AIUI, the idea is that a solution is a path through a "hard-to-pebble graph", and that the consensus advances by awarding rewards to blocks that take the shorte

AIUI, the idea is that a solution is a path through a "hard-to-pebble graph", and that the consensus advances by awarding rewards to blocks that take the shortest path. The combination of a short transaction window and the massive search space work to reduce the probability that multiple miners find equivalently acceptable solutions. Some schemes have additional features that make the next block dependent on the earlier blocks, so that the chains can't easily be forked long after the fact.