Treebeard's Stumper 1 November 2002

Fifteen Men on the Dead Man's Chest

Arrgh, there be fine pirates a' plenty at yesterday's Dunn Middle School Treasure Island Halloween party! Now me fifteen mateys be standing around this dead man's circular chest giving the evil eye between each pirate and every other. Ye can almost see the lines cut across that circle between those swabs! For fifteen pirates, how many lines connect them, and how many spaces be marked out there on the treasure chest? Avast me, ye need a boon? Start with two and three and then four poor souls and make your count to discover the secret keys, but don't heave-to too soon or ye be scuttled!

Here are 5 pirates giving each other the evil eye across the dead man's chest. That's easy. I count 10 lines and 16 regions. But what about 15 men (or any number) around that same circle?

It's fun talking like a pirate! See the funny Dave Barry column (8sep02) for inspiration and this Pirate Jargon page for vocabulary.

At Dunn Middle School, we all read Robert Louis Stevenson's Treasure Island last summer, and we just had our Treasure Island Halloween festival. We all made costumes based on the book, the kids created booths and activities, and we invited younger kids from nearby Family School. Last year we did the same with The Wizard of Oz. (I've heard rumors about reading Tom Sawyer next year, but I'm promoting Alice in Wonderland for the costume possibilities!) This was the perfect middle school Halloween activity. That's me in the photo with a wig and a few dozen old AOL Cds sewn to my shirt: "Arrgh, I be a music pirate!" (And I made sweet noise and rainbows, like a walking disco ball!)

Of course my stumper is really a problem in combinatorial geometry. How many chords can be drawn between 15 (or n) points on a circle, and how many regions do they define? There's no requirement that the points be equally spaced. In fact, you might miss a region if you do space the points equally since any three lines intersecting at a single point will obscure a potential region. Consider that a constraint.

Here are a few related stumpers:

• By lining up the points just so, what's the fewest number of regions you can define?
• I got the idea to make a string art copy of the solution with nails and kite-string. It's too awkward to tie every line separately, but is it possible to loop a single string around all 15 nails to show every line without going over the same line twice?
• What is the real origin of that great pirate song from the book:

  Fifteen men on the dead man's chest --
  Yo-ho-ho, and a bottle of rum!
  Drink and the devil had done for the rest --
  Yo-ho-ho, and a bottle of rum!
  

Answer

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