. We color the non-zero rational numbers in two colors: white and black. We assume that 1 is colored white, that $x$ and $x+1$ are never colored the same color, and that $x$ and $\frac{1}{x}$ always have the same color. What is the color of $\frac{1543}{275}$?
. By applying the definition several times, it follows that $\frac{1543}{275}$ has the opposite color of $\frac{1543}{275}-5=\frac{168}{275}$, which in turn has the same color as $\frac{275}{168}$. We continue the reasoning in the same way: $\frac{275}{168}$ has the opposite color of $\frac{275}{168}-1=\frac{107}{168}$, which has the same color as $\frac{168}{107}$,
which has the opposite color of $\frac{168}{107}-1=\frac{61}{107}$, which has the same color as $\frac{107}{61}$, which has the opposite color of $\frac{107}{61}-1=\frac{46}{61}$, which has the same color as $\frac{61}{46}$, which has the opposite color of $\frac{61}{46}-1=\frac{15}{46}$ which has the same color as $\frac{46}{15}$, which has the opposite color of $\frac{46}{15}-3=\frac{1}{15}$ which has the same color as 15, which itself has the same color as 1, that is, white. By retracing, the color of $\frac{1543}{275}$ is also white.