micromass said:
Well, the problem I'm having is that some 10 beads are longer than other 10 beads. So you probably won't get a nice metric here...
But this is a philosophical argument really

, that means that everybody can be correct...
I checked which subforum we're in and apparently it's "General Math".
I think a discussion about metrics is not out of place here.
I wouldn't want to post in the wrong forum. :)
A metric on a set X is a function (called the distance function or simply distance)
d : X × X → R
(where R is the set of real numbers).
For all x, y, z in X, this function is required to satisfy the following conditions:
1. d(x, y) ≥ 0 (non-negativity)
2. d(x, y) = 0 if and only if x = y (identity of indiscernibles. Note that condition 1 and 2 together produce positive definiteness)
3. d(x, y) = d(y, x) (symmetry)
4. d(x, z) ≤ d(x, y) + d(y, z) (subadditivity / triangle inequality).
I think these conditions are all satisfied.
And if some 10 beads are "longer" than some other 10 beads, we'll simply have to round it to an integer number of beads.