Pirates

The Workings:

Today’s divertimento features a pirate who uses mathematics to hide, and then retrieve, his treasure. A good pirate captain had to know enough about the geometry of the sphere to be able to sail without getting lost, but there doesn’t seem to be many pirates who have made any contribution to the development of mathematics. After a brief stroll through Wikipedia, the name of the Englishman Sir Walter Raleigh (1552-1618), sailor, privateer, writer, courtier and politician, who popularised tobacco in Europe, after an initial attempt to colonise what is now North Carolina (whose capital is named Raleigh in his honour), comes to mind. A hero to the English (although he was executed on the orders of James I Stuart and John Lennon called him a “prick” in the song I’m so tired), Raleigh is the epitome of pirate to the Spanish: he took part in the sack of Cadiz in 1596 and seized Guyana from Philip III.

And the mathematics? Tradition holds that Sir Walter asked Thomas Harriot, whom he had hired as a mathematical tutor and assistant navigator, how he could calculate the number of cannonballs in a stack, which led Harriot to wonder how to stack equal spheres, be they cannonballs, oranges or tennis balls, in such a way that they would take up as little space as possible. Harriot wrote about this to Johannes Kepler, who concluded that the best way to do it was the way fruit sellers did it: by arranging the spheres in a first layer so that the centres form squares, and the same in the second layer but shifting the centres so that each sphere in the second layer is tangent to four spheres in the second layer, and so on. The problem of proving this to be true has since become known as Kepler’s conjecture.

The Kepler conjecture has an interesting history per se and has been definitively resolved in the paper A formal proof of the Kepler Conjecture, by T. Dales, M. Adams, G. Bauer and TD Dang, published earlier this year 2017 in the journal Forum of Mathematics. Another recommended reading on this topic is the post Maryna finds the magic function in this very blog.

The Fun:

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?

 

 

 

 

Solution:

We encourage the readers to try to solve the divertimento for themselves. Whether you succeed or not, you can always consult the solution in this link.