John Norwood's reasoning as reported by Kent Schenkel when coupled with
Maxwell's perjury query to my reasoning prompts some additional thoughts.

That is, if, as Maxwell states, deduction yields an inescapable truth based
upon other items which are true.  Maxwell then gives an example which builds
assertions of truth (which if true) lead to truth.

My own prior answer raises the issue of "states of  mind" as an item which must
be true for building deductive reality.  In this test, the King's very purpose
of the test is to detect non-accurate states of mind.  This may insert some
circularity into the deduction process.

The three knights' raised hands proves that each sees and each sees "1 or more"
red beanies (assuming the other knights are truthful, a trait which may or may
not be correlated with "smartest" depending upon your criteria for "smartest").

However, the lack of standing by each of the other two knights is not as
dispositive as the raising of hands since standing requires the knight to
demonstrate being "smart".

Saving "face" is a quite common trait.  Seeing is such a simple task for one of
the three "smartest" knights of the realm, that all of the three must respond
accurately to what they see or be revealed as a quitter.  However, internal
processes of the mind are not as easily evaluated by observers, and a knowing
knight may opt to loose by not standing.  A knowing knight may choose to opt
out of the competition based upon the additional information which only became
available after starting the competition.

But I am not sure that is what is going on in the other knights' minds.

Let me use R for a red beanie with a raised hand and r for a red beanie and a
lowered hand, likewise for W and w.  The options for the beanies are as
follows:

1:RRR  2:RRW  3:RWR  4:rWW

5:WRR 6:WrW  7:WWr  8:www

Let us treat the first letter as the standing knight, based on what the
standing knight sees the standing knight can rule out options 2, 3, 4, 6, 7,
and 8.  Which leaves options 1 and 5.

Norwood's first point is that each knight can reach an inescapable conclusion
of their own beanie (assuming truthful hand raising) for options 4, 6, and 7.
Importantly, Norwood's first point does not rule out option 5 (nor the
symmetric patterned options 2 and 3).  Note that option 8 yields an unambiguous
result.

I am curious how to deduce "red".

Michael

Kent Schenkel wrote:

> John Norwood got it right:
>
> Knight #3 (hero) reasons as follows:
>
> 1.      If  any knight sees two hands raised and at least one white beanie,
> he will stand.
>
> 2.      Neither knight stands.
>
> Neither knight sees a white beanie. (10 second delay is to see if any other
> knight stands)
>
> Kent Schenkel
> University of North Carolina at Wilmington
> [log in to unmask]