Alien Finger Puzzle

Introduction

I found this problem as part of the Mensa Logic Brainteasers book.

Problem

There is a number of aliens in a room, more than one. Each alien has more than one finger on each hand. All aliens have the same number of fingers as each other. All aliens have a different number of fingers on each hand. If you knew the total number of fingers in the room, you would know how many aliens were in the room. There are between 200 and 300 alien fingers in the room.

How many aliens are in the room?

Solution

We know that all aliens have the same number of fingers as each other. Let's call this value $f$. Since each alien has more than one finger on each hand, $f>1$. Furthermore, let $a$ be the number of aliens in the room. Since there is more than one alien in the room, $a>1$. The number of alien fingers is then $a\cdot f$, where $200\le a\cdot f\le 300$.

The key thing to notice is that if we were told the total number of alien fingers in the room, we would be able to uniquely determine $a$, the number of aliens in the room. Since $a$ is a factor of the total number of alien fingers in the room, it means that in order to be able to uniquely determine its value, it needs to be the only factor. In other words, $a$ has to be a prime number, and $a=f$.

So the total number of alien fingers in the room is a value between $200$ and $300$ that is the square of a prime number. The only number that meets this criteria is $289={17}^2$, which means there are $17$ aliens in the room.