ON THE ALBANY-MADISON TOWN LINE, N.H.—Paul King hiked through deep woods and scrambled over boulder-strewn hills, hunting for his next clue.

“It’s still here,” he exclaimed at the sight of a bent red pine tree, one of eight landmarks demarcating the border between two northern New Hampshire towns: Albany, population 735, and Madison, home to 2,500 people and a famously large rock.

The...