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Council of Magic - 5 Years Later:
Five years ago, you managed to gain the support of the Council of Magic. However, now the King of Puzzlington needs their aid once again. As it worked last time, he sends you once again. However, 5 years have passed and things have changed. The old members of the council have retired and new ones have been elected.

There are now 5 types of spellcasters in Puzzlington: Wizards, Witches, Priests, Warlocks and Sorcerers. The first four work exactly the same as last time, but I'll repeat them here.

Wizards:
Wizards follow one of two paths: the path of Fire or the path of Water.
Wizards on the Path of Fire always tell the truth when asked a question.
Wizards on the Path of Water always lie when asked a question.

Witches:
Witches come in two styles: Light Witches or Dark Witches.
Light Witches tell the truth during the day and lie at night.
Dark Witches lie during the day and tell the truth at night.

Priests:
Priests worship one of two Gods: Yes, god of life or No, god of death.
When asked a question, instead of answering, priests just say their God's name.
That is, a Priest of Yes will always answer "Yes" to any question.

Warlocks:
Warlocks are unpredictable tricksters.
When asked a question, Warlocks will tell the truth or lie, as they wish.
However, Warlocks have now learned to withhold answers and remain silent.

Sorcerers:
Sorcerers draw their power from one of two star signs: Taurus or Gemini.
Taurus Sorcerers tell the truth if the question has an odd number of words.
Gemini Sorcerers tell the truth if the question has an even number of words.
Sorcerers will never lie; if the word count is wrong, they remain silent.

The Council:
The Council of Magic now consists of 6 powerful spellcasters. The spellcaster's names are Aries, Bandana, Cathy, Darrin, Eve and Francis. The Council has one Wizard, one Witch, one Priest, one Warlock and one Sorcerer; you do not know which Council member is which type of spellcaster.

In addition, the Council now has a High Mage. The High Mage can be any spellcaster type; however, you know the High Mage is not a Warlock. You do not know who the High Mage is or what spellcaster type they are.

You do not know the Path of the Wizard, Style of the Witch, God of the Cleric or Sign of the Sorcerer. You do, however, know the High Mage does not have the same sub-type as the other council member of the same type.

The members of the Council have full knowledge of each other. That is, they know which member is which type of spellcaster and which sub-type each is; they also know which member of the Council is the High Mage. The sub-type of a spellcaster is their Path, Style, God or Sign, as appropriate.

You arrive at the palace at noon. You can, once every 12 hours, ask any one member of the Council any one question that can be answered Yes or No (that is, after each question, it switches from night to day or day to night). If a spellcaster is asked a question they can't answer (because they don't know the answer), they remain silent. Priests are an exception to this rule; to them, their god is always the answer, no matter what the question.

The new Warlock hates you as much as the old one; when asked a question, they will choose to tell the truth or lie, whichever they think will hurt you the most.

Your task is the same as before. Learn the type and sub-type of all Council members as fast as you can; you will also need to learn the identity of the High Mage.


To list the changes carefully: There is a new spellcaster type that either tells the truth or says nothing, there is a duplicate spellcaster on the council and warlocks can choose silence.

I'm not yet sure how many questions it will take this time. More is all I can say with certainty.

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  • 6
    $\begingroup$ 5 years ago? It was only posted yesterday! :P $\endgroup$ – Beastly Gerbil Sep 3 '16 at 20:41
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    $\begingroup$ @BeastlyGerbil. Nope. It has been 5 years. It's hard to tell you this, but you've been in a coma for the last 5 years. $\endgroup$ – Marius Sep 3 '16 at 20:57
  • 3
    $\begingroup$ @Marious, Oh no! I've missed 5 years of puzzling! :P $\endgroup$ – Beastly Gerbil Sep 3 '16 at 20:58
  • $\begingroup$ I would likely get hurt by asking a priest which god he thought was fake. I'd probably get killed as to not have to receive an answer xD $\endgroup$ – ObviouslyJake Sep 4 '16 at 3:39
  • $\begingroup$ If Warlocks do not know the answer, will they reliably remain silent, or can they answer anyway? $\endgroup$ – celtschk Sep 17 '16 at 16:15
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I can do it in:

11 questions at most

background:

First, note that the theoretical minimum for the worst-case number of questions is 10. No strategy can ever require less than 10 questions in the worst case. Proof: Each question has 3 possible answers - yes, no or silence. A perfectly constructed question will evenly split all the remaining possible configurations at the time into 3 groups, corresponding with each answer, thus dividing the possibility space by 3. A perfect strategy will do this for each question. The number of possible configurations at the start = 6! * 2*2*2*2*4 = 46080. log(base3, 46080) = 9.77, therefore 10 questions is the theoretical minimum, whether it can be achieved is a different question ( I think it could be by improving my stage 1 questions, but I leave that to someone else).

Let's recap two important facts from the previous incarnation of this question. First, anyone who is known to either reliably lie or reliably tell the truth, can be forced to tell the truth with a nested question, e.g: "what would you say if I asked you "does 1+1=2?"". This can be used to extract truth from any of (wizard of any type, witch of any type, sorcerer of known type), which we'll call the target-group. Secondly , Etoplay's answer to the previous council of magic question (link at the top) contains a general strategy which I will use, which is optimal after you know the identity of someone in the target group. It's a 2-stage strategy: first use normal questions to establish a target-group member, then use what Etoplay calls the 'last few questions' to narrow down to the one true solution. Anyone who wants to understand this is advised to read Etoplay's answer as it's complex and I won't repeat it here.

strategy:

As mentioned, stage 2 of my strategy is the same as Etoplay's 'last few questions' section. The only thing remaining is to detail the first stage, along with logarithmic calculations for the number of questions required in each variation. Stage 1 proceeds as follows. Let the council members be numbered 1-6, let P = the number of possible configurations remaining. Calculating P is tricky combinatorics which I won't detail, but all values have been verified with a script.

first question, to person 1:
"If I asked the warlock whether two plus two equals four, what would he say?".
If silent: 1 is either witch, wizard or sorcerer of even-truth. All of these are in the target-group, so proceed to stage 2. P = 24000, ceiling(log(base3, P)) = 10, so 10 more questions required in this case, 11 total.
If not silent: 1 is either warlock or priest, with subtype depending on what the answer was. Now ask second question.
second question is the same, but to person 2:
If silent: 2 is in target group, move to stage 2. P = 7680 given everything we know, 9 more questions required, total 11.
If not silent: 2 is either warlock or priest, with subtypes depending on both answers. Now ask third question.
third question is the same, but to person 3:
If silent: 3 is in target group, move to stage 2. P = 1680 in the worst case (ie. first 2 answers were different). 7 questions required, 10 total.
If not silent: 3 is either warlock or priest, with person 1-3 subtypes highly narrowed down based on answers 1-3. Now ask question 4.
fourth question, to person 4:
"are you alive?"
if silent: 4 is a sorc of even-truth. move to stage 2. P = 32 given everything we know by now. 4 more questions required, 8 total.
If not silent: 4 is witch or wizard. move to stage 2. P = 64 based on everything we know. 4 more questions required, 8 total.

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  • 1
    $\begingroup$ umm, i know I'm coming to the party late, but isn't this answer flawed? The warlock is now able to give silence as an answer. Anyone giving a silent answer is therefore potentially a warlock. You don't yet have someone guaranteed to give a true answer when you get your first silence, so you can't yet resort to a trinary search to optimize the remaining questions. $\endgroup$ – dsollen May 19 '17 at 16:31
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Since posting this puzzle, I've crafted a bad answer. It takes 16-19 questions to sort out everything, depending on the High Mage's type. I'm sure it can be done much faster, but I'm not great at solving these puzzles.

Ask A "Does one plus one equal two?" (even)
Ask B "Does one plus one equal two?" (even)
Ask C "Does one plus one equal two?" (even)
Ask D "Does one plus one equal two?" (even)
Ask E "Does one plus one equal two?" (even)
Ask F "Does one plus one equal two?" (even)
Ask F "Are you a council member?" (odd)
Ask E "Are you a council member?" (odd)
Ask D "Are you a council member?" (odd)
Ask C "Are you a council member?" (odd)
Ask B "Are you a council member?" (odd)
Ask A "Are you a council member?" (odd)

Divide everyone into two groups:
Group A gave the same answer twice.
Group B gave different answers.
Group C gave "yes" and silence, in any order.

Group A will have the Wizard and Priest.
Group B will have the Witch.
Group C will have the Sorcerer.
The High Mage will be with the matching spellcaster.
Presumably, the warlock will mimic one of the other types.

If the High Mage is a Wizard or Priest, either Group B or C will have just one character, as there are only three characters left.
You know that character's type and subtype from their prior answers.
Ask that character questions until you know everything.

If the High Mage is a Witch or Sorcerer, the warlock will likely mimic either a Witch or Sorcerer (if they mimic a Wizard or Priest, the above answer works). If they mimic the same type as the High Mage, the same answer still works, so they likely won't.
In that case, there are two characters in each of group B and C.
Make sure the apparent subtypes of the characters in each group are different. If a group has matching subtypes, the Warlock is in that group; discard that group and just ask the other group questions until you know everything.
Ask each member of group B "What would X say if I asked them 'are you a council member?'?" X is the other member of Group B.
If one fails to answer, the Warlock is in group B. Ask group C questions until you know everything.
Otherwise, the Warlock is in group C; ask group B questions until you know everything.
This uses 12-14 questions to identify a source. Afterward, it will take 3-7 more questions.
For example, assuming these are the groups and Group B has our source: Group A: A, B
Group B: C, D
Group C: E, F

Just from knowing we can trust Group B, we know C and D are Witches and we know their Subtype.
Ask C: Is F the Warlock?
We now know the Warlock; we also know the Sorcerer and their sub-type.
Ask C: If A a Wizard?
This tells us the Wizard, the Priest and their Subtypes.
Ask C: Are you the High Mage?
The High Mage must be in the group we trust, so this tells us the High Mage. All information is known in 17 questions.

If the High Mage/Warlock is in Group A, it will take more questions to clean up. Assume these are the groups (if group C has only one member, some question will need tweaking to please the Sorcerer):
Group A: A, B, C
Group B: D
Group C: E, F

Ask D: Is the Warlock A or B?
Ask D: Is the Warlock B or C?
If the Warlock isn't in Group A, ask D: Is E the Warlock?
This just sorts out where the Warlock is.

If the Warlock is in Group C, ask the following:
Ask D: Is A a Wizard?
Ask D: Is B a Wizard?

If both are true, C is a Priest.
If neither is true, C is a Wizard.
If exactly one is true, ask D: Is C a Wizard?

Ask D: Is X the High Mage?
X is a member of group A whose type is repeated. All information is known in 18-19 questions.

If the Warlock is in Group A, ask the following (I'll assume C is the Warlock):
Ask D: Is A a Wizard?
Ask D: Is E the High Mage?
All information is known in 16 questions.

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  • $\begingroup$ How do you conclude: "Ask each member of group B "What would X say if I asked them 'are you a council member?'?" X is the other member of Group B. If one fails to answer, the Warlock is in group B." The Warlock could just pretend to be the High Mage and mimic the other type of Witch in group B right? A Witch never fails to answer. $\endgroup$ – Hans Janssen Sep 9 '16 at 12:06
  • $\begingroup$ Witches can fail to answer. They alternate between truth and lies, but if they don't know the answer, they will remain silent. $\endgroup$ – qwertyu63 Sep 9 '16 at 14:01
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The key is to extract as much information as possible with every question, that means to ask them what each other would say. Note that you don't need to limit yourself to "if I asked him what is 1+1, what would he say", you can also say "if I asked him what she would say he would say".

Start with a three person question, if they remain silent then you've narrowed the warlock to half the group. If they don't then repeat the question to the other group.

If both groups answered then you talked to a Priest or the Warlock. Switch who you're talking to and switch groups and repeat.

The real key is you never want to be asking a question which just addresses one piece of information, you want to be cutting the number of possibilities that remain for the entire set in half.

As for figuring out how many questions this takes, who is who (6), who is high (2), and the various flavors (2^4), so I think ideally 8 questions... but that's off hand.

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