A Classic Logic Puzzle:
Vampires, Insanity & Transylvanian Sisters
As you can probably tell, I’ve been on a bit of a logic kick this week. And why not? Logic is one of the most useful forms of mathematics there is!
Whether you’re using formal, symbolic logic to sort out a math or CS problem, or simply using sound logic to sort through a series of statements it does us good to exercise our reasoning faculties.
And given the amount of noise in the media these days, I think we can all use to sharpen up on our critical thinking and logical reasoning skills. So here’s a fun little puzzle problem for your problem solving pleasure.
Let’s give it a go!
Inspector Craig Visits TransylvaniaInspector Craig of Scotland Yard was called to Transylvania to solve some cases of vampirism. Arriving there, he found the country inhabited both by vampires and humans. Vampires always lie and humans always tell the truth. However, half the inhabitants, both human and vampire, are insane and totally deluded in their beliefs: all true propositions they believe false, and all false propositions they believe true. The other half of the inhabitants are completely sane: all true statements they know to be true, and all false statements they know to be false. Thus sane humans and insane vampires make only true statements; insane humans and sane vampires make only false statements. Inspector Craig met two sisters, Lucy and Minna. He knew that one was a vampire and one was a human, but knew nothing about the sanity of either. Here is the investigation: Craig (to Lucy): Tell me about yourselves. Lucy: We are both insane. Craig (to Minna): Is that true? Minna: Of course not! From this, Craig was able to prove which of the sisters was the vampire. Which one was it?— From Logician Raymond Smullyan
The Solution
Step 1: Outline All Possible Scenarios
We know that each sister can be either Vampire or Human, as well as Sane or Insane. We’ll begin by listing all possible cases.
I’m also going to include a T or L to indicate whether the combination yields true statements or lies.
- Lucy = Sane Human (T), Minna = Sane Vampire (L)
- Lucy = Sane Human (T), Minna = Insane Vampire (T)
- Lucy = Insane Human (L), Minna = Sane Vampire (L)
- Lucy = Insane Human (L), Minna = Insane Vampire (T)
- Lucy = Sane Vampire (L), Minna = Sane Human (T)
- Lucy = Sane Vampire (L), Minna = Insane Human (L)
- Lucy = Insane Vampire (T), Minna = Sane Human (T)
- Lucy = Insane Vampire (T), Minna = Insane Human (L)
Step 2: Take the Statements Into Consideration
We know from inspector Craig’s conversation with Lucy and Minna that they have opposing arguments:
- Lucy: “We are both insane!”
- Minna: “Of course not!” → i.e. “We are not both insane.”
Both of these statements cannot be true because they contradict each other. We also know that these statements cannot both be lies because that will also be contradictory.
So remove all scenarios that result in Lucy and Minna either both telling the truth or both lying.
This leaves us with 4 potential scenarios:
- Lucy = Sane Human (T), Minna = Sane Vampire (L)
- Lucy = Insane Human (L), Minna = Insane Vampire (T)
- Lucy = Sane Vampire (L), Minna = Sane Human (T)
- Lucy = Insane Vampire (T), Minna = Insane Human (L)
Step 3: Run the Remaining Scenarios
Let’s play out each of these four cases to see if we find any other logical inconsistencies.
Case 1: If Lucy is a Sane Human, then she is telling the truth. That means that both her and her sister are insane. But in case 1, both Lucy and Minna are sane. That’s a contradiction, so this combination can’t be true.
Case 2: If Lucy is an Insane Human that means she is lying, and so they are not both insane. But in this scenario, Minna is also insane. So this is obviously untrue since they both are insane.
Case 3: If Lucy is a Sane Vampire, she is lying. Which implies that they are not both insane. In this case Minna is a Sane Human. Because Minna is also sane, Lucy’s lie holds. And if Minna is a Sane Human, she tells the truth when she says that they’re not insane. This case holds!
Case 4: If Lucy is an Insane Vampire, she is telling the truth, which means they are both insane. In this scenario, Minna is an Insane Human so the claim that they are both insane holds. And lastly, because insane humans lie, Minna is lying when she claims they’re not insane. So this case also holds!
Step 4: Determining the Vampire
Out of the original 8 scenarios, we ended up finding two that didn’t cause contradictions. In both of these cases Lucy is a Vampire and Minna is a Human, although their sanity flip-flops. Fortunately Inspector Craig is only looking for the Vampire and doesn’t really care who is sane or not.
Therefore, we can happily conclude that Lucy is the Vampire.
