## Wednesday, December 09, 2009

Another question submitted by Rich:

Alice is an ant that lives on a stick. Alice is no different from other ants : All ants walk at a speed of 1 metre per minute, are perpetually walking, and upon reaching the end of sticks or meeting other ants they turn and walk back the other way (they do not fall off nor can pass one another). We now place a finite number of ants on Alice's stick (again with Alice in the middle) and start them walking. We do not know how they are oriented. Could there be an ant in the middle of the stick after a minute? If so when would this ant be Alice?

1. Apologies, forgot to include that Alice's stick is 1 metre long! Is quite fundamental to the question actually as well.

2. So does this false choice of governance think the ant will be back in the middle after a minute? Or would the ant be Obama instead of Alice?

Oh, wait...that's a conspiracy too.

3. Here's Rich's solution:

After a minute there will always be an ant in the middle, regardless of how many we placed on there or how they were oriented. This can be visualised if we give Alice a baton to begin with and every time an ant meets another, as well as saying hello it passes the baton on, so the baton will move at a speed of one metre per second until it reaches the end of the stick and will come back the other way, thus the baton will be in the middle of the stick after a minute and will obviously be accompanied by an ant.

The lucky ant holding the baton after a minute will be Alice if and only if there are an equal number of ants either side of her when we start. To prove this it is required to notice that after a minute the ants (as a collective) will have moved into positions that were a mirror image of how they started, so if Alice were in the middle after a minute and we started with L ants to the left and R ants to the right of her after a minute there would be R ants to the left of her and L ants to the right of her, but since ants cannot pass one another L must equal R for Alice to be the one in the middle.

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