I noticed a funny but tenuous relationship between some finite algebraic gadgets that are kind of like fields, and combinatorial objects that are like necklaces of a certain type.
The algebraic gadget is: it's got the multiplicative structure of a cyclic group of n things (just like a finite field of n things would) and an additive structure which distributes over the multiplication (and has a unit and is commutative) but isn't necessarily associative (!!). Do these have a standard name? Am I fooling myself that any of these exist that aren't fields? Anyway, this includes all fields at least.
The combinatorial object is: take n dots in a circle, and choose one of them. And choose two of them that are adjacent (i.e. 1 apart) on the circle. And choose two of them that are 2 apart from each other on the circle. And choose two that are 3 apart, etc. until there's only one or zero dots left.
For an example, consider GF(8). Pick a generator g. We know g^{7} = 1.
1 + g has to equal g ( 1 + g^{6} ) = g + 1 by commutativity and distributivity of addition.
1 + g^{2} has to equal g^{2} ( 1 + g^{5} ) in the same way
So if we write out the first row of a possible additiontable for GF(8)
x  1  g^{1}  g^{2}  g^{3}  g^{4}  g^{5}  g^{6}

x+1  0 
g^{4} 
g^{1} 
g^{5} 
g^{2} 
g^{6} 
g^{3}

I see
4 is ahead of 3 by 1,
1 is ahead of 6 by 2 (mod 7), and
5 is ahead of 2 by 3.
Those are the choices of pairs on the necklace.
So it's possible to make a necklacewithpairings on a pentagon, in exactly one way: if the pentagon is ABCDE, choose A, and the pair CD (1 apart) and the pair BE (2 apart). And this seems to give rise to some distributive algebraic structure like I said above, but its addition isn't associative  and it couldn't possibly be, because there's no such thing as GF(6). 