# Ant and Triangle Problem

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 none of the ants collide?

Puzzle Solution:

So let’s think this through. The ants can only avoid a collision if they all decide to move in the same direction (either clockwise or anti-clockwise). If the ants do not pick the same direction, there will definitely be a collision. Each ant has the option to either move clockwise or anti-clockwise. There is a one in two chance that an ant decides to pick a particular direction. Using simple probability calculations, we can determine the probability of no collision.

P(No collision) = P(All ants go in a clockwise direction) + P( All ants go in an anti-clockwise direction) = 0.5 * 0.5 * 0.5 + 0.5 * 0.5 * 0.5 = 0.25

### 0 Thoughts on “Ant and Triangle Problem”

1. Amit Gaur on July 1, 2013 at 10:21 am said:

A simple way to solve.
P(#collision)= # of way when there is no collision/ total number of way ants can move
= 2/2^3
= 1/4

there is only 2 way when there is no collision when all 3 ants move clockwise or anticlock wise
and there are 3 ants and each can move in 2 direction so there will be 2^3 way they can move.

2. nice explanations both.

3. Each ant has 2 possibilities of moving in a direction.. clockwise or anti-clockwise. Now as there are 3 ants .. we have 8 total possibilities of the ants moving in a direction.

Now out of 8 we can have only 2 possibilities where they will not collide.
i.e if all the ants travel clockwise or all the ants travel anti-clockwise.

so the probability is 2/8 = 1/4 = 0.25

4. Ravinder Singh on May 10, 2015 at 3:25 pm said:

Let the 3 ants x, y, z
anticlockwise- A & clockwise C
{xA,yA,zA}
{xA,yA,zC}
{xA,yC,zC}
{xC,yC,zC}
{xC,yA,zA}
{xC,yC,zA}
{xC,yA,zC}
{xA,yC,zA}

Here in 8 cases only in 2 cases there will be no collision when all the ants will travel in same direction either clockwise or anticlockwise.
i.e
{xA,yA,zA}
{xC,yC,zC}

so 2/8= 1/4 = 0.25

• shubankar on August 25, 2015 at 2:04 am said:

nice

• Best explanation. I was confused which are those 8 moves. Thanks for listing those 8 available options.

5. Ketan Kankapure on October 16, 2015 at 1:26 pm said:

Three Ants:
0–>clockwise
1–>Anticlockwise
All combinations will be :

A B C

0 0 0 [No collision]
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1 [No collision]

Probability = 2/8=0.25

6. Question lacks a lot of useful info. Speed of ants should be same. Also, they should start moving simultanesouly, not randomly.

7. gauti on July 20, 2016 at 5:16 pm said:

what is probability of colliding???

8. 6/8