Doing this with ints instead of colors is a great idea, carl.
Finally, after all of these years, someone acknowledges the benefits of Resolution Debating (seeking resolution, not competition. Precisely upon what do we agree and disagree). 
Yes, certainly. You have 3 members, all wearing blue bands. Forgiving that the puzzle stated that there were multiples colors, none of them could LOGICALLY DEDUCE that they were all wearing blue bands. They could easily RATIONALLY INFER that they were. But they would have to be God or as in the Blue-eyes puzzle, be “Perfect Logicians” in order to eliminate ALL alternatives. The Master would be wrong.
The fact that a person can only think of or currently comprehend one solution merely reflects his limited intelligence. It certainly does not constitute logical deduction.
If the Master had said, “You are God and this puzzle is solvable”, then he could have been right … except for that small “actually being God” issue.
“It’s turtles all the way down, because there couldn’t be any other solution.” The question of what was holding up the Earth really was solvable. But they could not imagine the thought of the Earth not requiring anything holding it up, “floating in space”. The Master Guru tells them that it is solvable. So obviously the solution is, “Turtles all the way down”.
Perhaps “unknowability” is useable because ambiguity is easy to ensure and verify. If there are two known possibilities, it doesn’t matter how many more there are, the problem is already “unknowable”. So the Master could say that the puzzle is UNsolvable and be right. But not because he made it so by saying it. And he cannot make it solvable by saying that it is.
My next proposal was going to be to substitute the colors with numbers in order to see the vast possibilities of algorithms that can be formed by “master logicians”. They don’t have to use only integers and cannot assume that only integers are present.
I disagree. It demonstrates that multiple potential solutions are probable and thus no true solution can be guaranteed.
Exactly. And as long as it is a “might”, then it cannot be LOGICALLY DEDUCED that it is “turtles all the way down”.
Realize that my Rational Metaphysics is very strict about ensuring that there is ZERO possibility of incorrect deductions. So this isn’t something that is new to me. There is basically only one way to ensure that literally all possibilities have been addressed. And I can see that the resolution is not applicable in this puzzle as it is stated.
By your proposed method of deduction (“can’t think of any other possibility”), it would have been solvable. Thus repetition of colors (or numbers) is not required nor is the ability to see your own color anywhere else.
I missunderstood your question, James. If each person saw that chart with only one color missing, that color would not be a part of the puzzle.
What do you mean by “not part of it”?? 
And why not?
Let me try and be thorough 
Is it even worth putting it in tabs anymore?
[tab]If you have a chart on the wall and say these colors are included in the problem, and you can’t see one of them, obviously that is the color of your band. You would get up and leaving thinking that this is probably the most boring math problem ever, although perhaps more suitable to your objection.
But now you are gone and the others are sitting there trying to figure out when to stand up. What should they do?
The person who can see a color with only one person wearing it must assume that they need to get up and go, because if anyone there was not able to see a color on the chart, they would have already left. From there, in sequence, people who can see 2 people with the same color, then 3, then 4…
So, not having that first person who can’t see any headbands with one color of the chart only changes the problem slightly by making it less boring. If you take down the chart, you still have a chart composed of all the colors you can see. Those colors of the headbands around you are your color group. If you can’t see a color, it is not in the group. The sequence begins to iterate at 2.[/tab]
There were two objections being discussed.
1) Is it necessarily true that you must be able to see your own color on someone else?
2) Having proposed a possible means for solving the puzzle, can it be proven that it is impossible to solve the exact same puzzle arrangement by any other means?
I quickly dispatched (1) merely by showing that a known or famous color chart would allow a person to guess at his color without ever seeing it on anyone else.
Objection (2) is the one that I propose makes the puzzle invalid. You propose a particular means to solve the puzzle. Now prove that no other means could ever be proposed that would also solve the same puzzle arrangement. Without being able to prove that no other means could ever exist, the proposed solution cannot be used and thus does not actually solve the puzzle. For example, perhaps there is a different color chart of which you were not aware that implies a different missing color or perhaps merely a more sophisticated math model would imply a different one of the colors that you do see.
Perhaps let me address this issue in a different manner.
Puzzle Key Question:
Is there a color arrangement for that scenario that would allow for the existence of an algorithm to resolve what color each person was wearing and also can be proven to be the only possible algorithm for that color arrangement?
You did not quickly dispatched 1 because the problem does not give you a chart. It doesn’t state “consider a group of colors (blue, green, yellow, orange, red)”. This information is not present, there is no color chart as a given in the problem premise. When the problem starts you don’t have a closed group of colors to work with. Part 1 of the problem is constraining all possible colors down to a group.
As to 2, it is evident that there is no other way to solve this puzzle. Assuming that the group of colors consists only of colors that you can see is necessary because otherwise you do not have a constrained group.
Once you have a group, you can write an algorithm:
for (bell = 1; bell>numberOfLogicians; bell++) {
for(numberOfColor = 1; numberOfColor>bell; numberOfColor++){
if (numberOfColor == bell){
System.out.println("logicians who can see this numberOfColor have to get up and leave");
//TODO: write a method for making logicians leave.
}
}
}
(numberOfLogiciansWearingHeadbandOfTheSameColorThatALogicianCanSee would be the correct name, but to make it readable I called it numberOfColor)
The puzzle doesn’t have to state that chart, just like it didn’t state the number of repeated colors. The challenge of the puzzle was to consider all possible color arrangements in order to find any that would allow the prediction. We are looking for any possible arrangement of colors that would allow the prediction of the color of an unseen band. If you know of the primary color chart and you see one of each of those colors except one, you can predict that one. And you can claim that “it is evident” that the color could not be predicted any other way.
“It is evident” (aka. “I can’t think of anything”) is the “turtles all the way down” solution. It does not constitute a premise for a logical deduction, merely a probability based upon the person’s intelligence.
You have to be able to prove that the proposed algorithm is the only possible means of solving the problem, else it is not a valid solution. You can’t merely state, “it is evident”.
It is evident that you can’t know a color that you can’t see.
Okay then;
It is evident that you cannot predict a color that you can’t see.
And it is evident that it is turtles all the way down.
[-o<
It is evident that you cannot predict a color that you can’t see.
Someone who can see your color (and all other colors present) states that the problem is solvable.
Therefore, you can see your color.
That would constitute a prediction. 
I added (and all other colors present) to the post above.
Where is the prediction? It’s an induction.
Make up your mind.
It is evident that you cannot predict a color that you can’t see.
Attempting to guess a color would be a prediction.
Someone who can see your color states that the problem is solvable.
He is not making a prediction because he can see all the colors.
He is giving you a clue that is an indication of what the color group contains.
Therefore, you can see your color.
Direct conclusion from the two sentences above.
Because you can see it, it isn’t a prediction.
The one who said it, the Master, is not the one making the prediction.
YOU are, based upon what he said, what you see, and your guess that there is only one way to resolve such a puzzle.
And you still haven’t proven that there is only one way to resolve that particular arrangement of colors. Saying “it is evident” does NOT constitute a proof. It was evident to people thousands of years ago that “it was turtles all the way down” because they couldn’t imagine anything else.
There are two ways to arrive at the answer, actually:
- arriving at the conclusion that your headband has the same color as another one in your field of vision.
- taking a guess at what your headband color is out of millions of possible answers.
The first one leads to an induction algorithm. The second one is not logic. Since the master said it is a solvable problem, one must conclude that guessing won’t be required.
Look at the reasoning that you are using for each of those:
-
“I can’t think of any way to know my color other than to assume that it is one of those that I am seeing.”
-
“ASSUMING that it is one of the colors that I am seeing, I CAN’T THINK OF ANY WAY to deduce my color other than by this one algorithm.”
In BOTH cases, you are assuming based upon the fact that you can’t [currently] think of any way to deduce your color other than the one way you learned.
In effect, it is like saying, “I know that I am right because I am too stupid to think of anything else!”
What if Science was run that way? “We think electricity works this way because we can’t think of anything else.”
The fact that such was so very often wrong, Science was developed around demonstrations so as to give more concrete evidence and not rely on people just being too stupid to think of anything else.
You are asking me to prove that guessing a color can’t be anything but guessing.