A Complex Treasure

 In the wake of this recent work, there has been a resumption of the sporadic debate over which imaginary numbers are real (pun intended) or not. Not so much by mathematicians, who tend to believe that they discover, rather than inventing, the structures of thought they employ, but physicists, who tend to think of the math they use as a different order of being than the actual physical world they describe.

From a pragmatic point of view it doesn't matter. If you have various mental tools, use the one that works best. The best meta-rule is Ockham's Razor.

In that spirit, I thought I would revisit a cute little puzzle that is often used to show how complex numbers are a bit more simple, nifty, or appropriate than just plain pairs of coordinates for solving a problem. The problem has nothing to do with the deeper properties of the quantum field; it's about a treasure map. 

It goes like this:

You have come into possession of a chart from a pirate long dead. It shows the location of a small island in the Spanish Main, and on the back are directions to find the treasure. "Pace from the cairn to the ash tree, turn right and pace the same, driving a stake. Again from the cairn to the bay tree, turning left and pacing the same, and a stake. Midway between your stakes is the treasure to be found."

You get to the island and find the ash and bay trees. Unfortunately in the meantime the island has been visited by a gang of kleptotaphophiles, who stole all the stones of the cairn, leaving it unmarked. How can you find the treasure?

Or, since this is really a math puzzle, how can you use complex numbers to prove your solution is correct?

The trick with a math problem is often to gain an intuition as to what the solution is, and then use the tools you have to show it is right. 

In my experience you are much more likely to be taught how to use the tools, and less likely to be taught how to gain an intuition. With that in mind, let's look at the map.

On our island it just so happens that the bay and ash trees are on an exact east-west line, and are exactly two furlongs apart. (A mathematician would phrase that, "Without loss of generality we may assume ..."). 

Now here's the essence of gaining an intuition. Take boundary cases, cases where something goes to 0 or 1, anything to simplify the problem without changing it overall. In the case of the treasure map, for example, start with cases where you can easily see the answer without doing any numerical geometry.

Let's call the point midway between the trees "Zero" and see what would happen if it were the cairn.

Pacing and staking, we get a simple diagram that shows the treasure would be exactly one furlong due south of Zero. Okay, what's another way to simplify?

Just pick the cairn as being one of the trees. Then the pacings for that tree are of zero length, and you drive the stake right there. When you pace the other tree you get:

Whaddya know, one furlong due south of Zero. And obviously it works the same starting from the ash tree.

What else can we eliminate? How about the string between the stakes? If they are in the same place, the centerpoint, and thus the treasure, will be right there. Put the cairn at one furlong due north of Zero:

Yep, it's at the same spot, one furlong south. And for a final flourish, what if we put the cairn right on the treasure?

Well, by now you will have gained the intuition that wherever you put the cairn, the treasure will be in the same place. And you even know where the place is. 

You now have your conjecture, and can prove it fairly straightforwardly using complex numbers.

But while you are doing that, I will have jumped to a conclusion, dug up the treasure, and escaped.