I have a D20 ring. It's a loop with the numbers 1-20 printed around it in what I initially thought was a random order. On closer inspection, I realized that they aren't random; but I'm having trouble discerning the underlying pattern.
Here's the number sequence:
1 6 19 14 9 4 17 12 7 2 15 20 5 10 13 18 3 8 11 16
My work so far
If you start counting at 19 instead of 1, then group the numbers into sets of 4 consecutive numbers, you get this:
$$ \begin{matrix} 19&14&9&4&\\ 17&12&7&2&\\ 15&20&5&10&\\ 13&18&3&8&\\ 11&16&1&6&\\ \end{matrix} $$
Each of these sets is a residue class of 5 (that is, each set contains all the numbers in $5\pmod k$). In order of the sets, $k$ has values of $4,2,5,3,1$.
Subtracting $k$ from each set yields:
$$ \begin{matrix} 15&10&5&0&\\ 15&10&5&0&\\ 10&15&0&5&\\ 10&15&0&5&\\ 10&15&0&5&\\ \end{matrix} $$
From here, I'm stumped. What significance does that order of $k$ have? Why are the multiples of 5 in each set not consistently organized? What might the designer's intent be for choosing this clearly-intentional sequence? Perhaps the pattern has some special significance for generating fair random numbers when the ring is spun?
This seems linked to the knight move in chess.
Starting with $11$ and repeating the sequence gives
(with $20$ going back to $1$)