## Suppose that you have a triangle with 3 ants on different vertices (corners) of the triangle. What is the probability that either 2 of the ants or all of the ants collide if all 3 ants start walking on the sides of the triangle?

Think about this puzzle for a minute. It should become clear that whether or not the ants collide depends on what direction the ants are walking in.

## Reverse the way the riddle is asked

Let’s start by reversing the way the riddle is asked and find the probability that there will ** not** be a collision instead. When will this happen?

Well, what if all of the ants are walking in the same direction? Then there will never be a collision between any of the ants because they are all walking in the same direction. And, the only time they will be walking in the same direction is if they are all walking either counter-clockwise around the triangle, or clockwise around the triangle. So, there are only two scenarios in which a collision will not happen between the ants.

## 3 Ants on a Triangle Riddle Solution – the probability

Now that we know that there are only two scenarios where the ants will not collide, we have to ask ourselves how many different ways are there for the ants to move on the sides of the triangle? Well, each ant can move in 2 different directions. Because there are 3 ants, this means that there are 2^{3} (which equals eight) possible ways that the ants can move. And since we already know that there are only 2 ways in which the ants can avoid collision entirely, this means that there are 6 scenarios where the ants will collide. And 6 out of 8 possible scenarios, means that the probability of collision is 6/8, which equals 3/4 or .75. Thus, the probability of the ants colliding is .75.