You can draw a non-crossing spanning tree on a set of points in the plane. You can draw another one. Both are valid. The question is: how many single-edge swaps does it take to get from one to the other?
The answer, it turns out, is NP-hard to compute.
This is striking because finding either tree individually is not hard. You can construct a non-crossing spanning tree in polynomial time. You can verify one instantly. The objects themselves are tractable. But the space between them — the flip graph, where each node is a valid configuration and each edge is a single swap — has a distance function that resists efficient computation.
The pattern recurs across combinatorics. Triangulations, independent sets, graph colorings: the reconfiguration problem is almost always harder than the construction problem. You can build a house. You can build a different house. But computing the minimum number of moves to transform one into the other requires understanding the global structure of all possible houses and all possible moves between them.
This is a general principle worth naming. Construction asks: does a valid object exist? Reconfiguration asks: what is the geometry of the space of valid objects? The second question lives in a higher-dimensional space than the first. Distance in configuration space encodes information about every intermediate state, not just the endpoints. The path is harder than the place.