Which is more important in determining author order: time spent or results obtained?

In my field, this would clearly be shared (first) authorship.

As for the exact positions, in my experience one of A and B would also have been spending a lot of time as a de facto project lead that determines the course of the project and does most of the writing. In fact, in my field this would almost always be B, unless A somehow has the capabilities of effectively leading multiple projects and delegating vast proportions of the work in those projects.

I don't place a lot of value on happening to find the right solution for a problem. The way you describe this, it seems almost stochastic: A proposed four solutions and got lucky, B proposed more solutions but didn't get lucky. Obviously the situation and value of contributions changes if B could not have found the solution, but A (possibly because of greater experience) could have.

Two other considerations: first, exact position on a paper can have very different values for different people. A first first authorship can be absolutely vital for scientists who are rounding off their PhD or postdoc, whereas scientists who aren't rounding off could also obtain this in a next project. Second, maybe there are ways to even out the author contributions? One way would be to do another project together and flip author positions for that one. Another way would be that the person who doesn't get author precedence can present the work at conferences for the first year. And maybe there are better ideas that someone in your lab can come up with.

I think the explorations that failed can be as valuable as the one that succeeded, and should be reported along with the success.

See What to do when you spend several months working on an idea that fails in a masters thesis?

This argues for B as lead author. which, along with the fact that they worked harder/longer might settle the question.

(I am fortunate that in mathematics the convention is alphabetical order.)

Weighting contributions in a fair way can be all but impossible. As you noticed, contribution is a vector with many dimensions (time, effort, results, novelty, and whatnot). All attempts on sorting complex contributions on a single dimension axis will need agreement

• not only about the respective extent (which is difficult to measure),

• but also about the weighting of the elements (which needs mutual consent, as there is no "correct" answer).

If the authors disagree about ordering of their names, they are obviously assuming different matrices for projecting the contribution vectors to a one-dimensional value (or are greedy).

To resolve conflicts like these, you can always mention the authors in alphabetical order - maybe including the dept. chair (and add a tiny notice to the paper in order to show you did that).