We know that if you start at the beginning of the list and take `i`

steps that you will end up at the meeting point, and we know that if you start at the meeting point and go `i`

steps that you will end up at the meeting point again. Another way to state that last part is that if you start at the beginning and go `2i`

steps that you will end up at the meeting point.

Now, imagine you take two tortoises and start them from the meeting point. Visualize them both taking one step backwards at a time, except one goes to the beginning of the list and the other one goes through the cycle. Notice that their paths diverge at the beginning of the cycle, and notice that if you keep stepping backwards until one reaches the beginning of the list, the other one ends up at the meeting point.

The breakthrough I had was being able to visualize this, and that’s when it made sense intuitively.

]]>“But think about it: they wouldn’t just meet at the meeting point node, they would meet for all of the nodes at the beginning of the cycle until they reached the meeting point node.”

Can you expand on why that is the case? It makes sense that they would meet at the meeting point node because the cloned tortoise goes `i`

steps and the original tortoise goes `2i`

steps, thus replicating what the original tortoise and the hare did, respectively. But why does it then follow that the cloned tortoise and the original tortoise would meet at the start of the cycle?

One intuitive reason that comes to mind: the original tortoise and the cloned tortoise move at the same speed, so if they _didn’t_ meet at the beginning of the cycle, then they would _never_ meet. So by proving that they _do_ meet at some point, then we implicitly show that they must meet at the start of the cycle, whereupon they move in lockstep until the `2i`

meeting point eventually arrives.

Dear Miss Kitterson,

People use those social media buttons? I thought those were just for decoration. I will consider it!

]]>Yours Truly,

Miss Kitterson