The Navigation Compass problem is a game involving 3 rings, $A, B,$ and $C$, which each have their own initial angle and rotation per step --- rings may rotate in multiples of $\frac{\pi}{3}$ either clockwise or counter-clockwise. Additionally, the rings only rotate in 3 patterns, e.g. ring $A$ may move simultaneously with ring $B$ and/or with ring $c$ or by itself. These patterns are not known prior to starting the problem. The goal of the player is to rotate the rings such that they are all oriented $\pi$ radians or $180\degree$. This can be formulated as solving the following system of equations. \begin{align} \cos(a_1 x_1 + a_2 x_2 + a_3 x_3 + a_0) &= -1 \nonumber \\ \cos(b_1 x_1 + b_2 x_2 + b_3 x_3 + b_0) &= -1 \nonumber \\ \cos(c_1 x_1 + c_2 x_2 + c_3 x_3 + c_0) &= -1 \end{align} \begin{align} a_1 x_1 + a_2 x_2 + a_3 x_3 + a_0 &= \pi \nonumber \\ b_1 x_1 + b_2 x_2 + b_3 x_3 + b_0 &= \pi \nonumber \\ c_1 x_1 + c_2 x_2 + c_3 x_3 + c_0 &= \pi \label{system} \end{align} The initial angle of ring $A$ is denoted as $a_0$. The amount it would rotate for a single step in each of the 3 movement patterns is $a_1, a_2,$ and $a_3$. All angles $a_0, a_1, a_2,$ and $a_3$ can only be multiples of $\frac{\pi}{3}$, so that each ring can only be in 6 distinct orientations. The variable to solve for, the number of times to rotate each pattern, is denoted by $x_1, x_2,$ and $x_3$. Assuming a configuration of a Navigation Compass puzzle is solvable, a solution may have fractional answers for the value of $x_1, x_2,$ and $x_3$. Since these variables --- which represent the finite number of times a pattern should be moved --- are discrete, any non-real number $x_1, x_2, x_3 \notin \mathcal{R}$ may or may not be an actionable answer, i.e. the puzzle has no solution.