The Unexpected Result of Twisting on the Magic Paving Stones

At the end of Twisting on the Magic Paving Stones, I noted that there was something unexpected.  Please look back over previous articles to see the full details but, in short, what I was discussing is a scenario in which, in each round of a sort of game, a number of magic paving stones (≥2) are inserted at random locations (with an evenly distributed likelihood) into a path, then a monster takes a step from one paving stone to the adjacent one closer to you (starting at one step closer to you than the last paving stone in the path) and then you eliminate one paving stone of your choice (and since you don’t want the monster to get you, you are going to eliminate one behind it, further away from you).

Set the number of randomly inserted paving stones to 2, the initial pathlength to a sufficiently large value (for which I choose 120, so the monster starts on paving stone number 119) and plot the output once every 200 rounds and you get this:

It might not be immediately obvious, but we’ve seen something like this before (or at least those of us who looked at the OE curve).  To emphasise this, here is that monster location curve with the OE curve below it – noting that smoothing has been turned off.

This is rather curious and was in fact something that I was struggling with – having previously posted two articles on the same sort of thing before retracting them (soon to be reinstated at Observable Events Curve - Is Double Dipping Essential? and Observable Events Curve - Not Quite a Drunkard's Walk).

Note that the time (or number of rounds) that is taken for the monster to get you (or “capture time”) is not set, there’s something akin to chaos happening in that the capture time very much depends on slightly different conditions early in the scenario, which – after the first round – are randomly imposed.

The OE Curve line in red above is based on time it takes for the separation to reach zero, t₀ (with a relatively insignificant offset to account for the fact that the initial separation is not zero, but rather an offset that we can call x_i).  The final equation becomes:

x'=ct((ct₀+x_i)-ct)/(ct₀+x_i)

If ct₀>>x_i and ct=x, this approximates, of course, x'=x(ct₀-x)/ct₀.