Here is my problem. Fix a color for the number $1$, for example yellow. Choose another color, for example green. Now, for a positive rational denoted $x$, there are two rules :
- $x$ and $1/x$ have the same color.
- $x$ and $x+1$ have different colors. (ie if $x$ is green, then $x+1$ is yellow; if $x$ is yellow, then $x+1$ is green)
One can see (using continued fraction) that you can find a color for every positive rational number $x$. My question is : is this coloring perfectly defined? Or using these rules, can we find a positive rational number that is yellow and green at the same time?
In fact, I saw a contest problem asking to find the color of a specific number (so one may think that this coloring is perfectly defined) and saying "Do not try to prove that this coloring is perfectly defined". I actually have no idea if it is a difficult problem or not.
As you suggest, writing $x$ in continued fractions $x=[x_0,x_1,\ldots x_k]$ determines univocally the color of $x$: if $a_0+a_1+\ldots +a_k$ is odd then $x$ is yellows, otherwise is green. It remains to see if this coloring satisfies the two rules:
i) if $x=[x_0,x_1,\ldots x_k]$, then $1/x=[0,x_0,x_1,\ldots x_k]$, so they have the same color
ii)if $x=[x_0,x_1,\ldots x_k]$, then $x+1=[x_0+1,x_1,\ldots x_k]$, so they have different colors.