Imagine at each lattice point of $\mathbb{Z}^2$ within $[1,3n]^2$,
with coordinates
$\equiv 2 \bmod 3$,
we place, with equal probability, one of these six patterns:
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![SixTemplates][2]
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The result is collection of disjoint "worm-like" paths, whose minimum length is $3$.
For example, here is an example for $n=10$:
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![RandDiskPaths10][1]
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The longest path here starts at $(1,14)$, and has length $27$.
My question is:
> **Q**. What is the growth rate of the longest path, with respect to $n$?
With $10$ random trials each, the average longest path for $n=10$ is actually
considerably smaller than $27$; it is in fact $18.3$.
Here is a graph up to $n=50$; it appears to grow
(with considerable variability) roughly proportional to $\sqrt{n}$.
Is there some relatively straightforward way to see what is the
expected growth rate of the longest path?
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![LongestPaths][3]
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<sub>
I gratefully acknowledge programming assistance from several users in response to
[this posting @Mathematica Stack Exchange](http://mathematica.stackexchange.com/q/51247/194).</sub>
[1]: https://i.sstatic.net/s5HAZ.jpg
[2]: https://i.sstatic.net/qr0Sc.jpg
[3]: https://i.sstatic.net/RkAqT.jpg