Solution: Pirates

We now publish our solution to the Pirates divertimento.

The divertimento:

Legend has it that a pirate sailing ship arrived at a remote island pursued by Spanish galleons, where the captain hid the loot he was carrying on board, the fruit of his raids. He landed with his henchmen on a deserted beach where there was a palm tree and a rock. He stuck his sword into the beach and, from there, walked in a straight line to the palm tree. While there he turned 90º anticlockwise and walked (always in a straight line) the same distance as before, where he drove in a stake. He returned to the position of the sword and walked, also in a straight line, to the rock and, turning 90º in the same clockwise direction, repeated the same distance, and in the same way, to a point where he drove another stake. He looked for the middle point between the two stakes and there he ordered the treasure to be buried. He immediately had the sword and the stakes picked up to protect the exact location of the treasure. He returned to the ship with his crew and continued his misdeeds… until ten years had passed. Then he returned to the island and unearthed the treasure.

How did he manage to locate the treasure with the help of only the location of the palm tree and the rock, which were still there?

The Solution:

Consider a Cartesian coordinate system, so that the palm tree is at \((-a,0)\) and the rock is at \((a,0)\).

The sword is at any point \((p,q)\). Following the instructions, the first stake is at \((-a+q,-a-p)\) and the second at \((a-q, -a+p)\). The midpoint of these two, where the treasure is, is \((0,-a)\), which is independent of \((p,q)\).

It is therefore enough to walk from the palm tree to the rock, reach the midpoint, turn 90º clockwise and walk the same distance in this direction.