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The Ant Walk


I don't claim to have made this puzzle, nor do I have a solution.

The is a circle of radius 3 meters with an ant in the centre. The ant picks a direction randomly, all angles have an equal probability. It walks in this direction for 1 meter before fogetting where it was going and chosing another direction randomly. This could go on forever, but on average how many meters has the ant travelled before it exists the circle?

Computers are allowed, but I do want to see a mathematical solution.

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This is not quite an answer to the question, but the square of the distance from the origin shold increase by 1 meter squared for each 1 meter travelled--so the suare of the expected distance would be the square of 3 meters after the ant has taken 9 of its 1 meter walks.

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The meandered without a definitive answer.

Since I contributed a question that that thread, I'll answer it after answering the OP.

Here the angles of successive steps are not quantized.

In this case, one can show the average square of the distance

one travels from center increases linearly with the number of steps.

In fact, the average square of the distance is just N times the step length,

where N is the number of steps taken.

So the simple result is, to escape a circle of radius 3, taking unit steps,

you need on average to take 32 = 9 steps.

I simulated this for circles of radii 3, 10 and 100 steps and got averages

ranging from 10-11, 103-106, and 9485-11380, respectively.

jim gives the same answer above. ;)

For a two-dimensional random walk on a grid:

Think of a drunk touring a city with streets and avenues equally spaced.

At each intersection a choice is make from four directions [including reversing direction].

After a sufficient number of steps, every intersection of an arbitrarily large city will be visited.

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