# Navigating a Field of Sprinklers

Everyday I walk to class and I have to walk through a sequence of sprinklers. I usually watch them for a second and try to plan a path in which I never have to stop or back track and will not get wet. It would be even better if I could maintain a constant speed.

## Question

If we consider sprinkler system to be an $n \times n$ lattice where the water extends a length $L$ from the lattice point, has random initial direction and each rotates at a constant angular velocity $v$, what is the smallest $L$ and $n$ such that there exists no smooth, path from $(0,0)$ (or more generally, from any $(a,b)$) to $(n,n)$ and $\frac{dx}{dt} \ge 0$, $\frac{dy}{dt} \ge 0$? Or, what conditions must we impose on this system to guarantee a solution (other than having them all rotating the same and having me sprint arbitrarily quickly)?

Example sprinkler system:

## Observations

Any case where $L \le .5$ is trivial since there is always a solution by following the outside of the circles formed by the sprinklers. Similarly, $L<\sqrt{2}n$ for any $n$. Of course there will not always be such an $n$ and $L$ like in the $2 \times 2$ case where there is a solution for all $L$ and any initial orientation. I feel like if $n$ and $L$ are sufficiently large, there will be no solution since you will eventually be caught inside of a polygon that is shrinking, but I do not have bounds on this conjecture.

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In your figure all the sprinklers rotate in the same direction (which I think is like reality) but in your text every other one (using what ordering?-a checkerboard?) rotates oppositely. – Ross Millikan Sep 27 '12 at 14:11

Observations for $L < \sqrt{2}$:
If the walker moves at a finite speed we need to consider the angular rotations. If all the angular rotations are rational (or the same), then there is a finite time at which the pattern will repeat itself. In your case all the sprinklers rotate at $\omega$ or $-\omega$, so this is guaranteed. Each intersecting pair will have a time at where they are intersecting or not. This gives a set of potentially $2^n+1$ different graphs, each existing for a known lifetime. For a given walker speed it is now possible to determine if a safe path exists by traversing the (now time-dependent) cyclic set of graphs.
• With a larger $L$ the same method should still hold, but I'm a bit unsure how the graph edges would be removed.