Puzzling Adventures: The Mother Lode of All Inheritances

A great-grandfather makes his family work for their land and fortune

Image: Cloe Liane Shasha

Readers of this column will recall from last month Sir Birnie, the aristocratic landowner whose bequests used geometry to define his heirs' inheritance. When his first great-grandchild, Emma May, came of age, there was again great excitement. The fortunes of the family had risen and his eldest grandchild, Johanna, had been able to buy out the southern neighbor, so the entire property shown in the original map now belonged to the family—although only after the former neighbor to the south had heavily logged his portion. YZ was now clearly marked and the property extended a full kilometer to the east of the intersection and a half-kilometer to the west.

See image here

The new letter read as follows:

My dear first great-grandchild,

With luck, your parents' generation has regained lost land and fortune. To do that, my first grandchild would have had to be an excellent geometer. You, however, have the possibility to capture a vastly larger treasure, enough to put you among the first families of the Kingdom. For this, you will need to be an even more formidable geometer. With that hope in mind, I will call the XR line Euclid and the YZ line Pythagoras in the following map.

See image here

Now measure out a point W that is 400 meters to the west of the intersection and E that is 200 meters to the east of the intersection.

If you walk north from the intersection of Euclid and Pythagoras, you will find a large pine P, and if you keep walking you will find a large maple M. Let L be the point of intersection between WM and EP and let R be the intersection between WP and EM. Now draw a "crossing line" through L and R until it reaches Euclid. That is where the treasure lies.

Emma May walked Pythagoras, but found several pines and maples as well as many stumps from the neighbor's logging. She was worried that she wouldn't know where to dig.

After some thought, however, she determined the exact spot where the treasure should be. Where was it and how did she determine the location?

Dennis Shasha is at the Courant Institute of Mathematical Sciences, New York University. His most recent puzzle book, Puzzles for Programmers and Pros, was published in May 2007 by John Wiley and Sons/Wrox.

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1. 1. kellyeliz 09:13 PM 7/3/08

I am absolutely livid as an artist to see the illustration that SciAm has used for this article. The perspective is completely flawed, the colouring is rough, there is no shading whatsoever. Was there a contest within the office to draw an illustration for the article?
I'm horribly disappointed.

2. 2. a4psciam in reply to kellyeliz 03:14 PM 7/5/08

It seems more likely, given the name of the artist, that there was a contest AT HOME; and that wife, daughter, mommy, sister, aunt or niece was the winner! Talk about naif.

3. 3. waterbergs 02:52 PM 7/6/08

The solution to this one is simple once one assumes that there is a solution! If this is so and the searcher can't determine which is the right pine and maple,but can still find the treasure, the site of the treasure must be independent of M and P. One can then choose M at infinity, EM and WM are then vertical. WL is then clearly twice EM, and so the treasure must be 600m east of E, i.e. 800m east of Pythagoras - 1 minutes work.

4. 4. waterbergs in reply to waterbergs 02:54 PM 7/6/08

Opps, I meant WL is clearly twice ER.

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