# Blog

## Friendship circles

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Difficulty: Tricky

Anas is making a map of his friend group. In his map, each circle is a person. A line joins 2 circles if and only if those 2 people are friends.

Using the clues below, can you figure out who goes where on the map?

1. Anas has 3 friends: Ruth, Mitra, and Buddy
2. Buddy and Cara are both friends with Mitra
3. Cara is Samir’s only friend

Scroll down or click for a hint, or the answer!

## Brainteaser hint

You might find this puzzle easier with a pen and paper handy.

Every friendship in the diagram is mentioned in the clues.

The first clue tells us that Anas has 3 friends, so Anas must be in one of the 2 circles with 3 lines connecting to it.

The first clue tells us that Mitra has a friend, and the second says she has 2 more. So Mitra’s circle also has 3 lines coming from it!

Don’t worry about using each clue in order! You can use them whenever you want, and you might want to use each of them several times.

Anas’ friendship network graph should look like this:

Let’s walk through the logical steps.

If we look at the map, there are 2 circles that have 3 lines coming from them, 2 circles with 2 lines, and 2 circles with only one line.

The first clue tells us that Anas has 3 friends, so he must be in one of the 2 circles with 3 connecting lines. The second clue tells us that Mitra has 2 friends in addition to Anas. This means she also has 3 total friends and so she must be in the other circle with 3 connecting lines:

We don’t know where their other friends go yet, but we’ll figure that out using the other clues.

The third clue is more straightforward than the second clue, so we can use it to fill in more information. It says “Cara is Samir’s only friend” meaning Samir must be in one of the circles with only one connecting line. The 2 ‘singleton’ circles are on the far left and far right.

But, looking closer, Anas or Mitra must be connected to the singleton circle on the right. The far right circle can’t be friends with Cara! This means that Samir must go in the far-left singleton circle next to Cara:

Now we can go back to the second clue, which says, “Buddy and Cara are both friends with Mitra.” Cara is only connected to Samir and to the middle circle. This means Mitra must take the middle circle and Anas must take the top circle:

Buddy must take the bottom circle so that he can be friends with both Anas (first clue) and Mitra (second clue):

And Ruth takes the last blank circle:

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