[psyang]

In fact, you are correct! Great work!

Spoiler!

The trick is to play the scenario backwards as you discovered. If you assume the path they take is the y-axis of a graph, then you can plot each of their positions over time (in minutes).

A will be a straight line from (0,0) to (60,4)
B will be a straight line from (0,0) to (60,3)
C will be some sort of line that bounces between A's line and B's line. The slope of line segments that make up C's line will be 10 or -10.

The main point is that C's line is bounded by A and B's line, and converges to (0,0).

If you were to place C anywhere between (60,4) and (60,3), C's line would still converge to (0,0).

So, in fact, C can be anywhere.

We get caught up because the scenario is set up to be deterministic. They all start at the same spot, they move at fixed speeds and follow a specific path. But in the immediate nanosecond time interval after they start, C's path is chaotic - C bounces between A and B an infinite number of times!

A few comments below.

Spoiler!

Originally Posted by GGG

This is like a backwards version of the fly on a train problem. Where two trains are travelling toward each other at different rates and the fly goes back and forth. The question there is how far does the fly travel which you can determine from the flys velocity times the time or by summing the series.

Yeah, I was thinking of posting that problem. It's made famous because it was posed (allegedly) to John Von Neumann and he immediately answered. The person said "oh, you figured out the trick!" to which he responded "what trick, I just summed the infinite series!".

I was at a social gathering several years ago, and was chatting with another guy who enjoyed math, so I told him the Von Neumann story. It was loud because there were 20 or so people in a smallish space all talking in their own conversations. When I got to the punchline, there was a sudden lull in the conversations, and in almost perfect silence I said with gusto "I just summed the infinite series!". Immediate laughter.

Originally Posted by GGG

The framing is different here as you start with an infinitely small distance to be travelled in an instantly small time like the paradox that states and journey can never end or begin because you always can only go half of the remaining distance. So I keep getting stuck in this reverse limit.

Yup, limits and infinity are hard to grasp.

I remember when I was younger (maybe elementary or junior high?) and I had learned about the math concept of a ray - a line going out to infinity from a point. I thought "What if you were on a train that took the path of a ray. It would go forever. Now what if you were on a train that went in the opposite direction of a ray. You would come from infinity and stop at a point. The conductor of the train would say "We've been travelling forever, but we finally made it". It made me laugh, but also blew my mind because I couldn't wrap my head around the idea of "coming from infinity" .

Originally Posted by GGG

So if I run the problem backwards such that Bob and Alice are 1 km apart walking home and the gap is closing at 1km/hr. In this scenario it doesn’t matter where the dog starts it just gets smushed after 1 hour when the gap disappears and since 1 hr has elapsed The dog has run 10k. But this means that dog can be anywhere between them.

I don’t think that’s quite right but I can’t resolve the paradox I’m in.

You were dead on! Laughed at the phrasing of the dog getting smushed - should be in a Sliver thread! - but I love how you got to the right conclusion even though you can't believe it's right because it doesn't mesh with a deterministic real-world scenario. Math is not always real world

My favorite quote is "In theory, there is no difference between theory and reality, but not in reality"