Four Sisters (Logic puzzle) - Solution

Since the sisters keep switching position without you knowing, it may seem difficult to get useful information. 

Like with most puzzles like this, it’s best to work backwards. In the final round, you’re going to have to identify all of the sisters and how many questions would it take to figure out the sisters, assuming you’ve identified some already before the switch?

If you’ve identified 1 sister, you’d need 5 questions to figure out who is who.

If you’ve identified 2 sisters, you’d need 4 questions to figure out who is who.

If you’ve identified all 4 sisters, you’d need 3 questions to figure out who is who.

It’s pretty obvious that identifying 1 sister isn’t good enough. But what about 4 sisters? Is it possible to identify them all in 7 questions? Yes, but only if you get lucky and it’s best to assume the worst possible outcome each time.

So, we know we need to identify 2 of the sisters and use 4 questions in the final round.

Round 1 - 3 questions

With the first round of questioning, it’s imperative to figure out what each of the four girls will say. You can do this by asking three of the girls the same question. There are three different questions you could ask.

Each of the girls can be divided into three pairs. Which you can get them into by eac of these questions:

Are you honest?: Becky and Daisy (“No”) vs Ava and Connie (“Yes”)

Are you delusional?: Ava and Becky (“No”) vs Daisy and Connie (“Yes”)

Is the sky blue?:  Becky and Connie (“No”) vs. Ava and Daisy (“Yes”)

To give a concrete example, let’s say that you ask the question: “Are you honest?”

Ava, who believes she is honest and is honest, will answer “yes”

Becky, who believes she isn’t honest, but is honest, will answer “no”

Connie, who believes she isn’t honest and isn’t honest, will answer “yes”

Daisy, who believes she is honest, but isn’t honest, will answer “no”

You only need to ask three of the girls this, because you can infer what the fourth will say. So after the first round of questions you will have something like this:

Girl 1: Yes

Girl 2: No

Girl 3: Yes

Girl 4: No

Round 2 - 2 questions

Two of the girls have switched places. So how are you going to be able to know what she said last time and if you are even talking to the same girl? As it turns out, it doesn’t really matter if you’re talking to the same girl or not. It just matters if she gives the same answer as before or not.

You’ll want to ask the girls who both said yes, or both said no, the same question as you asked in round 1. 

So let’s say that you asked girl 1 and girl 3 “Are you honest?” there are four possible results:

Round 2 answers

Yes and Yes

This is the easiest results to get. This means that either girls 1 and 3 switched position or girls 2 and 4 switched positions. Either way, the other pair will switch position after this round and in all cases, it doesn’t matter. Because regardless of who swapped with who, the answers to the first question hasn't changed.

No and no

This result is actually impossible. Because this would mean that all of the girls switched places.

Yes and no or no and yes

Who says yes and who says no doesn’t actually matter. They’re essentially the same result. So let’s focus on girl 1 saying yes and girl 3 saying no.

Round 1 answers

Girl 1. Yes

Girl 2. No

Girl 3. Yes

Girl 4. No

Round 2 answers

Girl 1. Yes

Girl 2. ?

Girl 3. No

Girl 4. ?

This tells us that girl 3 swapped with either girl 4 or girl 2. We also know that at the end of the round girl 1 will swap with either girl 4 or girl 2. That girl said “no” to the previous question. So at the start of round 3 you will know what everybody said and will look like this.

Start of round 3

Girl 1. No

Girl 2. Yes

Girl 3. No

Girl 4. Yes

Round 3 - 1 question

You only need one  question. Since you know what they answer to being honest, you’ll have narrowed each girl down to two pairs.

Girl 1. Daisy/Becky

Girl 2. Ava/Connie

Girl 3. Daisy/Becky

Girl 4. Ava/Connie

So you simply need to ask one of them “Is the sky blue?”. Let’s say, you ask girl 2 and she replies “yes”, meaning that she is Ava. That also lets you know that girl four has to be Connie

Girl 1. Daisy/Becky

Girl 2. Ava

Girl 3. Daisy/Becky

Girl 4. Connie

Round 4 - 4 questions

Now three girls have swapped position. But after this you will know exactly who everybody is.

Question 1. Girl 2, are you honest?

Because three people were swapped, it means that at least one of Ava and Connie moved. However, since exactly three people were swapped, it means it was impossible for them to switch with each other (The same goes for Daisy and Becky). That means that one of them will answer yes, while the other has to answer no.

Let’s break it down.

Girl 2 says “yes”

Connie and Ava are the only ones who would say “yes” to that question. If girl 2 is Ava, it means that Connie switched and either Daisy or Becky is girl 4. But what if Connie is girl 2? Girl 4 would still have to be Becky or Daisy. Because if it wasn’t, that would mean that Connie and Ava switched with each other and that would mean that either all 4 girls switched position or just the two of them did.

Girl 2 says “no”

We know that Girl 2 is either Daisy or Becky. This means that Ava or Connie is girl four. We know this because if both girls 2 and 4 were Daisy and Becky, that would mean that all four of the girls switched positions, which is impossible.

Question 2. Girl 3, are you honest?

The same principle as above.

After your first two questions, you will have narrowed the identity of each girl down to two people and can ask any two questions to split the remaining pairs.