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Ants on circle

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4/14/12
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You have a circle with a number of ants scattered around it at distinct points. Each ant starts walking at the same speed but in possibly different directions, either clockwise or anticlockwise. When two ants meet they immediately change directions, and then continue with the same speed as before. Prove that the time the ants can take to be back to their original positions simultaneously equals to the time one ant takes to complete a full circle once (ignoring the earlier times at which the ants are back to their positions simultaneously).
 
Very interesting problem.

When two ants A and B meet, they change directions.
Lets view this as when two ants A and B meet, they continue in the same direction but their "original point" gets changed by the reflection property by the radius at the point of intersection as the mirror.

Now, keeping that in mind, we see that in one circle time, an ant will cross another ant at either 0 points or at 2 points diametrically opposite to each other.

By commutative property of reflection for mirrors all crossing through the centre of the circle, after one circle time, all original points will be at their original locations. Hence, all the ants will complete one full circle at one circle time.

Hope that helps.
Thanks

P.S.: I am not really sure how to explain "By commutative property of reflection for mirrors all crossing through the centre of the circle, after one circle time, all original points will be at their original locations." very well.

Pratik
CSE Blog - quant, math, cse puzzles
http://www.pratikpoddarcse.blogspot.com

Similar puzzles:
http://pratikpoddarcse.blogspot.in/2009/10/walking-ants.html
http://pratikpoddarcse.blogspot.in/2009/11/three-ants-hc-verma.html
 
how can you prove that two ants hit each other twice if there are n other ants on the circle?

Very interesting problem.

When two ants A and B meet, they change directions.
Lets view this as when two ants A and B meet, they continue in the same direction but their "original point" gets changed by the reflection property by the radius at the point of intersection as the mirror.

Now, keeping that in mind, we see that in one circle time, an ant will cross another ant at either 0 points or at 2 points diametrically opposite to each other.

By commutative property of reflection for mirrors all crossing through the centre of the circle, after one circle time, all original points will be at their original locations. Hence, all the ants will complete one full circle at one circle time.

Hope that helps.
Thanks

P.S.: I am not really sure how to explain "By commutative property of reflection for mirrors all crossing through the centre of the circle, after one circle time, all original points will be at their original locations." very well.

Pratik
CSE Blog - quant, math, cse puzzles
http://www.pratikpoddarcse.blogspot.com

Similar puzzles:
http://pratikpoddarcse.blogspot.in/2009/10/walking-ants.html
http://pratikpoddarcse.blogspot.in/2009/11/three-ants-hc-verma.html
 
When two ants meet they immediately change directions, and then continue with the same speed as before.
This is a similar problem to the ants walking on a rod. When two ants meet they immediately switch directions. But we can also think about this as the two ants meet and keep walking. So they get back to the original points simultaneously.
 
Since the relative position of all the ants never change and they are all traveling at the same speed.
 
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