\section{Pulling back on a pedal}

Imagine that I am steadying a bicycle to keep it from falling over,
but without preventing it from moving forward or back if it decides that it
wants to.
The reason it might want to move is that there is a string tied to the
right-hand pedal
(which is to say, the right-foot pedal),
which is at its lowest point, so that the right-hand crank
is vertical.
You are squatting behind the bike, a couple of feet back,
holding the string so that it runs
(nearly) horizontally from your hand forward to where it is tied
to the pedal.

\begin{problems}
\item
Suppose you now pull gently but firmly back on the string.
Does the bicycle go forward, or backward?
Remember that I am only steadying it, so that it can move
if it has a mind to.
No, this isn't a trick;
the bike really does move one way or the other.
Can you reason it out?
Can you imagine it clearly enough so that you can {\em feel}
the answer intuitively?
\item
Try it and see.
\end{problems}

John Conway makes the following outrageous claim.
Say that you have a group of six or more people,
none of whom have thought about this problem before.
You tell them the problem, and get them all to agree to
the following proposal.
They will each take out a dollar bill,
and announce which way they think the bike will go.
They will be allowed to change their minds
as often as they like.
When everyone has stopped waffling, you will
take the dollars from those who were wrong, give some
of the dollars to those who were right,
and pocket the rest of the dollars yourself.
You might worry that you stand to lose money if there are
more right answers than wrong answers, but
Conway claims that in his experience this {\em never} happens.
There are always more wrong answers than right answers,
and this despite the fact that you tell them in advance that
there are going to be more wrong answers than right answers,
and allow them to bear this in mind during the waffling process.
(Or is it {\em because} you tell them that there will be
more wrong answers than right answers?)
%
%Fluff.  PGD January 2006
%
%\begin{problem}
%Try this trick for yourself, and see if it works.
%Of course the fact that this trick invariably works for Conway doesn't
%mean it will {\em necessarily} work for you; mileage may vary.
%In other words, do not expect anyone to cover your
%losses if you mess this trick up.
%\end{problem}
