Why Can't We Make Exactly 7 Friends?

We asked to provide a proof for the following problem in our last newsletter. We are publishing the best explained response. Lookout for our future newsletter. Your response can also be published. It’s important to continue to learn by proofs.

Prove that we can’t make exactly 7 friends

“CKSTEM has 25 students in a class, and each student is friends with exactly seven classmates.” Why can’t that statement be true?

Proof BY Ansh Agarwal

First, let’s assume that each student can make friends with 7 students. If each student puts a checkmark in their notebook for each friend they have, there would be 7 checkmarks for each student, and 7 * 25 = 175 checkmarks in total. Now, let’s look at what happens when 2 students become friends. Each person puts a checkmark in their book, so the total amount of checkmarks goes up by 2. Since the total amount of checkmarks starts at 0, the total amount of checkmarks is always even. Since we got an odd number, our previous assumption was false, and so, it is not possible to have 25 students in a class, where everyone is friends with 7 students.

We can generalize our observation. If class size is 25 (odd) then all students can’t have equal but odd number of friends, however, it’s possible to have have equal but even number of friends for all students.

What happens if a class has even number of students? You know how to put checkmarks in your notebooks - right? It’s your turn to analyze this situation. Understanding parity (odd and even) can help you in solving many cool problems.

Thanks Ansh Agarwal for sending us a creative proof. You can learn more about Ansh here. You can also add your comments if you have additional thoughts on proving this.