I was remembering today a contest problem I encountered back in 1999-2000 about pirates and buried treasure. At first, I couldn’t remember the problem, then when I found multiple on-line versions of it (physicsforums.com, mathpages.com, Bradley University, geometer.org, the mathfactor podcast, and University of Georgia), I couldn’t remember how I had solved it! It apparently appears originally in George Gamow’s One, Two Three,…Infinity.
The essential features of the problem are that two pirates arrive on a desert island on which there are three prominent trees. Choosing the most prominent tree as their starting place, the pirates walk first to one tree, counting their steps. When they arrive at the tree, they turn 90 degrees to the right and walk the same number of steps and mark the spot.
Then they return to the starting place and walk to the other tree, again counting steps. When they arrive at the second tree, they turn 90 degrees to the left and walk the same number of steps and mark that spot. They then bury their treasure mid-way between the two marked spots.
Upon returning to the island some time later, they find that the most prominent tree that was their starting point is now gone. Can they still find the treasure?
Once I was sure I had the right problem, I remembered finally that I had originally used a coordinate proof.
The comments in the Math Factor podcast contain a nice solution using coordinate geometry with a nifty ending that makes finding the treasure fairly simple.
The solution in Gamow’s original apparently uses complex numbers to solve it.
The University of Georgia site asks students to find four separate proofs using 1) complex numbers, 2) Euclidean geometry, 3) coordinate geometry and 4) vector algebra.