Let $n=8m+1, m\in\mathbb{N}$. Does the set of nonzero elements of $\mathbb{Z}_n$ split into disjoint octets of the form $8_k=\{\pm a_k,\pm b_k,\pm a_k\pm b_k\}$?
The computer tells me it is possible if $n$ is not a multiple of 3, up to $n=113$. Larger $n$ have many solutions; $n=73$ has thousands of solutions.
For example, $Z_{17}\backslash\{0\}=\{\pm 1,\pm 4,\pm 1\pm 4\}\cup\{\pm 2,\pm 8,\pm 2\pm 8\}=\{\pm1,\pm4,\pm3.\pm5\}\cup\{\pm2,\pm8,\pm10,\pm6\}$
My motive is this: I am idly wondering whether there are $m$ patterns like this:
- Each pattern has an $n$-omino. This $n$-omino has $n$ different coloured tiles, and it tiles the plane by translations.
- Different patterns may have different $n$-ominoes, but the same $n$ colours.
- Any two colours are adjacent (edge or corner) in exactly one pattern.
- A solution would be to colour tile at $(c,d)$ in pattern $k$ with colour $ca_k+db_k(\bmod n)$.