A young man was going through the attic of his grandfather's house and found a paper describing the location of a buried treasure on a particular island. The note said that on the island one would find a gallows, an oak tree, and a pine tree. To locate the treasure one would begin at the gallows, walk to the pine tree, turn right 90 degrees and walk the same number of paces away from the pine tree. A spike was to be driven at that point. Then return to the gallows, walk to the oak tree and turn left 90 degrees and walk the same number of paces away from the oak tree. Drive a second spike in the ground. The midpoint of a string drawn between the two spikes would locate the treasure.
 
The young man and his friends mounted an expedition to the island but found the oak tree and the pine tree but no gallows. It had eliminated years ago without a trace. They returned home with the map below and no treasure. Show them where to look for the treasure.
Island Treasure
 
Richard Francisco & Molly McKee
 
Map of the Island
Choose an arbitrary point for the location of the gallows
Following the instructions provided, first draw a path from the gallows to the pine tree. We now need to turn right 90o and walk the same distance away from the pine tree.
 
This can be achieved in Geometer Sketchpad by selecting the segment from the gallows to the pine tree and then rotating this segment 90o.
Next draw a path from the gallows to the oak tree. We now need to turn left 90o and walk the same distance away from the oak tree.
 
Select the segment from the gallows to the oak tree. Since we are turning left this time instead of right, we must rotate the segment -90o or 270o.
Now draw the segment between the two points.
The midpoint of this segment will be the location of the treasure.
Notice that no matter where the gallows are located, the treasure is always buried in the same location.
 
Finding Buried Treasure
 
Geometer Sketchpad version
Our GSP sketch appears to show that the treasure will be located at a static point regardless of the location of the gallows! However, we must still prove this result! We will use the complex plane to elucidate the ninety degree rotations, there are known methods to compute rotations (scaling by i, etc) in the complex plane. Let ∂ be the location of the gallows, some complex number (remember that complex numbers are of the form a+bi, but for this proof we do not need to decompose ∂). We are going to locate the trees at specific values that make the calculations easier, namely the pine tree as the complex number -1+0i, and the oak tree as 1+0i. The mapping of these trees to these points on the real axis will not change the solution to the problem, as we can always map the layout of the island to these dimensions. The complex plane provides simple calculations for the distance between complex numbers: The distance between the pine tree and the gallows is -1-∂ and the distance between the oak tree and the gallows can be represented as 1-∂. To rotate the distance to the pine tree 900 clockwise, we multiply by -i, since -i is the appropriate rotation to move in this direction in the complex plane. Similarly, we will multiply the distance 1-∂ by i, since i will rotate 900 counterclockwise in the complex plane. Therefore we have that the first spike will be driven into the ground at (-i)(-1-∂) = i + i∂. Similarly, the second spike will be driven at (i)(1-∂) = i - i∂. To drive a stake in the midpoint of these values, we add the complex numbers and divide by 2:  Therefore, for any location of the gallows, ∂, the midpoint of the respective segments will ALWAYS be mapped to the value i, which is a static location in the complex plane!!