common?