# Three Ants on The Corners of a Triangle

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Three ants are sitting at the three corners of an equilateral triangle. Each ant randomly picks a direction and starts to move along the edge of the triangle. What is the probability that none of the ants collide?

How about other shapes, Square? N sides?

Nishant Guru Asked on 8th August 2015 in

In order that there is no collision we require that all the ants move in the same direction. Either all clockwise or all anticlockwise.

We assume the ants have a 50/50 chance of picking either direction. So the probability of them all deciding to go clockwise is given by ½•½•½ = 0.125

The probability of them all deciding to go anticlockwise equally is given by ½•½•½ = 0.125

Either of these will do so we can add the probabilities to make 0.25

There is another approach that perhaps requires slightly less understanding of probability. With three things each having two choices we have 2x2x2 = 8 possible configurations. If ‘A’ indicates anticlockwise and ‘C’ clockwise they are AAA, AAC, ACA, ACC, CAA, CAC, CCA & CCC. Of these 8 only 2 are of use to us. AAA and CCC. 2 of 8 is ¼ or 0.25

## Square, N sided Polygon

Using the first approach for the triangle we had 2•½•½•½ or 2•(½^n) or 1/2n-1 or 2-(n-1) where n was equal to 3. We can see trivially that for a square the answer will be 1/8

Using the other approach we have that there are 2n configurations, of which 2 will be useful to us. 2/2n brings us to 1/2n-1

## Assumptions

I think it’s fairly clear that there are no real ants, the ants are just a device for explaining the puzzle. Nonetheless assumptions might be that the ants direction picking is unbiased, and that they move with the same speed.

Guddu Curious Answered on 8th August 2015.

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