Ant and Triangle Problem


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 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

    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.

    so 2/8= 1/4 = 0.25

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


    • [email protected] on January 24, 2017 at 8:55 pm said:

      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:
    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???

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