# Treasure Hunt with Complex Numbers

Fig. 1

In his popular book “One, Two, Three … Infinity“, physicist George Gamow told a story:

Once upon a time, there was a young man who found among his great grandfather’s papers a piece of parchment that revealed the location of a hidden treasure. It read:

“Sail to ____ North Latitude and ____ West longitude where you will find a deserted island. There is a large meadow on the north shore of the island where stand an oak and a pine. You will see also an old gallows on which we once used to hang traitors. Start from the gallows and walk to the oak counting your steps. At the oak, you must turn right by a right angle and take the same number of steps. Put a spike in the ground there. Now you must return to the gallows and walk to the pine counting your steps. At the pine, you must turn left by a right angle and see that you take the same number of steps, and put another spike into the ground. Dig half way between the spikes; the treasure is there.”

So the young man charted a ship and sailed to the South Seas. He found the island, the meadow, the oak and the pine, but to his great sorrow the gallows was gone. Unlike the living trees, the gallows has long since disintegrated in the weather, and not a trace of it or its location remains.

Unable to carry out the rest of the instructions (or so he believes), the young man fell into despair. In an angry frenzy he began to dig at random all over the field. But all his efforts were in vain; the island was too big!

Needless to say, the young man sailed back empty handed. And the treasure is still there.

This is a sad story, but what is sadder still is the fact that the young man might have gotten the treasure, if only he had known some mathematics, and specifically the use of complex numbers.

How come?

Consider the island as a plane of complex numbers; Place the origins of three rectangular coordinate systems at the location of oak( $o_1$), pine ( $o_2$) and half way between them ( $o$). $\Gamma, x_1, x_2, y_1, y_2$ and $d$ are complex numbers. Notably, $d$ is the half way point between the spikes.

Fig. 2

From Fig. 2, we see that $-1 + x_1 =\Gamma\implies x_1=\Gamma+ 1,\quad\quad\quad(1)$ $1 + x_2 = \Gamma \implies x_2=\Gamma-1.\quad\quad\quad(2)$

By the fact (see Exercise-1) that

(1) The multiplication by $\bold{i}$ is geometrically equivalent to a counterclockwise rotation by a right angle

and

(2) The multiplication by $\bold{-i}$ is geometrically equivalent to a clockwise rotation by a right angle, $y_1-(-1)=\bold{i}\cdot x_1\implies y_1=-1 + \bold{i}\cdot x_1\overset{(1)}{=}-1+ \bold{i}\cdot(\Gamma+1)=-1 + \bold{i}\cdot\Gamma + \bold{i},\quad\quad\quad(3)$ $y_2-1=-\bold{i}\cdot x_2\implies y_2=1+\bold{(-i)}\cdot x_2\overset{(2)}{=}1-\bold{i}\cdot(\Gamma-1)= 1-\bold{i}\cdot\Gamma + \bold{i}.\quad\quad\quad(4)$

Since the treasure is halfway between the spikes, we have $y_1-d = \frac{1}{2}s,\quad\quad\quad(5)$ $d-y_2 = \frac{1}{2}s.\quad\quad\quad(6)$

Subtracting (6) from (5) gives $y_1+y_2 -2d=0.$

It means $2d = y_1+y_2 \overset{(3), (4)}{=} (-1 + \bold{i}\cdot\Gamma+\bold{i})+ (1-\bold{i}\cdot\Gamma + \bold{i}) = 2\bold{i}.\quad\quad\quad(7)$

Therefore, $d = \bold{i}.\quad\quad(8)$

We see that the unknown position of gallows denoted by $\Gamma$ fell out in (7), and (8) tells regardless where the gallows stood, the treasure must be located at the point $\bold{i}$ of rectangular coordinate system with origin $o$.

And so, had the young man done the simple math shown above, he could have looked for the treasure at the point indicated by the cross in Fig. 1 and found it there.

Exercise-1 Prove

(1) The multiplication by $\bold{i}$ is geometrically equivalent to a counterclockwise rotation by a right angle.

(2) The multiplication by $\bold{-i}$ is geometrically equivalent to a clockwise rotation by a right angle.

Exercise-2 Locate the treasure using a computer algebra system (hint: see “A Computer Algebra Aided Proof in Plane Geometry“).