## Puzzle #7 (Follow-up): Treasury Island, Part 1

In Puzzle #7: The Mysterious Islands, I told the story of two islands where each individual’s remaining lifespan can be perfectly predicted.

The lifespans on Uniformia are evenly distributed, and their average remaining life expectancy will gradually decline over time.

Ignoramia is different.  Their average life expectancy is actually going to increase for a few years, but only Franz knows it.

He uses this knowledge to gain power over his fellow islanders.  They will credit Franz for the improved life expectancy, but he did not cause the change.  He was like an ancient astronomer predicting an eclipse to everyone else’s amazement.

The Ignoramians are so pleased by the extension of their average life expectancy that they ignore its alarming cause: the die off rapidly.  At the beginning of the story, there are one hundred Ignoramians, with an average life expectancy of 1.6 years.  The lifespans of the Ignormanians were distributed as follows:

 Lifespan (years) Number of Ignoramians 1 70 2 20 3 5 4 2 9 3

There are 30 Ignoramians with lifespans of two years or more.  The average of their initial lifespans is 3 years.  If you combine this with the one-year lifespan of the other 70 islanders, you get an overall average life expectancy of 1.6 years.  The unlucky 70 pull down the average.

After a year, the first 70 have perished.  But for the survivors, this is great news.  True, the average life expectancy of the 30 declines from three years to two, but the other 70 are gone and no longer suppress the average.  So the average life expectancy improves from 1.6 years to 2 years.  This is cause for celebration on the island.

The Ignoramians are happy because they only think about a single statistic.  As long as this number improves, they will ignore their own rapid extinction.

Solutions

I also posted the puzzle at Wilmott.com, where they tore through it faster than piranha on a drunken squid.  On this blog, the first solution came from Jamie Wolfe, an old friend.  ΣΗΦ used linear algebra, which got me thinking about a way to use matrices to explain the relationship between average life expectancy and the mortality distribution.   Aman  gave a very clear explanation of how to solve the problem.

There is more than one solution to the puzzle.  For example, instead of three deaths in the ninth year, we could have one each in years eight, nine and ten.

In Part 2 of this post, I will discuss what the puzzle is really about.