There are ${6\choose2}=15$ different pairs $jk$, and each day consumes three of them. Consider these pairs as sides and diagonals of a regular hexagon $H$ with vertices $\\{1, \ldots,6\\}$. You have produced the first day list $d_1=(12,34,56)$. Rotate $d_1$ by $60^\circ$, meaning: add $1$ to each entry, and obtain $d_2=(23,45,61)$. Now take the pattern $d_3=(14,26,35)$, and rotate it $2$ times by $1$. In this way you obtain $d_4=(25,31,46)$ and $d_5=(36,42,51)$. Now each side or diagonal of $H$ appears exactly once in a $d_i$ $(1\leq i\leq 5)$.