# Three Ants on The Corners of a Triangle

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?

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 1/2^{n-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 2^{n} configurations, of which 2 will be useful to us. 2/2^{n} brings us to 1/2^{n-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.

### Your Answer

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