Match dimension, distance model and asymptotic regime. An improved construction establishes a lower bound; an upper bound constrains all constructions. Neither automatically gives the exact optimum.
Packing identical oranges is familiar in three dimensions. Replace the room by a space with many independent coordinates and the same question becomes much harder: how much of the space can disjoint equal balls occupy?
Put many points on paper and connect every pair exactly one unit apart. How many lines can you get? An OpenAI model found examples that beat a long-standing guess about the best arrangement.