Many years ago, a large jetliner crashed into the ocean. No one could ever find the plane or any of its passengers or crew. The outside world did not know that there were many survivors. They made their way to two separate islands.
The islands, known as Uniformia and Ignoramia, were uninhabitated before the survivors arrived. They made pleasant lives for themselves. There is always plenty of food and water. The weather is always good. No one ever leaves, there are never any newcomers, and there will never be a rescue. By now, everyone is too old to reproduce. There are never any accidents, murders, or suicides.
People on both islands die only from natural causes, but in a very unnatural way. If anyone dies in a particular year, it happens at noon on September 1st. It is even possible to find out exactly when each person will die, but most would rather not know.
It is now the afternoon of September 1, 2012. One hundred people remain on Uniformia. They will die at a constant rate over the next ten years: ten in 2013, ten in 2014, and so on through 2022. The average life expectancy of the people on Uniformia is now 5.5 years. Their life expectancy will decline as each year passes.
There are also one hundred survivors on the island of Ignoramia. Their easy island life has never demanded abstract thinking, which is good since most Ignoramians hate mathematics. In particular, most believe that algebra is unnecessary.
Franz is the only Ignoramian who likes math. He calculates the average life expectancy of the people on his island. It is only 1.6 years. This is alarming news to the other Ignoramians.
Franz offers them a deal. If they put him in charge of the island, he promises that next year the average life expectancy will improve to 2 years. The year after that, it will be 3 years, and then 4, and then 5.
The Ignoramians do not understand how this is possible, but they trust Franz and agree to his deal.
As the years pass, the average life expectancy changes exactly as Franz promised:
| Date (September 1) | Average Life Expectancy (years) |
| 2012 | 1.6 |
| 2013 | 2.0 |
| 2014 | 3.0 |
| 2015 | 4.0 |
| 2016 | 5.0 |
The Ignoramians are grateful to Franz. What did he do to improve their life expectancy?
Copyright 2012. All Rights Reserved.
The Well-Tempered Spreadsheet 
There are many ways to increase life expectancy. I would guess that Franz had to decrease the life span of people that were going to die sooner. He could also improve overall standards of living each year. I am curious to learn the correct answer.
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Thanks Ivan. Remember, Franz cannot change the lifespan of any individual since these are fixed.
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Can people move btwn the two islands? If so, swap them.
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No, they can’t do that.
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Is the average increased due to surviving outliners (few people living longer than expected)?
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You are getting warmer.
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How can you have outliers if you know exactly when each person will die?
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My next guess is that he used Mode or a Weighted Average which changes as times goes by and can increase.
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The average life expectancy is a weighted average.
Can you work out exactly what happened?
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Win, Starting with 100 people, if 70 die year 1, 20 in year 2, 5 in year 3 and 2 in year 4 leaving three people with an average of five years left, the math works.
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By the way, I’m curious if you used algebra or some other method to find the answer.
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Will, slow response but if you remember the old puzzle about the farmer delivering apples and making three stops and leaving half his apples plus half an apple at each stop. In the end he has one apple left. The trick is to start at the end and work backwards. I did the same with your puzzle. I started with one person in 2016, math didn’t work, two people, didn’t work, three people it worked. You could argue it was algebra. I’m glad to see you have Mr Page exercising his brain as well.
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Jamie, that is correct! Thank you for working on the puzzle.
So Franz didn’t have to do anything, but he still got credit for the improving life expectancy.
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I am a little lost. Is it a simple averaging exercise or are we to assume any probability distribution?
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Technically it’s not a probability distribution, since all the dates of death can be known in advance. But the problem is to figure out the distribution of deaths that fits the average life expectancies that are given. Does that help?
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OK I got it now 🙂 By solving the linear system with LHS matrix
1 1 1 1 1
-0.6 0.4 1.4 2.4 7.4
0 -1 0 1 6
0 0 -2 -1 4
0 0 0 -3 2
and RHS vector
100
0
0
0
0
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ΣHΦ: That’s a great approach. Linear algebra can provide a concise way to describe the problem. You inspired me to work out the general formulas to find to find the mortality distribution from the life expectancies and vice versa. I will write these up when I get a chance (unless you want to do it).
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Follow-up question: What is this puzzle really about?
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Since nothing can be done about improving life expectancy. It’s all got to be naturally increasing. Total man-years now = 160, if we need avg to be 2 years after an year, it’s 3 years for those who would live at present time. so number of people to die next year = x
3(100-x) + x = 160
x = 70
and so on…
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Aman, that is correct and very clear. Thank you – Win
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Jamie, I can’t seem to reply directly to your last comment, but thanks for sharing how you solved the problem. And Mr. Page’s brain, like mine, needs all the exercise it can get. Thanks, Win
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Denote a puzzle solution by the vector “x” with five components:
x[1], …, x[4] = # of people who die in 2013, …, 2016, respectively
x[5] = # of people who die in 2017+
For example, the solution to the puzzle as stated is x=(70, 20, 5, 2, 3)
Now suppose that we allow the life expectancy in 2012 to vary, but hold all other puzzle parameters fixed. Then there are exactly two additional puzzles which admit a solution:
2012 Life Expectancy = 2.2 with solution x=(40, 40, 10, 4, 6)
2012 Life Expectancy = 2.8 with solution x=(10, 60, 15, 6, 9)
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Nice work, Elijah! Thanks for the variation on the puzzle.
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