# 3 Ants and Triangle Puzzle

Three ants are sitting at the three corners of an equilateral triangle. Each ant starts randomly picks a direction and starts to move along the edge of the triangle. What is the probability that 2 of the ants collide?

## 3 Ants and Triangle Puzzle Solution

Let’s consider one of the ants as the reference point, and assume that it moves in a fixed direction (say, clockwise). Then there are two cases for the other two ants:

1. They move in the same direction as the reference ant. In this case, they will never collide with each other, because they will always be moving in parallel along the edges of the triangle.
2. They move in different directions than the reference ant. In this case, there are two possibilities:
• They move towards each other, in which case they will collide at the midpoint of the edge between them.
• They move away from each other, in which case they will never collide.

So, the only way for two ants to collide is if they move in different directions than the reference ant and move toward each other. Since there are two directions for each ant, there are a total of 2^3 = 8 possible configurations. However, we have already seen that 6 of these configurations (corresponding to cases 1 and 2 above) will not result in a collision. So, there are only 2 configurations that lead to a collision.

Therefore, the probability of 2 ants colliding is 2/8 = 1/4 = 0.25.

## 2 thoughts on “3 Ants and Triangle Puzzle”

1. Hi Rohit, you are considering only two of the three ants, but at point 2 one of them will certainly collide with the third, which has a fixed direction. I think you should raise the problem in a different way to make it more understandable.

The question was What is the probability that 2 of the ants collide?
0.25 is the probability that NONE of the ants collide with each other. Which are the configurations when the three of them are going in the same direction right or left.
Therefore, the probability of 2 ants colliding is 6/8 = 3/4 = 0.75, as Tony said.

2. Hi Rohit,

Are you sure 6 of these configurations will *not* result in a collision? The way I see it, only 2 configurations will *not* result in a collision – if all ants move clockwise or all move anti-clockwise. Therefore, the probability of 2 ants colliding is 6/8 = 3/4 = 0.75.

Tony