Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: [Phys-L] just for fun



Here's my answer to part (a) of the second question:

By way of background: When playing chess, you choose the
move that is best for you, assuming the other guy counters
with the move that is best for him ... *not* assuming the
other guy is going to help you out. This is called a
/minimax/ strategy.

In that spirit, if we want a /reliable/ strategy for figuring
out the light/switch problem, we should not assume the most
favorable conditions. We seek a strategy that works even
under unfavorable (but not unreasonable) conditions ... i.e.
a minimax strategy.

To say the same thing another way: Prudent planning requires
considering *all* the plausible hypotheses.

In that spirit, we consider the hypothesis that the bulbs
are up in the ceiling, out of reach, and that they are LED
bulbs anyway, so they don't heat up. This means that the
hot/warm/cold strategy suggested by Chris Gould -- which is
an excellent strategy under other conditions -- does not work
here.

We further make the not-at-all-worst-case assumption that
the switch/bulb mapping is one-to-one, and each switch has
a well-documented "on" state, so we are just looking for
one of the six possible permutations. That means we start
out with log2(6) i.e. 2.6 bits of entropy. We need 2.6 bits
of information to complete the task. Assuming the bulbs
have only two observable states (on versus off) the most
information we can get from a single observation is log2(3)
i.e. 1.6 bits of information. We are exactly one bit shy of
what we need.

We conclude that the original goal is not reliably achievable
unless some helpful assumptions are made.

Under mild restrictions, if we make an intelligent guess we
will be right exactly half the time.

--> If the mapping is not one-to-one, all bets are off.
This includes the case where some of the switches are
double-throw ("three way") switches. It also covers
the possibility of non-working circuit elements. In
such cases the initial entropy is dramatically higher.

Note that the information-theory entropy is *exactly* the same
thing as the thermodynamic entropy. The basic principles of
thermodynamics are not restricted to the large-N limit ... or
the large-S limit. The basic principles apply just fine to
small systems.

See also the celebrated 12-coins puzzle:
http://www.av8n.com/physics/twelve-coins.htm