The Khan Academy has a 17-minute “Alien Abduction Brain
Teaser” video that offers an interesting example of mathematical induction.
Salman likes to do these lessons himself, drawing sketches
on a virtual blackboard, producing inexpensive animation that requires no film
editing before publication on YouTube. His voice always sounds comforting. He would be a good classroom math teacher if
he wanted to be.
In this problem, ten people have been abducted, and each
represents the welfare of one-tenth of the world’s population. The aliens put a
hat, either blue or purple, on the back
scalp of each person, who can see those in front but not behind. The people (as
in the IFC film “Exam”) have 24 hours to come up with a strategy. Each person, in reverse order, will be asked
the color of his hat. If he guesses right, he and his people are saved, from being converted to mushroom or fungus farms. (This
sounds like a notorious Bible story, doesn’t it.) There is a way, involving counts of even or
odd occurrences of a color, to answer in such a way that all people but the
rear person can survive. The rear person
is exposed to a possible Isaac-like “sacrifice”.
The puzzle would work, of course, for any finite number of
people.
Why is this important to me?
“Mathematical induction” (here combined with a little number theory) is
certainly an important way to prove many mathematical propositions. More nebulous is inductive reasoning, and statistical
prediction, and settings of the “scientific method”. (I suspect Khan has videos on these.)
In my own life, in a few areas (“self-broadcast” and
personal relationships), I certainly have experienced a dose of having to listen,
sometimes after the fact, of what people “want” from me, and thing is morally “right”
for me. Some of this has to do with my
own dependence on “fantasy” and interest in self-broadcast without the expected
willingness to accept intimate or personal contact with other people on their
terms rather than mine – before being noticed globally. I can extrapolate the “moral principles”
involved, but then I start noticing that people want contradictory
results. Even so, “induction” from the
possible color of my own “hat” can cover a lot of moral territory.
There are ways to complicate the experiment. Imagine that all the abductees are young men,
and are each expected to step into or dive into separate swimming eddies, some
of which contain chemical depilatories. Afterwards,
they can’t see themselves or those behind, but can still see those in
front. (Believe it or not, a health club
in Dallas had an “accident” like this with its whirlpool in the 1980s.)
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