Finding Treasure with a Gallows & Two Trees

[jackiefrost]

I came across this amusing treasure hunt problem in a book on the history of complex numbers. The author got it from George Gamow's classic, "One, Two, Three, Infinity". I'll paraphrase.

Directions:
Long ago pirates buried a treasure on a certain desolate island. No longer desolate, while vacationing on the island you discover the original directions for locating the treasure. The directions only involve a gallows (pirates, you know) and two conspicuous trees. One tree is an oak, the other a pine.

The directions for locating the treasure are as follows:
Walk from the gallows straight to the oak, counting your steps. Turn exactly 90 degrees to the right and walk in a straight line the counted number of steps. Drive a stake at that location. Now, go back to the gallows and walk to the pine tree, again, counting your steps. At the pine tree turn exactly 90 degrees to the left. At the pine walk the distance that you just determined by step counting. Drive a second stake. The treasure is buried under the midpoint between the two stakes.

That's pretty easy except for one ugly problem; you know precisely where the two conspicuous trees are located on the island but any trace of a pirates' gallows has long since disappeared. Can you still find the treasure with only the trees and the directions to go on?

[The intention of both authors was convey awe at the "magical" utility of complex numbers but when a treasure is involved I guess anything goes]

 

[robert Ihnot]

It 's found in the above book, and on the internet. Concerns the fact that multiplication of a point in the complex plane connected to the origin by i turns it 90 degrees left, and by -i turns 90 degrees clockwise. Place the elm and oak conveniently at (-1,0) and (1,0).

Before the thread vanishes, I will indicate the method. Suppose we take the one connected with oak at (1,0) and the right turn. That's -i(G-1), but we want that connected to the point (1,0), so we get -i(G-1)-1. The other is i(G+1)+1, and the midpoint is {-i(G-1)-1 +i(G+1)+1}/2 = i.

Which shows that to know the position of the gallows is unnecessary.

 

[Chris Hillman]

Incidently, this is Gamow of Big Bang fame. He was also a famous prankster and a gifted expositor.

 

[robert Ihnot]

I wish to add, he defected from the Soviet Union in 1934 and was once described by a United Press reporter as "The only scientist in America with a real sense of humor"