Alice in Wonderland

[Mr. P-1]

Originally Posted by wblipp

I'll make my final stand on this analysis for the simplified problem of two people. Show me the fault and I'll concede. Where is the fault in this logic chain?

A. There are at most two colours.
B. If there are exactly two colours, the problem is unsolvable.
C. The problem is solvable.
D. There must be exactly one colour.
E. The one colour must be the colour I see on the other logicians head.
F. My dot is the colour of the other logician's.

The problem is that C is a false premise. A valid argument from false premises is unsound, and may lead to false conclusions.

[Mr. P-1]

Originally Posted by CRGreathouse

That's not my point of disagreement, no.

You've given one world-state in which the problem is solvable, and concluded that this must be the world-state because the problem is solvable. But there are other world-states in which the problem is solvable, even though the number of colors used is not 1.

World-state (unsolvable): The wizard has three colors of paint, and uses them randomly.
World-state (solvable): The wizard has only one color of paint (and this is common knowledge).
World-state (solvable): The wizard has two colors of paint, red and blue, but never repeats a color (and this is common knowledge).
World-state (unsolvable): The wizard has three colors of paint, red, green, and blue, and never repeats a color (and this is common knowledge).
World-state (solvable?): The wizard has two colors of paint, red and blue, and always uses red in the morning and blue otherwise (and this is common knowledge).
World-state (unsolvable): The wizard has only one color of paint, but the participants do not know this.
World-state (solvable): The wizard has an unlimited number of colors of paint, and always matches the eye color of the person painted (and this is common knowledge).
etc.

Since there are multiple world-states which are solvable, the participants cannot conclude which one they are in with certainty.

Actually it is we, not the participants, who cannot conclude with world-state the participants are in. If the participants are in any of the solvable world-states as you have defined them, then it is common knowledge (among the participants) that they are in that world-state.

[wblipp]

Originally Posted by CRGreathouse

Since there are multiple world-states which are solvable, the participants cannot conclude which one they are in with certainty.

I get your point. Although the only way I know to solve the puzzle is the "one colour" method, the participants might know another method. Hence it is incorrect for me to conclude the dots are one colour just because the wizard has told us the problem is solvable.

I'll think about whether or not I consider this is a legitimate objection or a trick solution. My first inclination is to dismiss this as "not playing fair," but that might be ego.

[chris2be8]

I've a few suggestions that may help narrow it down:

Johnny asked if the problem was solvable. So he did not have enough information to solve it until the Professor said it was solvable. Which limits the amount of information he could have had that was not stated in the puzzle.

The bell rang 4 times before Johnny and his sister left the table, at least once when they did and at least once afterwards. So it must have rung at least 6 times.

So if they could have all left after 6 rings with no extra common knowledge that must be the solution. I'm assuming none of them are colour blind though.

Chris