There is a puzzle that has been around for several years called
something like "Nuts To You" or "Drives You Nuts" or some variation
thereof. It consists of 7 hexagonal pieces (each resembling a nut,
hence the name) each with the numbers 1 through 6 inscribed around its
perimeter in some order. The idea is to play a kind of hexagonal
dominoes with these pieces; a solution is found when all 7 pieces are
arranged in a hexagon (the puzzle comes complete with a hexagonal
frame for arranging them, it also resembles a nut) such that every
pair of adjacent pieces are labeled the same along their common edge.
For example one piece might look like this:

   _____
  /  6  \
 /3     1\
/         \
\         /
 \2     5/
  \__4__/
                                6
which I shall abbreviate by:  3   1
                              2   5
                                4

which might participate in the following solution:

          5
        6   4
  6     1   2     3
3   1     3     2   4
2   5     3     1   6
  4     5   1     5
  4     2   6     5
5   2     4     6   1
3   1     4     3   2
  6     1   3     4
        5   2
          6

Each piece can be rotated (of course), but it cannot be flipped over.
(This restriction is enforced by simply not printing the numbers on
the other side!)

The seven pieces I have represented in my sample solution are
definitely NOT the ones that make up the commercially sold puzzle. I
don't have that anymore, I just made these up. It did have the
property that no two pieces were alike, just like my hypothetical set.

The commercial set of pieces seemed to have the property that there
was JUST ONE possible solution, although I cannot be 100% certain that
this was the case.

Now the puzzle itself is a bit dry. But what I wonder about is the
following meta-problem:

Given that there are many (how many?) different sets of 7 pieces that
can be chosen, how "interesting" a property is it for there to be only
ONE solution? Do you have to try hard to achieve that property? Or
do most of them have it? Indeed, do ANY of the have it (remember I
only SUSPECT that the commercial set has it!).

How about if we relax the restriction about duplicate pieces?

-Alan