How Many Jokers

Logic Level 2

On a certain island, there are only knights, who always tell the truth, knaves, who always lie, and jokers who can do either. You come across 3 of them, Al, Bo and Cat. They each say the following:

Al: Bo always lies.

Bo: Cat is a knave.

Cat: Al is a knave but Bo is not.

Later, Al adds: Exactly one of us is a knight.

What is the maximum possible number of jokers?


The answer is 3.

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2 solutions

Jay B
Feb 15, 2019

If Cat is a knave, then Bo is a knave. Then Cat is not a knave. This is impossible. If Cat is a knight, then Al is a knave, Bo is not a knave but Bo still lies. But this means that exactly one of them is a knight contrary to Al being a knave. Then Cat is a joker.

Suppose that Cat is a joker but did not lied. Then Bo is not a knave but he lied. Since Al would be a knave, this brings us up to 2 jokers. Let's suppose that Cat lied. Then either Al is not a knave, or Bo is a knave. If Al is not a knave, let’s assume that he’s a joker. Then Bo doesn’t always lie but he did this time. Since this fits, its possible to have 3 jokers.

Saya Suka
Feb 27, 2021

Al : Bo always lies.
Bo : Cat is a knave.
Cat : Al is a knave but Bo is not.
Later, Al adds : Exactly one of us is a knight.


Essentially, everyone is accusing the next person in a cyclic turn as a knave, but if any of those were really the truth, we may only have one knave among them at most (if any), else one of the knaves would be telling the truth (by accusing a real knave) and contradicting their nature to always lie. By the same argument, we may only have one Knight among them at most (if any), else one of the Knights would be telling a lie by a false allegation against a comrade and contradicting their nature to always tell the truth. In any scenario, you can always find at least a Joker amongst these three.

Other than the standard repeated allegations, Cat declared that Bo is not a knave even though she's the one who accused her earlier, and Al added later on that there's exactly one Knight among the three.

Because Bo and Cat are being 'half-bakedly' contradictory (I mean, Bo accused Cat of lying but somehow, Cat's own words made it seems like she is letting Bo get away with the slanderous allegation), then it must be because there's at least a Joker between these two. If it was a scenario of either a Knight accusing a knave or vice versa, then there would have been total contradictions amongst them.

If we considered Bo's not knave statement by Cat as true, then Cat can't be a knave who always lie (not while Cat just told an assumed truth) while Bo also can't be a knave since the statement was assumed to be true. Then Bo must have lied earlier when she accused Cat, just not as a knave but a deceptive Joker. With this, Cat really told the truth about Bo and is possibly another Joker or even a Knight. Al accused a Joker Bo as knave, so he lied either as a knave or another Joker, but anyway his one Knight statement cannot be true, else Al himself would be an honest knave which is an oxymoron. So Cat is a confirmed Joker and Al could either be a Joker or a knave.

Bo = lying Joker all the way
Cat = lying then truthful Joker, OR truthful Joker respectively
Al = lying Joker, OR knave respectively

If we considered Bo's not knave statement by Cat as false, then Cat can't be a Knight who's always honest (not while Cat just told an assumed lie) while Bo can only be a knave in this scenario. Therefore, Bo's accusation is invalid and Cat cannot be a knave, either. Here, a Joker Cat and a knave Bo is fixed. Al accused a knave Bo as knave, so he told the truth either as a Knight or another Joker, and his one Knight claim might be true or false depending on who's saying it.

Bo = knave all the way
Cat = lying Joker all the way
Al = truthful then lying Joker, OR Knight respectively

Scenario { Truth Value } Bo Cat Al Number of Joker(s)
1️⃣ { F F [ F T ] F } Joker Joker Joker 3
2️⃣ { F F [ T T ] F } Joker Joker knave 2
3️⃣ { T F [ F F ] F } knave Joker Joker 2
4️⃣ { T F [ F F ] T } knave Joker Knight 1

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