Uh...isn't this just binary?
a ring is 0
a ring,ding is 1
The ring represents a doubling of its contents. The special case for the leading bit could be avoided by always having an innermost empty ring. That would also give you a better (i.e. visible) representation for 0. The addition rule ensures that you always have exactly 1 leading 0.
> The rings must be nested. The innermost ring must have a ding. No region can have more than one ding. Those are the only rules.
According to these rules the intermediate results are not valid numbers
(They correspond to things you can do with the expanded algebraic representation of a base-2 number, Sum a_k 2^k, but that's a necessary prerequisite for such a number system to work. After all, a way of writing numbers that did not ultimately correspond to the existing ways of writing numbers would be wrong.)
1×2 + 1×4 + 0×8 + 1×16, 1×8
and substitute the second for every 1 in the first,
8×2 + 8×4 + 0×8 + 8×16
then finally un-distribute,
(1×8)(1×2 + 1×4 + 0×8 + 1×16)
and that's why putting small copies of the second number in the place of every "ding" in the first number results in multiplication.