I'd like to find eight subsets $S_1$, $S_2$,$\ldots$,$S_8$ of $\{1,2,3,\ldots,8\}$ with the following properties:
1) Each $S_i$ has size 3, and each $i$, $1\leq i\leq 8$, is in precisely three of the $S_i$.
2) The $S_i$ have a unique "system of distinct representatives". That is, there is only one way of choosing elements $x_i\in S_i$ such that all the $x_i$ are distinct, or equivalently that $\{x_1,x_2,\ldots,x_8\}=\{1,2,3,\ldots,8\}$.
I don't know whether such subsets should exist. I tried starting with $S_i=\{i,i +1,i+2\}$ (modulo 8), so e.g. $S_5=\{5,6,7\}$ and $S_8=\{8,1,2\}$; this satisfies (1), and then I tried messing around changing a few elements about to get (2) to work, but I failed. There's a finite field of size 8 but I couldn't see a natural way of constructing eight subsets of size 3 that would work. Similarly there's a finite field of size 9 but I couldn't see a natural way of constructing eight subsets of its multiplicative group of size 3 that would work. This sort of thing is a long way from my area and I thought I should ask in case I'd missed something.