Frankie has stored all of her food on lily pad A. However, her food has a tendency to “fly” away. Every second, the food that’s on every lily pad splits up into six equal portions that instantaneously relocate to the six neighboring pads.
At zero seconds, all the food is on lily pad A. After one second, there’s no food on pad A, and $1/6$ of the food is on each of the surrounding six pads. After two seconds, $1/6$ of the food is again on pad A, while the rest of the food is elsewhere.
After how many seconds $N$ (with $N > 2$) will pad A have less than $1$ percent of its original amount?
Here we can code up the six possible directions and let the probability distribution evolve through time. Let's say that at time $t$ we have $\pi(t, a, b) \in (0,1)$ proportion of all of Frankie's food at the lily pad with center at $a\zeta + b\xi.$ Then we see that this means that it recursively should have received 1/6 of the food on lily pads $(a\pm 1, b)$, $(a, b\pm 1),$ and $(a \pm 1, b\mp 1)$ at time $t-1,$ that is \begin{align*}\pi( t, a, b ) &= \frac{1}{6} \Biggl( \pi (t-1, a+1, b) + \pi( t-1, a-1, b) \\ & \quad\quad\quad +\pi( t-1, a, b+1) + \pi (t -1, a, b-1) \\ &\quad\quad\quad\quad + \pi( t -1, a+1, b-1) + \pi (t-1, a-1, b+1) \Biggr),\end{align*} for all $a, b \in \mathbb{Z}, t \in \mathbb{N},$ with initial condition $\pi(0,0,0) = 1$ and $\pi(0,a,b) = 0$ for $a\ne b \ne 0.$
Coding this up in Python and then solving for $T = \min \{ t \gt 2 \mid \pi(t, 0,0) \lt 0.01 \}$ yields that Frankie's food on lily pad A will be less than $1\%$ of the total after $T = 17$ seconds.
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