# Ian and Dog

Ian has a very energetic dog. Ian’s dog runs faster than Ian can walk.

At the start of the day, Ian leaves his house, with his dog, and walks towards his barn along a fixed path. Ian walks at a constant speed. His crazy dog sprints for the barn along the same path. The dog also travels at his own constant (faster) speed.

The dog gets to the barn, turns around immediately, and then heads back to meet Ian. The dog meets Ian one minute after leaving the house. Not exhausted, the dog immediately heads back off to the barn for a second time, turns around, comes back, and meets Ian for a second time. This process continues. When the dog meets Ian for the

*third*time, Ian notices that exactly two minutes have passed since they first left the house.Based on this, what percentage of the distance to the barn had Ian walked when he first encountered the dog?

What percentage of the distance to the barn had Ian walked when he first encountered the dog?

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## Solution

Let’s define the distance between the house and the barn to be one unit, and Ian’s speed to be

*i*of these units per minute.At the end of the first minute, Ian will have walked a distance

*i*, and his dog will have run*1 + (1 - i) = 2 - i*units.Recursively, we’re now back to a self-similar problem. From this new position, it’s like Ian is starting again, except this time the distance to the barn is

*(1-i)*units, not one unit. Similarly for the third meeting, he and the dog start off at the some location, but the distance to the barn is now just*(1-i)*units.^{2}Because it took one minute for the pair to meet the first time, the time it will take to meet when the distance is

*(1-i)*, instead of one, is just*(1-i)*minutes. The distance to the barn for the third trip is*(1-i)*units, and so the time between meetings on this trip is^{2}*(1-i)*minutes.^{2}We know they met for the third time at the two minute mark (which is one minute

*later*than the first meeting). So, the time for dog’s second and third expeditions must total one minute.*(1 - i) + (1 - i)*

^{2}= 1Expanding, solving the quadratic, and taking the only sensible solution yields:

When Ian meets his dog for the first time, he will be approximately 38.2% of his way to the barn.

When Ian meets his dog for the first time, he will be approximately 38.2% of his way to the barn.

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