Treasure Island
22 Oct 2008
Puzzle

An old parchment has directions to a treasure chest buried in an island:

“There is an unmarked grave and two tall oak trees. Walk from the grave to the left tree, counting the number of steps. Upon reaching the left tree, turn left by 90 degrees and walk the same number of steps. Mark the point with a flag. Return to the grave. Now, walk towards the right tree, counting the number of steps. Upon reaching the right tree, turn right by 90 degrees and walk the same number of steps. Mark this point with another flag. The treasure lies at the midpoint of the two flags.”
A party of sailors reached the island. They find a pair of tall oak trees merrily swaying in the wind. However, the unmarked grave is nowhere to be found. They are planning to dig up the entire island. It'll take a month. Can they do any better?

Source

Heard from a fellow student at IIT Delhi in 1991--1995. In Jan 2011, I learnt that this puzzle appeared in the book One Two Three . . . Infinity: Facts and Speculations of Science (352 pages, 1998) by George Gamow.

Solution

(by Yuri Robbers) It does not matter much where the grave is! There are only two places where the treasure can be. If you construct a square with the two trees at two diametrically opposite corners, then the treasure can be on either of the other two corners. There is a degenerate case where the grave would be on the same line as the trees. in this case the treasure would be buried exactly between the trees. This degenerate case is, however, excluded by the wording of the instructions, since in that case there would not be a left and a right tree.

Proof with vectors: Let the unmarked grave be the origin. Let A and B denote vectors to the two oak trees. The first flag is at A + iA. The second flag is at B - iB. The mid-point is ½ (A + B) + ½ (A - B)i, which may be rewritten as A + ½(B - A) - ½(B - A)i. Note that -iX denotes counterclockwise rotation of vector X by 90 degrees. Now the last expression has three summands, which can be visualized as follows: "A" denotes moving from the unmarked grave to the left oak tree, "½ (B - A)" denotes moving half the distance between the two oak trees from A to B, and "- ½(B-A)i" denotes turning counterclockwise by 90 degrees and moving the same amount as the previous step. Finally, the unmarked grave could be on either side of the line joining A and B. So there are two points where the treasure might be.