Me Again,

        Sorry, I mixed my "raising hands" with"standing up" in my previous
submission.  Here is a revised version:


                              These INductive vs. DEductive problems are very
SEductive, so I'll chime in with what I hope is a PROductive answer (or at
least one that can be consider a "serious attempt".

As Michael O'Hara points out, DEductive reasoning cannot solve the problem
because if ANY two knights have a red beanie, then all will raise their hands.

Here are the possibilities:

There are no red beanies, no hands are raised
There is one red beanie, in which case one knight does not have a hand raised,
and two have hands raised.
There are two red beanies, in which case all three knights have hands raised,
because each sees at least one red beanie.
There are three red beanies,  in which case all three knights have hands
raised, because each sees two red beanies.

Thus, seeing two red beanies and three hands raised IS NOT proof of the
existence of a third red beanie, but there is a 50/50 chance.  This is of no
comfort to the knight, because after all of this deductive reasoning, he has no
better odds of guessing correctly than when he began -- the king told him his
beanie would be white or red, and simply tossing a coin would give him a 50/50
chance of getting it right!!!

Perhaps it helps to rephrase it in the negative:  If any knight sees two white
hats, he will NOT raise his hand.  Since all have hands raised, none is
observing two white hats.  But this does not prove that ALL the hats are red
because it could come about two ways:

All the hats are red, or
Only one hat is white.

Each knight has insufficient information to know whether his own hat is white -
only a 50/50 chance.

I think the clue comes from the king, whose construction of the problem
presumes that ALL knights will raise their hands.  His statement is:  "When you
deduce the color of the beanie on your own head, then lower your hand and
rise."  Now the king might be smart enough, and may have thought this through
enough, to realize that any two red beanies will cause all 3 knights to raise
their hands.  But the surest way to ensure that all 3 knights will have their
hand raised is to put a red beanie on EACH of the knights.  So the knight could
reach "a conclusion, which is not necessarily true but which has a  certain
probability of being true" and Keith has defined induction.

Was that a serious enough attempt?

Rick Kunkel
University of St. Thomas
St. Paul, MN